energy stable high-order methods for...

ENERGY STABLE HIGH-ORDER METHODS FOR SIMULATING UNSTEADY,

VISCOUS, COMPRESSIBLE FLOWS ON UNSTRUCTURED GRIDS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF

AERONAUTICS AND ASTRONAUTICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

David M. Williams

June 2013

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/tj082qz5745

© 2013 by David Michael Williams. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/tj082qz5745

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Antony Jameson, Primary Adviser


Charbel Farhat


Robert MacCormack

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

High-order methods have the potential to dramatically improve the accuracy and

efficiency of flow simulations in the field of computational fluid dynamics (CFD).

However, there remain questions regarding the stability and robustness of high-order

methods for practical problems on unstructured triangular and tetrahedral grids. In

this work, a new class of ‘energy stable’ high-order methods is identified. This class

of schemes (referred to as the ‘Energy Stable Flux Reconstruction’ class of schemes)

is proven to be stable for linear advection-diffusion problems, for all orders of ac-

curacy on unstructured triangular grids in 2D and unstructured tetrahedral grids

in 3D. Furthermore, this class of schemes is shown to be capable of recovering the

well-known collocation-based nodal discontinuous Galerkin scheme, along with new

schemes that possess explicit time-step limits which are (in some cases) more than 2x

larger than those of the discontinuous Galerkin scheme. In addition, the stability of

the Energy Stable Flux Reconstruction schemes is examined for nonlinear problems,

and it is shown that stability depends on the degree of nonlinearity in the flux and on

the placement of solution and flux points in each element. In particular, it is shown

that choosing the solution and flux point locations to coincide with the locations of

quadrature points promotes nonlinear stability by minimizing (or eliminating) non-

linear aliasing errors. A new class of symmetric quadrature points is identified on

triangles and tetrahedra for this purpose. Finally, the Energy Stable Flux Recon-

struction schemes and the new quadrature points are applied to several nonlinear

problems with the aim of assessing how well the schemes perform in practice.

iv

Acknowledgements

I would like to begin by thanking my adviser, Prof. Antony Jameson. Prof. Jameson’s

dedication to airplane design and the field of computational fluid dynamics has been

incredibly inspirational to me. I admire him a great deal for his passion and his

wealth of expertise. During my time with him, he has been very generous in sharing

his wisdom and advice, both professional and personal in nature. It has been an

honor to work alongside him for the last 4 years.

I would like to thank Prof. Peter Vincent and research scientist Patrice Castonguay

for supervising my research and serving as mentors. Peter has been very influential

in encouraging my research interests in high-order methods. During his time with the

lab, he did a great job of leading by example, working very hard to publish results and

to raise grant money for the lab. His efforts are greatly appreciated, and will not soon

be forgotten. Patrice has also significantly aided my research. He was very patient

and persistent in solving the many challenges, both analytical and computational in

nature, that frequently arose during the course of my research. I admire him for his

composure and the quick-thinking that he displayed throughout our time together.

I would like to thank Manuel Lopez who served as a co-teaching assistant and research

collaborator. His hard work in organizing lab events, preparing course work, and

performing research has been invaluable. I am incredibly grateful for his assistance.

He has been like a brother to me, and I will miss the good times that we had while

hanging out both inside and outside of lab.

I would like to thank research collaborators Prof. Guido Lodato and research scientist

Lee Shunn. It has been a pleasure working alongside them, and I greatly appreciate

their contributions, and their patience with my many questions.

I would like to thank Professors Juan Alonso, Sanjiva Lele, Robert MacCormack,

Charbel Farhat, Michael Saunders, Parviz Moin, Philipp Birken, and Brian Cantwell

for their contributions to my education and my research at Stanford. Each of them

has had an important influence on my understanding of numerical methods and com-

pressible fluid mechanics.

I would like to thank the National Science Foundation and the Stanford Graduate

v

vi

Fellowship (SGF) for their financial support. In particular, I would like to thank the

Morgridge family, who directly funded my Ph. D. research through the SGF program.

I am forever grateful for their generosity.

I would like to thank Andre Chan for proof-reading large parts of my thesis. His

diligent efforts in this regard have been extremely valuable.

I would like to thank my ‘first generation’ of lab mates: Edmond Chiu, Matt Culbreth,

Lala Yi, Yves Allaneau, Kui Ou, and Aniket Aranake. Thank you for welcoming me

to the lab, and providing me with your assistance with research and homework. I will

always remember the fun that we had in classes, and at seminars and conferences.

I would like to thank my ‘second generation’ of lab mates: David Manosalvas, Ab-

hishek Sheshadri, Kartikey Asthana, Jerry Watkins, Joshua Romero, Cyrus Liu,

Mehul Oswal, and Alex Fickes. Their enthusiasm for computational fluid dynam-

ics has been inspirational to me. I greatly enjoyed our times doing research together,

and our frequent visits to Panda Express, Tree House, Coho, and the Axe and Palm.

Finally, I would like to thank my faith, friends, and family. Without each of them,

none of this would have been possible. In particular, I would like to thank my Lord and

Savior Jesus Christ. I would like to thank my friends Adam Nicholl, Colin Miranda,

Grady Chang, Timothy Williams, Ethan Kung, Cory Combs, Matt Tang, Eric Chu,

Austin Zheng, Aaron Adcock, Brianna Cardiff-Hicks, Wendy Quay-Honeycutt, Justin

Li, Angie Zhu, Julia Jang, and Michael Shu. I would like to thank my grandparents

Myrtle Harrington, Frank Williams, and Cleopatra Williams. I would also like to

thank my siblings Jonathan, Dayna, and Kimberly. Lastly, I would like to thank my

parents Michael and Diana, for their encouragement and support during this process.

I will eternally appreciate the sacrifices that they have made for my education. This

thesis is dedicated to them.

Contents

Abstract iv

Acknowledgements v

1 Introduction 1

1.1 The Potential of High-Fidelity, High-Order Methods . . . . . . . . . . 2

1.2 Efforts to Improve High-Order Methods . . . . . . . . . . . . . . . . . 6

I Stability Theory for Linear Advection-Diffusion

Problems 11

2 Energy Stable Flux Reconstruction for Advection-Diffusion Prob-

lems in 1D 12

2.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 FR Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17


lems on Triangles 32

3.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Flux Reconstruction Approach for Advection-Diffusion Problems on

Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 VCJH Flux Correction Fields on Triangles . . . . . . . . . . . . . . . 44

vii

CONTENTS viii

3.4 VCJH Solution Correction Fields on Triangles . . . . . . . . . . . . . 46

3.5 Visualization of VCJH Correction Fields . . . . . . . . . . . . . . . . 48

3.6 Proof of Stability of VCJH Schemes for the Linear Advection-Diffusion

Equation on Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.6.2 The Stability of VCJH Schemes . . . . . . . . . . . . . . . . . 52

3.7 Linear Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . 65

3.7.1 Orders of Accuracy and Explicit Time Step Limits . . . . . . 67


lems on Tetrahedra 71

4.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Flux Reconstruction for Advection-Diffusion Problems on Tetrahedral

Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.2 The FR Procedure . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Proof of Stability of VCJH Schemes for Linear Advection-Diffusion

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4 Constructing the Energy Stable (VCJH) Correction Fields . . . . . . 100

4.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4.2 Constructing the Correction Fields φf,l . . . . . . . . . . . . . 101

4.4.3 Constructing the Correction Fields ψf,l . . . . . . . . . . . . . 103

4.5 Choosing the Parameterizing Coefficients for the VCJH Schemes . . . 105

4.5.1 Symmetry Considerations Pertaining to Derivative Operators . 105

4.5.2 Defining the Coefficients . . . . . . . . . . . . . . . . . . . . . 109

CONTENTS ix

4.6 Visualizing the Symmetric VCJH Correction Fields . . . . . . . . . . 110

4.7 Equivalence of VCJH Schemes and Certain Filtered DG Schemes . . . 112

4.7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.7.2 Simplified Formulation of VCJH Schemes and Recovery of Fil-

tered DG Schemes . . . . . . . . . . . . . . . . . . . . . . . . 115

4.8 Constructing the Energy Stable (VCJH) Filter Matrices . . . . . . . . 116

4.8.1 Procedure for Forming the Filter Matrices . . . . . . . . . . . 117

4.8.2 Sparsity Patterns of the Filter Matrices . . . . . . . . . . . . . 119

4.9 Maximizing the Explicit Time-Step Limits of the VCJH Schemes for

Linear Advection Problems . . . . . . . . . . . . . . . . . . . . . . . . 120

4.9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.9.2 Results of Numerical Analysis . . . . . . . . . . . . . . . . . . 124

4.10 Numerical Experiments on the Linear Advection-Diffusion Equation . 126

5 Reformulation of Energy Stable Flux Reconstruction on Triangles 130

5.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.2 Choosing Parameterizing Coefficients on Triangles . . . . . . . . . . . 131

5.3 Proof of Stability and Formulation of a Class of Filtered DG Schemes

on Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

II Stability Theory for Nonlinear Advection-DiffusionProblems 135

6 Energy Stable Flux Reconstruction for Nonlinear Advection-Diffusion

Problems on Tetrahedra 136

6.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

CONTENTS x

6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7 Quadrature Points for Energy Stable Flux Reconstruction Schemes

on Triangles and Tetrahedra 157

7.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.3 Overview of the Theory of Quadrature Rules on Simplexes . . . . . . 163

7.3.1 Preliminary Definition of the Problem . . . . . . . . . . . . . 163

7.3.2 Quadrature Rule Requirements . . . . . . . . . . . . . . . . . 164

7.3.3 Enforcement of Quadrature Rule Requirements . . . . . . . . 165

7.4 Computation of Optimal Quadrature Rules . . . . . . . . . . . . . . . 174

7.4.1 Reformulating the Inequality and Equality Constraints . . . . 174

7.4.2 Forming the Objective Function and Solving the Optimization

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

III Nonlinear Numerical Experiments 183

8 Governing Equations 184

8.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8.2 Definitions of the Navier-Stokes Equations . . . . . . . . . . . . . . . 185

8.3 Definitions of Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . 189

9 Results of Numerical Experiments 191

9.1 Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

CONTENTS xi

9.1.1 Orders of Accuracy and Explicit Time-Step Limits (2D) . . . 198


9.2 Flow Generated by a Time-Dependent Source Term . . . . . . . . . . 207



9.2.3 Aliasing Errors in 2D and 3D . . . . . . . . . . . . . . . . . . 217

9.3 Flow Around Circular Cylinder . . . . . . . . . . . . . . . . . . . . . 221

9.3.1 Time-Averaged Results . . . . . . . . . . . . . . . . . . . . . . 224

9.4 Flow Around SD7003 Airfoil and Wing at 4 Degrees Angle of Attack 227

9.4.1 Lift and Drag Results (2D) . . . . . . . . . . . . . . . . . . . 234

9.4.2 Lift and Drag Results (3D) . . . . . . . . . . . . . . . . . . . 237

9.5 Flow Around SD7003 Wing at 30 Degrees Angle of Attack . . . . . . 239

9.5.1 Isosurfaces and Time-Step Limits . . . . . . . . . . . . . . . . 242

10 Conclusion 244

10.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

10.2 Future Avenues for Research . . . . . . . . . . . . . . . . . . . . . . . 246

Appendix 248

A. Energy Stable (VCJH) Correction Functions in 1D . . . . . . . . . . . 249

B. Mapping the Indices of Quadrature Points on the 2-Simplex . . . . . . 252

C. Quadrature Rules on the 2-Simplex and 3-Simplex . . . . . . . . . . . 255

D. Experiments on a Steady Linear Advection-Diffusion Problem . . . . . 265

Bibliography 270

List of Tables

1.1 Accuracy of the 2nd-order FV scheme (with p = 1) and the high-order

DG schemes (with p = 2 and p = 3) for the vortex propagation problem. 5

1.2 Normalized execution times of the 2nd-order FV scheme (with p = 1)

and the high-order DG schemes (with p = 2 and p = 3) for the vortex

propagation problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1 Reference values of c and κ for p = 2 and p = 3, for the 3-stage,

3rd-order Runge-Kutta scheme (RK33), the 4-stage, 4th-order, Runge-

Kutta scheme (RK44), and a 5-stage, 4th-order, Runge-Kutta scheme

(RK54) [16]. These values were computed using von Neumann analysis

(similar to the analysis performed in [52]) of the VCJH schemes on

uniform grids of right triangular elements. . . . . . . . . . . . . . . . 67

3.2 VCJH scheme accuracy properties and explicit time-step limits for the

model linear advection-diffusion problem on triangles with a = 0, b =

0.1, and p = 2. Values of λ = 1, β = ±0.5n−, and τ = 0.1 were used

in the experiments. The time-step limit (∆tmax) and absolute errors

(L2 and L2s err.) were obtained on the grid with N = 32. . . . . . . . 69











xii

LIST OF TABLES xiii

4.1 Values of c+ for the 3-stage, 3rd-order RK scheme (RK33), the 4-stage,

4th-order RK scheme (RK44), and a 5-stage, 4th-order RK scheme

(RK54) [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.2 Accuracy of VCJH schemes for the advection-diffusion problem with

a =(1/2, 1/2,

√2/2)and b = 0.01 on the grids with N = 4 to N = 64.

The advective numerical flux was computed using a Lax-Friedrichs flux

with λ = 1 and the diffusive numerical flux was computed using a LDG

flux with τ = 1 and β = ±0.5n−. . . . . . . . . . . . . . . . . . . . . 128

4.3 Time-step limits of VCJH schemes for the advection-diffusion problem

with a =(1/2, 1/2,

√2/2)and b = 0.01 on the grids with N = 24 and

64. The advective numerical flux was computed using a Lax-Friedrichs

flux with λ = 1 and the diffusive numerical flux was computed using a

LDG flux with τ = 1 and β = ±0.5n−. . . . . . . . . . . . . . . . . . 129

7.1 Order of accuracy estimates for quadrature rules on the 2-simplex with

Np = 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and 66 points. Estimates are

provided for quadrature rules that are consistent and inconsistent with

the 2-SCP configurations. . . . . . . . . . . . . . . . . . . . . . . . . 181

9.1 Accuracy of VCJH schemes for the Couette flow problem on structured

triangular grids, for the case of p = 2. The inviscid and viscous nu-

merical fluxes were computed using a Rusanov flux with λ = 1 and a

LDG flux with τ = 0.1 and β = ±0.5n−. . . . . . . . . . . . . . . . . 198


triangular grids, for the case of p = 3. The inviscid and viscous nu-

merical fluxes were computed using a Rusanov flux with λ = 1 and a

LDG flux with τ = 0.1 and β = ±0.5n−. . . . . . . . . . . . . . . . . 199

9.3 Accuracy of VCJH schemes for the Couette flow problem on unstruc-

tured triangular grids, for the case of p = 2. The inviscid and viscous

numerical fluxes were computed using a Rusanov flux with λ = 1 and

a LDG flux with τ = 0.1 and β = ±0.5n−. . . . . . . . . . . . . . . . 199

LIST OF TABLES xiv


tured triangular grids, for the case of p = 3. The inviscid and viscous



9.5 Explicit time-step limits (∆tmax) of VCJH schemes on the structured

triangular grid with N = 8 for the Couette flow problem, for the

cases of p = 2 and 3. The inviscid and viscous numerical fluxes were

computed using a Rusanov flux with λ = 1 and a LDG flux with

τ = 0.1 and β = ±0.5n−. . . . . . . . . . . . . . . . . . . . . . . . . . 200

9.6 Explicit time-step limits (∆tmax) of VCJH schemes on the unstructured

triangular grid with N = 256 for the Couette flow problem, for the

cases of p = 2 and 3. The inviscid and viscous numerical fluxes were

computed using a Rusanov flux with λ = 1 and a LDG flux with

τ = 0.1 and β = ±0.5n−. . . . . . . . . . . . . . . . . . . . . . . . . . 200


tetrahedral grids, for the case of p = 2. The inviscid and viscous















LIST OF TABLES xv


tured tetrahedral grids, for the case of p = 2. The inviscid and viscous















9.15 Explicit time-step limits (∆tmax) of VCJH schemes for the Couette flow

problem on the structured tetrahedral grids with N = 8 for the cases

of p = 2 and 3, and N = 3 for the cases of p = 4 and 5. The inviscid

and viscous numerical fluxes were computed using a Rusanov flux with

λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−. . . . . . . . . . 205

9.16 Explicit time-step limits (∆tmax) of VCJH schemes for the Couette

flow problem on the unstructured grids with N = 12288 for the cases

of p = 2 and 3, and N = 648 for the cases of p = 4 and 5. The inviscid



9.17 Accuracy of VCJH schemes for flow generated by a time-dependent

source term on triangular grids, for the case of p = 2. The inviscid and

viscous numerical fluxes were computed using a Rusanov flux with


LIST OF TABLES xvi


source term on triangular grids, for the case of p = 3. The inviscid and



9.19 Explicit time-step limits (∆tmax) of VCJH schemes for flow generated

by a time-dependent source term on the triangular grid with N = 48,

for the cases of p = 2 and 3. The inviscid and viscous numerical fluxes

were computed using a Rusanov flux with λ = 1 and a LDG flux with

τ = 0.1 and β = ±0.5n−. . . . . . . . . . . . . . . . . . . . . . . . . . 213


source term on tetrahedral grids, for the case of p = 2. The inviscid




source term on tetrahedral grids, for the case of p = 3. The inviscid



9.22 Explicit time-step limits (∆tmax) of VCJH schemes for flow generated

by a time-dependent source term on the tetrahedral grid with N = 48,

for the cases of p = 2 and 3. The inviscid and viscous numerical fluxes


τ = 1.0 and β = ±0.5n−. . . . . . . . . . . . . . . . . . . . . . . . . . 215

9.23 Comparison of errors produced by experiments with the quadrature

points and the α-optimized points for the VCJH scheme with c = c+

and κ = κ+, for the problem with flow driven by a time-dependent

source term on triangular grids, for the cases of p = 2 and p = 3. The

inviscid and viscous numerical fluxes were computed using a Rusanov

flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−. . . . . 219

LIST OF TABLES xvii

9.24 Comparison of errors produced by experiments with the quadrature

points and the α-optimized points for the VCJH scheme with c = c+

and κ = κ+, for the problem with flow driven by a time-dependent

source term on tetrahedral grids, for the cases of p = 2 to p = 5. The



9.25 Values of the time-averaged drag coefficient and Strouhal number for

the circular cylinder in flows with M = 0.3 and M = 0.5. The flows

were simulated using the VCJH schemes with p = 2, c = cdg, κ = κdg

and c = c+, κ = κ+ in conjunction with the Rusanov flux with λ = 1

and the LDG flux with β = ±0.5n− and τ = 0.1 on the unstructured

triangular grid with N = 63472 elements. . . . . . . . . . . . . . . . . 224

9.26 Time-averaged values of the lift and drag coefficients for the SD7003

airfoil in flows with Re = 10000, Re = 22000, and Re = 60000. The

flows were simulated using the VCJH schemes with p = 2, c = cdg,

κ = κdg and c = c+, κ = κ+ in conjunction with the Rusanov flux

with λ = 1 and the LDG flux with β = ±0.5n− and τ = 0.1 on the

unstructured triangular grid with N = 25810 elements. . . . . . . . . 234

9.27 Time-averaged values of the lift and drag coefficients for the SD7003

wing-section in a flow with Re = 10000. The flow was simulated using

the VCJH schemes with p = 3, c = cdg, κ = κdg and c = c+, κ = κ+ in

conjunction with the Rusanov flux with λ = 1 and the LDG flux with

β = ±0.5n− and τ = 1.0 on the unstructured tetrahedral grid with

N = 711332 elements. . . . . . . . . . . . . . . . . . . . . . . . . . . 238

C.1 Quadrature rules with Np = 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and 66

points for the triangle (the 2-simplex). Note that the quadrature point

locations for these rules are consistent with the 2-SCP configurations. 259

C.2 Quadrature rules with Np = 45, 55, and 66 points for the triangle (the

2-simplex). Note that the quadrature point locations for these rules

are inconsistent with the 2-SCP configurations. . . . . . . . . . . . . 262

C.3 Quadrature rule with Np = 84 points for the tetrahedron (the 3-simplex).264

List of Figures

1.1 Examples of unsteady flow phenomena that arise during the ascent

and descent of commercial jets. The top and bottom photographs were

taken by Daniel Umana [7] and Vik Sridharan [8], respectively. The

photographs are under copyright by their respective photographers,

and have been reprinted here with permission. . . . . . . . . . . . . 3

2.1 An example of the Np = 3 solution points (denoted by circles) that

reside within the domain of the kth element Ωk for the case of p = 2. 14

2.2 An example of the approximate solution uD defined on the reference

element ΩS for the case of p = 2. . . . . . . . . . . . . . . . . . . . . 18

2.3 The flux points (denoted by squares) on the edges of the reference

element ΩS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 An example of the solution values uDL and uDR defined on the left and

right edges of the reference element ΩS for the case of p = 2. . . . . 19

2.5 An example of the solution values uDL−, uDR−, u

DL+, and u

DR+ defined on

the left and right edges of Ωk for the case of p = 2. . . . . . . . . . . 19

2.6 An example of the solution values u⋆L and u⋆R defined on the left and

right edges of Ωk for the case of p = 2. . . . . . . . . . . . . . . . . . 20

2.7 An example of the solution values u⋆L and u⋆R defined on the left and

right edges of the reference element ΩS for the case of p = 2. . . . . 20

2.8 An example of uC defined on the reference element ΩS for the case of

p = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9 Examples of the correction functions gL and gR for κ = κdg, for the

case of p = 2, where κdg is defined in Appendix A. . . . . . . . . . . . 22

2.10 Examples of the correction functions gL and gR for κ = κsd, for the

case of p = 2, where κsd is defined in Appendix A. . . . . . . . . . . . 22

xviii

LIST OF FIGURES xix

2.11 Examples of the correction functions gL and gR for κ = κ+, for the

case of p = 2, where κ+ is defined in Appendix A. . . . . . . . . . . . 23

2.12 Examples of u and uk defined on the reference element ΩS and the

physical element Ωk (respectively) for the case of p = 2. . . . . . . . . 24

2.13 An example of qD defined on the reference element ΩS for the case of

p = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.14 An example of the approximate flux fD defined on the reference ele-

ment ΩS for the case of p = 2. . . . . . . . . . . . . . . . . . . . . . 25

2.15 An example of the flux values fDL and fD

R defined on the left and right

edges of the reference element ΩS for the case of p = 2. . . . . . . . 26

2.16 An example of the flux values fDL−, f

DR−, f

DL+, and f

DR+ defined on the

left and right edges of Ωk for the case of p = 2. . . . . . . . . . . . . 26

2.17 An example of the flux values f ⋆L and f ⋆


edges of Ωk for the case of p = 2. . . . . . . . . . . . . . . . . . . . . 27

2.18 An example of the flux values f ⋆L and f ⋆


edges of the reference element ΩS for the case of p = 2. . . . . . . . 28

2.19 An example of fC defined on the reference element ΩS for the case of

p = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.20 Examples of f and fk defined on the reference element ΩS and the

physical element Ωk (respectively) for the case of p = 2. . . . . . . . . 30

3.1 Example of solution point locations (denoted by circles) in the reference

element for the case of p = 2. . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Example of the numbering convention for the faces and flux points on

the reference element for the case of p = 2. The flux points are denoted

by squares and the flux point index increases counterclockwise along

an edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Example of a vector correction function hf,j associated with flux point

f = 2, j = 2 for the case of p = 2. . . . . . . . . . . . . . . . . . . . . 43

LIST OF FIGURES xx

3.4 Plots of the VCJH correction fields φf,j for the case of p = 2, f = 1,

and c = c+ = 3.13× 10−2. . . . . . . . . . . . . . . . . . . . . . . . . 49


and c = c+ = 3.13× 10−2. . . . . . . . . . . . . . . . . . . . . . . . . 50


and c = c+ = 3.13× 10−2. . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1 Example of the Np = 10 solution point locations (denoted by spheres)

in the reference element for the case of p = 2. . . . . . . . . . . . . . 74

4.2 Example of the numbering convention for the faces on the reference

element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Example of the numbering convention for the flux points on the ref-

erence element for the case of p = 2. The flux points (denoted by

squares) are shown for the face f = 1. . . . . . . . . . . . . . . . . . 79

4.4 Example of a vector correction function gf,l associated with flux point

f = 1, l = 2 for the case of p = 2. . . . . . . . . . . . . . . . . . . . . 80

4.5 Contours of a correction field ψf,l (where ψf,l ≡ ∇ · gf,l) associated

with flux point f = 1, l = 2 for the case of p = 2. . . . . . . . . . . . 81

4.6 Plots of the VCJH correction fields φf,l for the case of p = 2, f = 1,

and c = c+ = 3.07× 10−2. . . . . . . . . . . . . . . . . . . . . . . . . 111


and c = c+ = 3.07× 10−2. . . . . . . . . . . . . . . . . . . . . . . . . 111


and c = c+ = 3.07× 10−2. . . . . . . . . . . . . . . . . . . . . . . . . 112

4.9 Regular grid of tetrahedral elements used in the von Neumann analysis.122

4.10 Plots of explicit time-step limits ∆t′lim vs. c obtained from the von

Neumann analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

LIST OF FIGURES xxi

7.1 Cubic closed packed (CCP) configurations on tetrahedra with Np = 1,

4, 10, 20, 35, 56, and 84 points. . . . . . . . . . . . . . . . . . . . . . 160

7.2 Cubic closed packed (CCP) configurations on triangles with Np = 1,

3, 6, 10, 15, 21, 28, 36, 45, 55, and 66 points. . . . . . . . . . . . . . . 162

7.3 Np = 15 points symmetrically arranged on the 2-simplex (triangle). . 170

7.4 Np = 84 points symmetrically arranged on the 3-simplex (tetrahedron). 171

7.5 Quadrature point locations for the quadrature rules with 2-SCP struc-

ture and Np = 1, 3, 6, and 10 points for the triangle (the 2-simplex).

The size of each point is scaled by the absolute value of the logarithm

of the associated quadrature weight. . . . . . . . . . . . . . . . . . . . 177


ture and Np = 15, 21, 28, and 36 points for the triangle (the 2-simplex).




ture and Np = 45, 55, and 66 points for the triangle (the 2-simplex).



7.8 Quadrature point locations for the quadrature rules with non-(2-SCP)

structure and Np = 45, 55, and 66 points for the triangle (the 2-

simplex). The size of each point is scaled by the absolute value of the

logarithm of the associated quadrature weight. . . . . . . . . . . . . . 180

7.9 Quadrature point locations for the quadrature rule with Np = 84 points

for the tetrahedron (the 3-simplex). The size of each point is scaled by

the associated quadrature weight. . . . . . . . . . . . . . . . . . . . 182

9.1 Boundary conditions for Couette flow on 2D and 3D computational

domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.2 Structured and unstructured triangular grids for the cases of N = 4

and N = 64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

LIST OF FIGURES xxii

9.3 Structured tetrahedral grid with N = 4. . . . . . . . . . . . . . . . . 196

9.4 Unstructured tetrahedral grid with N = 1536 elements. . . . . . . . . 197

9.5 Contours of density obtained using the VCJH scheme with c = c+

and κ = κ+ on the unstructured grid with N = 64 for the case of

p = 3. The inviscid and viscous numerical fluxes were computed using

a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.201

9.6 Contours of energy obtained using the VCJH scheme with c = c+ and

κ = κ+ on the unstructured grid with N = 64 for the case of p = 3. The



9.7 Contours of density obtained using the VCJH scheme with c = c+




9.8 Contours of energy obtained using the VCJH scheme with c = c+




9.9 Boundary conditions on 2D and 3D computational domains. . . . . . 210

9.10 Structured triangular grid for the case of N = 24. . . . . . . . . . . . 211

9.11 Structured tetrahedral grid for the case of N = 24. . . . . . . . . . . 212


κ = κ+ on the triangular grid with N = 32 for the case of p = 3. The




κ = κ+ on the tetrahedral grid with N = 32 for the case of p = 3. The



LIST OF FIGURES xxiii

9.14 Placement of the solution points (circles) and flux points (squares) at

the locations of the quadrature points described in Chapter 7 for the

case of p = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

9.15 Placement of the solution points (circles) and flux points (squares) at

the locations of the α-optimized points for the case of p = 3. . . . . . 218

9.16 Contours of energy obtained via the VCJH scheme with c = c+, κ =

κ+, and the solution and flux points placed at the locations of the α-

optimized points, on the triangular grid with N = 4 for the case of



9.17 Contours of energy obtained via the VCJH scheme with c = c+, κ =

κ+, and the solution and flux points placed at the locations of the α-

optimized points, on the tetrahedral grid with N = 8 for the case of



9.18 Computational domain for the circular cylinder with associated bound-

ary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

9.19 Faraway views of the unstructured triangular grid with N = 63472

elements around the circular cylinder geometry. . . . . . . . . . . . . 223

9.20 Close-up views of the unstructured triangular grid with N = 63472

elements around the circular cylinder geometry. . . . . . . . . . . . . 223

9.21 Density and pressure contours for the flow with M = 0.3 around the

circular cylinder. The flow was simulated using the VCJH scheme

with c = c+, κ = κ+, and p = 2 in conjunction with the Rusanov flux



LIST OF FIGURES xxiv

9.22 Mach number and vorticity contours for the flow with M = 0.3 around

the circular cylinder. The flow was simulated using the VCJH scheme




9.23 Density and pressure contours for the flow with M = 0.5 around the

circular cylinder. The flow was simulated using the VCJH scheme




9.24 Mach number and vorticity contours for the flow with M = 0.5 around

the circular cylinder. The flow was simulated using the VCJH scheme




9.25 SD7003 airfoil section (left) and a magnified view of its rounded trailing

edge (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

9.26 SD7003 wing-section (left) and a magnified view of its rounded trailing

edge (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

9.27 Boundary conditions for 2D and 3D computational domains for sim-

ulating the SD7003 airfoil and wing-section. Note: The domains are

not drawn to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

9.28 Faraway views of the unstructured triangular grid with N = 25810

elements around the SD7003 airfoil geometry. . . . . . . . . . . . . . 231

9.29 Close-up views of the unstructured triangular grid with N = 25810

elements around the SD7003 airfoil geometry. . . . . . . . . . . . . . 232

9.30 Faraway views of the unstructured tetrahedral grid with N = 711332

elements around the SD7003 wing geometry. . . . . . . . . . . . . . . 232

9.31 Close-up views of the unstructured tetrahedral grid with N = 711332

elements around the SD7003 wing geometry. . . . . . . . . . . . . . . 233

LIST OF FIGURES xxv

9.32 Temporal variation of the lift and drag coefficients for the SD7003

airfoil in flow with Re = 10000. The flow was simulated using the

VCJH scheme with c = c+, κ = κ+, and p = 2 in conjunction with

the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− and

τ = 0.1 on the unstructured triangular grid with N = 25810 elements. 234











9.35 Density and vorticity contours for the flow with Re = 10000 around

the SD7003 airfoil. The flow was simulated using the VCJH scheme














LIST OF FIGURES xxvi


wing-section in flow with Re = 10000. The flow was simulated using

the VCJH scheme with c = c+, κ = κ+, and p = 3 in conjunction with


τ = 1.0 on the unstructured tetrahedral grid with N = 711332 elements.238

9.39 Density and vorticity isosurfaces colored by Mach number for the flow

with Re = 10000 around the SD7003 wing-section. The flow was sim-

ulated using the VCJH scheme with c = c+, κ = κ+, and p = 3 in



N = 711332 elements. . . . . . . . . . . . . . . . . . . . . . . . . . . 239

9.40 Boundary conditions for the computational domain for simulating the

SD7003 wing-section. Note: The domain is not drawn to scale. . . . . 240

9.41 Views of the unstructured tetrahedral grid with N = 311958 elements

around the SD7003 wing-section. . . . . . . . . . . . . . . . . . . . . 241

9.42 Density and vorticity isosurfaces colored by Mach number for the flow

with Re = 100000 around the SD7003 wing-section. The flow was

simulated using the VCJH scheme with c = c+, κ = κ+, and p = 2 in



N = 311958 elements. . . . . . . . . . . . . . . . . . . . . . . . . . . 242

B.1 Baseline convention for numbering quadrature points on a triangle for

the case of Np = 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

B.2 Potential orientations of quadrature point numberings on a triangle for

the case of Np = 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

D.1 L-shaped 3D computational domain. . . . . . . . . . . . . . . . . . . 266

D.2 Unstructured tetrahedral mesh with N = 144847 elements. . . . . . . 267

LIST OF FIGURES xxvii

D.3 Velocity profiles at y = 0.3 and z = 0 for the steady linear advection-

diffusion problem, for the cases of p = 2 to p = 5. The inviscid and



D.4 Velocity profiles at x = 0.9 and z = 0 for the steady linear advection-

diffusion problem, for the cases of p = 2 to p = 5. The inviscid and



D.5 Solution contours for the steady linear advection-diffusion problem, for

the cases of p = 2 to p = 5. The inviscid and viscous numerical fluxes


τ = 1.0 and β = ±0.5n−. . . . . . . . . . . . . . . . . . . . . . . . . . 270

Chapter 1

Introduction

1

CHAPTER 1. INTRODUCTION 2

1.1 The Potential of High-Fidelity, High-Order

Methods

It is generally believed that higher fidelity numerical methods are needed for solving

many of the unsteady, compressible, viscous flow problems that arise in industrial

settings. Currently, such problems are frequently solved using 2nd-order finite vol-

ume (FV) methods [1, 2]. These methods are well-suited for simulating a wide range

of compressible, viscous, steady flow problems, including (most notably) high-speed

flows around aircraft flying at cruise conditions [3, 4, 5]. These methods are par-

ticularly robust for steady problems because they produce appreciable amounts of

numerical dissipation that tend to dampen spurious oscillations, thereby encouraging

the flow to converge towards a steady state. Furthermore, the implementation of these

methods is straightforward on unstructured meshes of triangular and tetrahedral el-

ements, making them well-suited for simulating flows around complex geometries for

which it may not be possible to create structured cartesian meshes. However, despite

these advantages, 2nd-order methods encounter significant challenges when they are

applied to unsteady flows, including flows around aircraft that are ascending, de-

scending, or experiencing gust, buffet, or flutter phenomena [6]. Figure (1.1) shows

examples of the unsteady flow phenomena that arise during the ascent and descent

of civilian aircraft. In this setting, the numerical dissipation produced by 2nd-order

methods tends to obscure important time-dependent flow features, including vortices

and low-amplitude acoustic waves [9, 10]. The current inability, in practice, to reliably

predict these unsteady flow phenomena contributes to the creation of unnecessarily

conservative aircraft designs, that, in turn, drive up the cost of flight. As a result,

there has been significant interest in developing high-order methods that are suitable

for unstructured meshes (cf. the recent review article by Vincent and Jameson [11]

and the recently assembled collection of articles edited by Wang [12]). For many of

the unsteady flow problems that arise in practice, these high-order methods have the

potential to produce less dissipation and to obtain more accurate results at lower

computational cost [13].

The potential advantages of using high-order methods in place of 2nd-order methods

can be illustrated by performing numerical experiments on the classic ‘vortex propa-

gation problem’ originally formulated by Hu et al. [14] and Gassner et al. [15]. This


Figure 1.1: Examples of unsteady flow phenomena that arise during the ascent and descentof commercial jets. The top and bottom photographs were taken by Daniel Umana [7] andVik Sridharan [8], respectively. The photographs are under copyright by their respectivephotographers, and have been reprinted here with permission.

problem involves the propagation of an inviscid, isentropic vortex in a quiescent fluid.

In this scenario, the vortex propagates indefinitely, and the exact solution can be

straightforwardly computed from the initial conditions. In 3D, the exact solution of

this problem takes the following form

ρ = ρ0

(1− γ − 1

2Λ2

) 1

γ−1

, (1.1)

ρu = ρ (u0 + rxc0 Λ) , (1.2)

ρv = ρ (v0 + ryc0 Λ) , (1.3)

ρw = ρ (w0 + rzc0 Λ) , (1.4)

E =p0

γ − 1

(1− γ − 1

2Λ2

) γγ−1

+ρ

2

(u2 + v2 + w2

), (1.5)


where

c0 =

√γp0ρ0

, (1.6)

Λ = Λmax exp

1−

(|r|r0

)2

2

, (1.7)

r = r× (x− x0 − u0 t) , (1.8)

and where ρ is the density, γ is the ratio of specific heats, u = (u, v, w) is the

velocity vector, E is the total energy, p is the pressure, c is the speed of sound, Λ

characterizes the strength of the vortex, r0 is the radius of the vortex, r = (rx, ry, rz)

and r = (rx, ry, rz) are the orthogonal orientation vectors for the vortex, x = (x, y, z)

is the position vector for the vortex, and t is time. Here, it should be noted that all

quantities subscripted by 0 denote values at initial time t0.

In order to obtain a numerical solution to the problem defined by equations (1.1) –

(1.8), it was necessary to approximately solve the Euler equations. The Euler equa-

tions are the partial differential equations (PDEs) that govern the behavior of inviscid,

compressible fluid flows. It should be noted that these equations can be obtained by

setting the viscosity µ equal to zero in the Navier-Stokes equations (cf. Chapter 8).

In a series of numerical experiments performed by the author, these equations (the

Euler equations) were approximately solved on a cuboid domainΩ = [−5, 5]×[−5, 5]×[−5, 5] using a 2nd-order FV method and 3rd and 4th-order discontinuous Galerkin

(DG) methods. In general, the truncation error for the methods was order p+1, where

p denoted the order of the polynomial basis used to represent the solution in each

element. For each method, the vortex propagation problem with initial conditions

given by ρ0 = 1, γ = 1.4, p0 = γ−1, Λmax = 0.4, u0 = (0, 1, 0), r0 = 1, r0 = (0, 0, 1),

and x0 = (0, 0, 0) was solved on a series of tetrahedral grids formed by splitting

cartesian grids withN = N3 hexahedral elements into grids withN = 6N3 tetrahedral

elements. The explicit 5-stage, 4th-order Runge-Kutta scheme of Carpenter and

Kennedy [16] was used to advance the solution in time (starting from t0 = 0), and

the time-step was chosen small enough to ensure that temporal errors were negligible

relative to spatial errors. The spatial errors were computed at t = 2.5 seconds using

a broken L2 norm of the difference between the exact and approximate solutions.


In order to minimize contributions to the error from the boundary conditions, exact

boundary conditions were imposed and the error was evaluated inside a box centered

around the vortex (with a domain defined by −1.25 ≤ x ≤ 1.25, −1.25 + t ≤ y ≤1.25 + t, −5 ≤ z ≤ 5). Table (1.1) shows the spatial error (in the energy E) on the

grids with N = 8, 16, 32, and 64. Results are tabulated for the 2nd-order FV scheme

with p = 1 and the DG methods with p = 2 and 3. Based on the tabulated data, it

is clearly evident that (as expected) the high-order methods are more accurate than

the 2nd-order FV method for a given mesh resolution N .

N p=1 p=2 p=38 9.26e-02 1.37e-02 3.46e-0316 1.96e-02 2.32e-03 2.08e-0432 3.80e-03 3.75e-04 1.20e-0564 8.46e-04 5.29e-05 8.03e-07

Table 1.1: Accuracy of the 2nd-order FV scheme (with p = 1) and the high-order DGschemes (with p = 2 and p = 3) for the vortex propagation problem.

It is important to note that, in certain cases, the high-order methods are also more

efficient than the FV scheme. Table (1.2) shows the ‘normalized’ execution times for

each numerical experiment, where each value of the execution time is normalized by

the execution time for the case with p = 1 and N = 8. Several of these times are

printed in bold-face in order to indicate that they are associated with experiments

that yielded roughly the same level of spatial accuracy.

N p=1 p=2 p=38 1.00 1.20 1.4416 3.68 6.05 9.0332 27.3 51.9 87.764 340 657 1177

Table 1.2: Normalized execution times of the 2nd-order FV scheme (with p = 1) and thehigh-order DG schemes (with p = 2 and p = 3) for the vortex propagation problem.

From these results, it is evident that the high-order methods are capable of producing

the same level of accuracy as the 2nd-order FV method while requiring less compu-

tation time. In particular, although the case with N = 8 and p = 3 and the case

with N = 32 and p = 1 produce roughly the same amount of error, results from the

case with N = 8 and p = 3 can be obtained approximately 19 times faster. This is


expected, as according to theory, the high-order methods (with p = 2 and p = 3)

produce significantly less dissipation than the 2nd-order method (with p = 1), and

thus accurately preserve the vortex structure for a longer period of time.

Based on these numerical experiments, it is clear that high-order methods have the

potential to outperform 2nd-order methods on unsteady flow problems. However, the

potential of high-order methods has yet to be fully realized in practice. For many

of the complex, nonlinear flow problems that arise in industrial settings, high-order

methods appear to be less robust and more difficult to implement than their 2nd-

order counterparts. Therefore, despite their promise, they have yet to be adopted by

the majority of fluid dynamicists.

1.2 Efforts to Improve High-Order Methods

There have been efforts to improve the flexibility and ease of implementation of high-

order methods, and in particular the well-known DG methods. These endeavors have

brought about a rise in popularity of high-order methods that omit the explicit use

of quadratures, allowing them to serve as simpler alternatives to the classical DG

methods that use a ‘method of weighted residuals’ formulation that requires complex

quadrature procedures (as described in [17, 18, 19]). In particular, various unstruc-

tured high-order schemes based directly on differential forms of the governing system

have emerged, such as ‘nodal’ DG methods [20, 13], Spectral Difference (SD) meth-

ods [21, 22] and, more generally, Flux Reconstruction (FR) methods [23]. For conve-

nience, such schemes will henceforth be referred to as ‘FR type’ schemes. FR type

schemes do not require integration to be performed, and hence their implementations

can (explicitly at least) omit quadratures. As such, efficient implementation of FR

type schemes is straightforward relative to their more traditional counterparts. Con-

sequently, it is envisaged that such FR type methods will become popular amongst a

wide community of fluid dynamicists within both academia and industry.

In what follows, a review of FR type methods is provided in order to establish the con-

text and motivation for the current work. First proposed by Huynh in 2007, FR is an

approach which generates new high-order schemes and recovers well-known schemes,

including a variety of collocation-based nodal DG and SD schemes [23]. The FR ap-

proach should not be confused with the classical concept of flux reconstruction that


appears in the context of traditional 2nd-order FV schemes (cf. [24, 25, 26, 27, 28]),

its closely related adaptations (cf. [29, 30, 31]), or the more recently developed flux

reconstruction procedures for formulating a posteriori error estimations for DG meth-

ods (cf. [32, 33, 34]). In the FR approach [23], the flux is subject to a reconstruction

procedure involving ‘correction functions,’ which are required to be polynomials of

one degree higher than the solution, as well as satisfying symmetry and boundary

constraints. The FR approach was originally formulated for advection problems in

1D and was extended via tensor products to quadrilateral elements in 2D [23]. In

2009, Huynh formulated an extension of the FR approach to diffusion problems [35]

in which both the solution and the flux are subject to reconstruction procedures. In

both [23] and [35], Fourier analysis was employed in an effort to evaluate the stabil-

ity of various FR schemes for linear advection and diffusion problems. Thereafter,

Huynh developed an extension of the FR approach to advection-diffusion problems

on triangles [36]. This approach makes use of scalar-valued, 2D correction functions,

which are required to satisfy symmetry conditions similar to those imposed on the 1D

correction functions. To the author’s knowledge, Fourier analysis of these schemes

has yet to be performed.

In 2009, Gao and Wang identified a closely related class of schemes for advection-

diffusion problems, referred to as Lifting Collocation Penalty (LCP) schemes [37, 38].

These schemes make use of ‘weighting functions’ which are different from correction

functions in that they are piecewise continuous polynomials that are of one degree

lower than the correction functions. The LCP approach involves multiplying the

governing equations by the weighting functions and integrating the result in order

to obtain ‘corrections’. These corrections are similar in form to those that arise

in Huynh’s FR approach [23, 35], and under certain circumstances in 1D, the two

approaches can be shown to be identical [39]. As a result, Wang, Gao, Haga, and

Yu have referred to the class of all FR and LCP schemes as ‘Correction Procedure

via Reconstruction’ (CPR) schemes [40, 39], although it is not yet clear whether all

FR schemes are in fact LCP schemes or vice versa. Wang, Gao, and Haga have

provided evidence for the stability of the schemes, successfully applying the FR and

LCP schemes to nonlinear advection-diffusion problems on grids of quadrilaterals and

triangles in 2D [37] and tetrahedra and prisms in 3D [41, 40].

In addition to being related to the LCP schemes, the FR schemes are also related


to the Summation By Parts Simultaneous Approximation Term (SBP-SAT) finite

difference schemes. The SBP-SAT and FR formulations are similar in that the SBP-

SAT formulation can recover the 2nd-order Galerkin Finite Element scheme on tri-

angles [42] which (for a linear solution basis) is akin to the collocation-based nodal

DG approach that the FR formulation recovers. However, SBP-SAT schemes of the

form given by [42] cannot recover Galerkin Finite Element methods for non-simplex

elements [43], and (thus far) have not been extended to yield compact high-order dis-

cretizations on triangles. For this reason, the SBP-SAT schemes will not be discussed

further, although the interested reader should consult [44, 45, 46, 47] for details per-

taining to these schemes. The remainder of this discussion will instead focus on FR

type schemes which are more closely related to the schemes of Huynh [23, 35, 36],

and Gao and Wang [37, 38].

Recently, efforts have focused on rigorously proving the stability of FR type schemes

using ‘energy methods’. Such methods offer advantages over von Neumann techniques

(i.e. Fourier analysis) since they automatically extend to all orders of accuracy, and

are valid for unstructured grids. In 2010, Jameson [48] used an energy method to prove

stability of a particular SD method for linear advection problems. Subsequently, in

2011, Vincent, Castonguay, and Jameson [49] used a similar approach to derive an

entire class of stable FR schemes for linear advection problems in 1D. These stable

schemes, referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes, are

parameterized by a single scalar. The scalar influences the analytical form of the

correction functions, and variations of the scalar lead to recovery of various known

numerical methods, including a collocation-based nodal DG method, the SD method

that Jameson [48] proved to be stable for linear advection, and Huynh’s so-called g2

method [23]. Recently, Castonguay and Williams et al. [50, 51] extended the VCJH

schemes, and proved their stability for advection-diffusion problems in 1D [51].

In 2D, Castonguay, Vincent, and Jameson used an energy method to identify a class

of VCJH schemes which they proved to be stable for linear advection problems on

triangles [52]. These schemes make use of vector-valued, 2D correction functions,

which are required to satisfy symmetry and orthogonality conditions. The VCJH

schemes on triangles are parameterized by a single scalar (as in the 1D case) which,

if set to the correct value, allows for recovery of the collocation-based nodal DG

scheme [52].


Note that the symmetry conditions used to define the VCJH correction functions due

to Castoguay et al. [52] are different than those used to define the correction functions

due to Huynh [36], and thus it does not appear that these two classes of schemes are

equivalent. In addition, because VCJH schemes due to Castonguay et al. [52] utilize

correction functions and LCP schemes due to Gao and Wang [38] utilize weighting

functions, the equivalence of the VCJH and LCP classes of schemes has yet to be

demonstrated.

In summary, a number of authors have proposed FR type approaches for the treatment

of advection and advection-diffusion problems in 1D and in higher dimensions on

triangles and tetrahedra. However, for linear advection problems, only the VCJH

approaches of [49] in 1D and [52] on triangles have been proven stable for all orders

of accuracy. Furthermore, for linear advection-diffusion problems on triangles, the

approach of [52] has not yet been proven stable. This thesis will extend the approach

of [52], to provide for the first time on triangles and tetrahedra, a provably stable

family of FR schemes for linear advection-diffusion problems.

The format of this thesis is as follows. Part 1, Chapters 2 – 5, describe the procedure

and linear stability theory associated with the energy stable FR schemes for advection-

diffusion problems. In particular, Chapter 2 presents an overview of the energy stable

FR schemes in 1D, Chapter 3 presents an overview of the energy stable FR schemes

on triangles and proves the stability of the schemes for linear advection-diffusion

problems in 2D, Chapter 4 presents an overview of the energy stable FR schemes

on tetrahedra and proves the stability of the schemes for linear advection-diffusion

problems in 3D, and finally, Chapter 5 presents a reformulation of the energy stable

FR schemes on triangles from Chapter 3, utilizing the theoretical advancements from

Chapter 4.

Part 2, Chapters 6 and 7, describe the nonlinear stability theory associated with

the energy stable FR schemes for advection-diffusion problems. In particular, Chap-

ter 6 discusses how to minimize aliasing errors that arise when the energy stable

FR schemes are applied to nonlinear advection-diffusion problems in 3D, and Chap-

ter 7 presents a new class of solution point locations that minimize aliasing errors,

where the solution point locations are set to coincide with quadrature point locations

associated with a new class of quadrature rules.


Finally, Part 3, Chapters 8 and 9, describe the results of numerical experiments that

involve applying the energy stable FR schemes to nonlinear problems. In particular,

Chapter 8 describes the nonlinear governing equations for fluid flow (the Navier-Stokes

equations), and Chapter 9 presents the results of experiments that involve applying

the schemes from Part 1 and the solution point locations from Part 2 to a range of

fluid dynamics problems.

Part I

Stability Theory for Linear

Advection-Diffusion Problems

11

Chapter 2

Energy Stable Flux Reconstruction

for Advection-Diffusion Problems in

1D

12

CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 13

2.1 Preamble

In order to provide a suitable context for the energy stable FR schemes (VCJH

schemes) on triangles in 2D and tetrahedra in 3D, this section presents a review

of the energy stable FR schemes in 1D.

2.2 Preliminaries

The important aspects of the energy stable FR schemes in 1D can be conveniently il-

lustrated via application of the schemes to the advection-diffusion equation. Consider

the 1D advection-diffusion equation which takes the following form

∂u

∂t+

∂

∂x

(f(u,∂u

∂x

))= 0, (2.1)

where u denotes the scalar solution, f denotes the scalar flux, and (as discussed

previously) x and t denote the spatial and temporal coordinates, respectively.

In following the standard approach of Cockburn and Shu [53], one may introduce an

auxiliary variable q into equation (2.1) as follows

∂u

∂t+

∂

∂x(f (u, q)) = 0, (2.2)

q − ∂u

∂x= 0. (2.3)

Through this operation, the 2nd-order PDE (equation (2.1)) is reformulated as a

system of 1st-order PDEs (equations (2.2) and (2.3)).

Solutions to equations (2.2) and (2.3) are sought in the 1D domain Ω with boundary

Γ. In following the conventional Finite Element approach, Ω can be divided into N

non-overlapping elements such that

Ω =

N⋃

k=1

Ωk, (2.4)

Ωi ∩Ωj = ∅ ∀i 6= j. (2.5)

Approximate forms of equations (2.2) and (2.3) can be constructed on the kth element


(Ωk). In particular, the solution u in equation (2.2) can be approximated by a function

uDk = uDk (x, t) which is required to be a polynomial of degree p within the kth element

and to assume the value of zero outside of the kth element. The polynomial portion

of uDk can be constructed as follows

uDk =

Np∑

i=1

(uDk)iℓi (x) , (2.6)

where(uDk)i

is the value of the approximate solution at solution point (‘nodal point’)

i, and ℓi (x) is the Lagrange polynomial of degree p which takes on the value of 1 at

solution point i and the value of 0 at all other solution points. Figure (2.1) shows an

example of the Np = p + 1 = 3 solution points that reside in the element Ωk for the

case of p = 2.

xk xk+1

Ωk

Figure 2.1: An example of the Np = 3 solution points (denoted by circles) that reside withinthe domain of the kth element Ωk for the case of p = 2.

In equation (2.6), the superscript D on uDk indicates that the approximate solution

may be discontinuous in the following sense: the sum of uDk and uDk+1, defined on

neighboring elements Ωk and Ωk+1, is not required to reside in C0 (Ωk ∪Ωk+1), i.e.

the solution is not required to be continuous at the interfaces between elements.

In a similar fashion, the flux in equation (2.2) can be approximated by a function

fDk = fD

k (x, t) which is required to be a polynomial of degree p within the kth element

and to assume the value of zero outside of the kth element. Evidently, the polynomial

portion of fDk can be constructed as follows

fDk =

Np∑

i=1

(fDk

)iℓi (x) , (2.7)

where(fDk

)iis the value of the approximate flux at solution point i. As before, the


superscript D on fDk indicates that the approximate flux may be discontinuous at the

interfaces between elements.

An approximate form of equation (2.3) can be obtained in the same way. In particular,

the solution u can be approximated (as before) by uDk , and the auxiliary variable q

can be approximated by a function qDk = qDk (x, t) which has the same properties (with

regard to degree and continuity) as uDk and fDk . As a result, the polynomial portion

of qDk can be defined as follows

qDk =

Np∑

i=1

(qDk)iℓi (x) , (2.8)

where(qDk)i

is the value of the approximate auxiliary variable at solution point i.

On rewriting equations (2.2) and (2.3) in terms of uDk , fDk , and qDk (defined in equa-

tions (2.6), (2.7), and (2.8)), one obtains the following

∂uDk∂t

+∂fD

k

∂x= 0, (2.9)

qDk −∂uDk∂x

= 0. (2.10)

It turns out that, in their current form, equations (2.9) and (2.10) do not comprise

a valid numerical scheme. For example, in equation (2.9), the solution in the kth

element (uDk ) depends entirely on quantities in the kth element, and there is no ex-

change of information with neighboring elements. In order to enable communication

between elements (and to allow for conservation of u), it is necessary to introduce

a continuous flux fk into equation (2.9) and to introduce a continuous solution uk

into equation (2.10). In particular, one may define fk as a degree p + 1 polynomial,

where the sum of fk and fk+1 is required to be in C0 (Ωk ∪Ωk+1), and one may define

uk as a degree p + 1 polynomial, where the sum of uk and uk+1 is required to be in

C0 (Ωk ∪Ωk+1). Upon replacing fDk with fk in equation (2.9) and replacing uDk with


uk in equation (2.10), one obtains

∂uDk∂t

+∂fk∂x

= 0, (2.11)

qDk −∂uk∂x

= 0. (2.12)

Equations (2.11) and (2.12) do not comprise a complete numerical scheme (as the

quantities fk and uk have yet to be precisely defined), however, they do comprise a

valid scheme in the sense that they allow for the exchange of information between

neighboring elements.

In what follows, the FR procedure for solving equations (2.11) and (2.12) will be

described. However, in order to enable a convenient description of this procedure,

it is first necessary to transform equations (2.11) and (2.12) from the domain of the

physical element Ωk = x | xk ≤ x ≤ xk+1 to the domain of the reference element

ΩS = x | − 1 ≤ x ≤ 1. Towards this end, one may define the following mapping

between the coordinates in Ωk and ΩS

x = Θk (x) =

(1− x2

)xk +

(1 + x

2

)xk+1. (2.13)

This mapping has a Jacobian denoted by Jk that is defined as follows

Jk =dΘk (x)

dx=

1

2(xk+1 − xk) . (2.14)

The mapping and its Jacobian can be utilized to transform the quantities uDk , fk,

qDk , and uk from physical space into the equivalent quantities uD, f , qD, and u in

reference space, as follows

uD = Jk uDk (Θk (x) , t) , f = fk (Θk (x) , t) , (2.15)

qD = J2k q

Dk (Θk (x) , t) , u = Jk uk (Θk (x) , t) . (2.16)

Upon substituting uDk , fk, qDk , and uk (as defined in equations (2.15) and (2.16)) into


equations (2.11) and (2.12), one obtains

∂uD

∂t+∂f

∂x= 0, (2.17)

qD − ∂u

∂x= 0. (2.18)

The energy stable FR procedure for solving equations (2.17) and (2.18) is described

in the next section.

2.3 FR Procedure

The energy stable FR procedure for solving advection-diffusion problems consists of

eight stages. However, it should be noted that the procedure can be described in

seven stages if one elects to combine the 3rd stage and the very brief 4th stage,

(cf. Chapter 3).

The first stage involves computing the discontinuous solution in the reference space,

which as mentioned previously, is denoted by uD. In order to compute this quantity,

one may multiply both sides of equation (2.6) by the Jacobian Jk, and substitute

equations (2.13) and (2.15) into the result in order to obtain the following

uD =

Np∑

i=1

(uD)iℓi (Θk (x)) =

Np∑

i=1

(uD)iℓi (x) , (2.19)

where(uD)i

= Jk(uDk)i

is the value of the discontinuous solution in the reference

space at solution point i, and ℓi (x) is the Lagrange polynomial in the reference space

that assumes the value of 1 at solution point i and the value of 0 at all other solution

points. Figure (2.2) shows an example of the approximate solution uD for the case of

p = 2.


ΩS

uD

Figure 2.2: An example of the approximate solution uD defined on the reference element ΩS

for the case of p = 2.

The second stage involves computing common values of the discontinuous solution at

the flux points located on the edges of the reference element. In 1D, the reference

element has two flux points, where one flux point is located on the left edge of the

element (at x = −1) and the other flux point is located on the right edge of the

element (at x = 1), as shown in Figure (2.3).

ΩS

−1 1

Figure 2.3: The flux points (denoted by squares) on the edges of the reference element ΩS .

The discontinuous solution in the reference space (equation (2.19)) can be evaluated

at the flux points on the left and right edges in order to obtain uDL = uD (−1) and

uDR = uD (1). Figure (2.4) shows an example of the approximate solution values uDLand uDR for the case of p = 2.


ΩS

uD

L

uD

R

Figure 2.4: An example of the solution values uDL and uDR defined on the left and right edgesof the reference element ΩS for the case of p = 2.

Next, uDL and uDR which reside in the reference space can be transformed into the

quantities uDL− and uDR− which reside in the physical space (utilizing equation (2.15)),

where the minus sign subscripts signify that these quantities belong to the physical

domain of the kth element. A similar procedure can be employed to compute uDL+

and uDR+, where the plus sign subscripts signify that these quantities belong to the

physical domains of the nearest neighbors of the kth element. Figure (2.5) shows an

example of the approximate solution values uDL−, uDR−, u

DL+, and u

DR+ for the case of

p = 2.

ΩkΩk−1 Ωk+1

uD

L−

uD

L+

uD

R−

uD

R+

Figure 2.5: An example of the solution values uDL−, uDR−, u

DL+, and u

DR+ defined on the left

and right edges of Ωk for the case of p = 2.

Thereafter, a common value of the solution at the flux point on the left edge (denoted

by u⋆L) can be computed from uDL− and uDL+, and a common value of the solution at

the flux point on the right edge (denoted by u⋆R) can be computed from uDR− and uDR+.


Figure (2.6) shows an example of the common solution values u⋆L and u⋆R for the case

of p = 2.

ΩkΩk−1 Ωk+1

u⋆

L

u⋆

R

Figure 2.6: An example of the solution values u⋆L and u⋆R defined on the left and right edgesof Ωk for the case of p = 2.

Finally, in preparation for the next stage, the common solution values u⋆L and u⋆R in

the physical space can be transformed to the reference space, yielding u⋆L and u⋆R,

in accordance with equation (2.15). Figure (2.7) shows an example of the solution

values u⋆L and u⋆R for the case of p = 2.

ΩS

u⋆

R

u⋆

L

Figure 2.7: An example of the solution values u⋆L and u⋆R defined on the left and right edgesof the reference element ΩS for the case of p = 2.

The third stage involves forming the continuous solution in the reference space (u)

by adding a ‘solution correction’ (uC) to the discontinuous solution in the reference

space (uD). In order to ensure that u is continuous, the sum of uC and uD is required

to equal the common solution values u⋆L and u⋆R at the flux points on the left and


right edges of the reference element, respectively. This requirement can be satisfied

by defining uC as follows

uC =(u⋆L − uDL

)gL (x) +

(u⋆R − uDR

)gR (x) , (2.20)

where gL (x) and gR (x) are ‘correction functions’ that are required to satisfy the

following point-wise constraints

gL (−1) = 1, gL (1) = 0, (2.21)

gR (−1) = 0, gR (1) = 1, (2.22)

and the following symmetry constraints

gL (x) = gR (−x) , gL (−x) = gR (x) . (2.23)

Figure (2.8) shows an example of uC for the case of p = 2.

ΩS

uC

Figure 2.8: An example of uC defined on the reference element ΩS for the case of p = 2.

Before proceeding further, it is important to discuss in more detail the correction

functions that are used to construct uC. There are an infinite number of such correc-

tion functions (which satisfy equations (2.21) – (2.23)), however, it is well-known that

not all of them yield stable FR schemes. This has driven efforts to identify a class of

‘energy-stable’ correction functions that are guaranteed to preserve the stability of the

FR schemes. The ‘VCJH correction functions’ comprise one such class of correction

functions. The VCJH correction functions are exceptional, in that they are required

to satisfy equations (A.1) – (A.4) in Appendix A, in addition to equations (2.21) –

(2.23). In order to satisfy these equations, the correction functions gL (x) and gR (x)


are required to be defined by equations (A.5) and (A.6). The VCJH definitions of

gL (x) and gR (x) are parameterized by a scalar constant κ, where it is important to

note that an infinite family of correction functions can be obtained by varying the

value of κ. Figures (2.9)–(2.11) show examples of the correction functions gL (x) and

gR (x) for different values of κ.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

(a) gL, κ = κdg

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

(b) gR, κ = κdg

Figure 2.9: Examples of the correction functions gL and gR for κ = κdg, for the case ofp = 2, where κdg is defined in Appendix A.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

(a) gL, κ = κsd

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

(b) gR, κ = κsd

Figure 2.10: Examples of the correction functions gL and gR for κ = κsd, for the case ofp = 2, where κsd is defined in Appendix A.


−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

(a) gL, κ = κ+

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

(b) gR, κ = κ+

Figure 2.11: Examples of the correction functions gL and gR for κ = κ+, for the case ofp = 2, where κ+ is defined in Appendix A.

Choosing gL (x) and gR (x) to be VCJH correction functions is beneficial because

(under a few minor assumptions) this ensures that the resulting FR schemes are

stable for linear advection-diffusion problems in 1D (as shown in [51]).

Now, having defined uC and the associated correction functions, one may construct

the continuous solution in the reference space (u) by summing uD and uC as follows

u = uD + uC = uD +(u⋆L − uDL

)gL (x) +

(u⋆R − uDR

)gR (x) . (2.24)

Figure (2.12) shows examples of u and uk for the case of p = 2, where uk =

J−1k u

(Θk (x)

−1) in accordance with equation (2.16).


ΩS

u

(a) Example of u

ΩkΩk−1 Ωk+1

uk

uk+1

uk−1

(b) Example of uk

Figure 2.12: Examples of u and uk defined on the reference element ΩS and the physicalelement Ωk (respectively) for the case of p = 2.

The fourth stage involves computing the discontinuous auxiliary variable in the refer-

ence space (denoted by qD). Towards this end, one may substitute u (equation (2.24))

into equation (2.18) and rearrange the result in order to obtain

qD =∂uD

∂x+(u⋆L − uDL

) dgLdx

+(u⋆R − uDR

) dgRdx

=

Np∑

i=1

(uD)i dℓi

dx+(u⋆L − uDL

) dgLdx

+(u⋆R − uDR

) dgRdx

. (2.25)

Figure (2.13) shows an example of qD for the case of p = 2.


ΩS

qD

Figure 2.13: An example of qD defined on the reference element ΩS for the case of p = 2.

The fifth stage involves computing the discontinuous flux in the reference space (de-

noted by fD). Towards this end, one may compute(fDk

)i, the value of the flux in the

physical space at each solution point i, using the values of(uDk)i

and(qDk)i

which can

be obtained by evaluating equations (2.19) and (2.25) at each solution point i, and

transforming the result to the physical space via equations (2.15) and (2.16). Once

each value of(fDk

)iis obtained, it becomes possible to construct the degree p poly-

nomial representation of fDk defined in equation (2.7). Upon transforming the flux in

equation (2.7) from the physical space to the reference space via equation (2.15), one

obtains the following expression for fD

fD =

Np∑

i=1

(fD)iℓi (x) . (2.26)

Figure (2.14) shows an example of fD for the case of p = 2.

ΩS

fD

Figure 2.14: An example of the approximate flux fD defined on the reference element ΩS

for the case of p = 2.

The sixth stage involves computing common values of the flux at the flux points


located on the edges of the reference element. The discontinuous flux in the reference

space (equation (2.26)) can be evaluated at the flux points on the left and right edges

in order to obtain fDL = fD (−1) and fD

R = fD (1). Figure (2.15) shows an example

of the approximate flux values fDL and fD

R for the case of p = 2.

fD

L

fD

R

ΩS

Figure 2.15: An example of the flux values fDL and fDR defined on the left and right edgesof the reference element ΩS for the case of p = 2.

Next, utilizing equation (2.15), fDL and fD

R can be transformed into the physical space,

where the resulting values are denoted by fDL− and fD

R−. A similar procedure can be

employed to compute the corresponding quantities fDL+ and fD

R+ in the neighboring

elements. Figure (2.16) shows an example of the approximate flux values fDL−, f

DR−,

fDL+, and f

DR+ for the case of p = 2.

fD

L−

fD

L+

fD

R+fD

R−

ΩkΩk−1 Ωk+1

Figure 2.16: An example of the flux values fDL−, fDR−, f

DL+, and f

DR+ defined on the left and

right edges of Ωk for the case of p = 2.


Now, due to the different physical behaviors of the advective and diffusive fluxes, it

is convenient to divide fDL− , fD

R−, fDL+ , and fD

R+ into advective and diffusive parts,

such that fDL− = fD

L−, adv + fDL−, dif , f

DR− = fD

R−, adv + fDR−, dif , f

DL+ = fD

L+, adv + fDL+, dif ,

and fDR+ = fD

R+, adv + fDR+, dif .

Thereafter, a common value of the flux at the flux point on the left edge (denoted by

f ⋆L) can be computed by summing the common values of the advective and diffusive

fluxes on the left edge (denoted by f ⋆L, adv and f

⋆L, dif ), which are themselves computed

from fDL−, adv, f

DL+, adv, f

DL−, dif , and f

DL+, dif . Likewise, a common value of the flux at

the flux point on the right edge (denoted by f ⋆R) can be computed by summing the

common values of the advective and diffusive fluxes on the right edge (denoted by

f ⋆R, adv and f

⋆R, dif), which are themselves computed from fD

R−, adv, fDR+, adv, f

DR−, dif , and

fDR+, dif . Figure (2.17) shows an example of the common flux values f ⋆

L and f ⋆R for the

case of p = 2.

ΩkΩk−1 Ωk+1

f⋆

L

f⋆

R

Figure 2.17: An example of the flux values f⋆L and f⋆R defined on the left and right edges ofΩk for the case of p = 2.

Finally, in preparation for the next stage, the common flux values f ⋆L and f ⋆

R in the

physical space can be transformed to the reference space, yielding f ⋆L and f ⋆

R, in

accordance with equation (2.15). Figure (2.18) shows an example of the flux values

f ⋆L and f ⋆

R for the case of p = 2.


f⋆

R

f⋆

L

ΩS

Figure 2.18: An example of the flux values f⋆L and f⋆R defined on the left and right edges ofthe reference element ΩS for the case of p = 2.

The seventh stage involves forming the continuous flux in the reference space (f) by

adding a ‘flux correction’ (fC) to the discontinuous flux in the reference space (fD).

In order to ensure that f is continuous, the sum of fC and fD is required to equal

the common flux values f ⋆L and f ⋆

R at the flux points on the left and right edges of

the reference element, respectively. This requirement can be satisfied by defining fC

as follows

fC =(f ⋆L − fD

L

)hL (x) +

(f ⋆R − fD

R

)hR (x) , (2.27)

where hL (x) and hR (x) are correction functions that are required to have the same

properties as gL (x) and gR (x) from stage 3, as they must satisfy the following point-

wise constraints

hL (−1) = 1, hL (1) = 0, (2.28)

hR (−1) = 0, hR (1) = 1, (2.29)

and the following symmetry constraints

hL (x) = hR (−x) , hL (−x) = hR (x) . (2.30)

Figure (2.19) shows an example of fC for the case of p = 2


ΩS

fC

Figure 2.19: An example of fC defined on the reference element ΩS for the case of p = 2.

It should be noted that the correction functions hL (x) and hR (x) (used in the con-

struction of fC) are required to be of VCJH type (i.e. they are required to satisfy

equations (A.8) – (A.10)) in order to ensure that the resulting FR schemes are stable

for linear advection-diffusion problems in 1D [51]. For the sake of completeness, the

VCJH formulations of hL (x) and hR (x) are given in equations (A.11) and (A.12).

Now, having defined fC and the associated correction functions, one may construct

the continuous flux in the reference space (f) by summing fD and fC as follows

f = fD + fC = fD +(f ⋆L − fD

L

)hL (x) +

(f ⋆R − fD

R

)hR (x) . (2.31)

Figure (2.20) shows examples of f and fk for the case of p = 2, where fk =

f(Θk (x)

−1) in accordance with equation (2.15).


ΩS

f

(a) Example of f

ΩkΩk−1 Ωk+1

fk

fk−1

fk+1

(b) Example of fk

Figure 2.20: Examples of f and fk defined on the reference element ΩS and the physicalelement Ωk (respectively) for the case of p = 2.

The final stage involves computing the divergence of the continuous flux f and uti-

lizing it to update the discontinuous solution uD. Towards this end, consider substi-

tuting equations (2.26) and (2.31) into equation (2.17), and evaluating the result at

solution point j in order to obtain

∂(uD)j

∂t= −

[Np∑

i=1

(fD)i dℓi (x)

dx

] ∣∣∣∣x=xj

+

[(f ⋆L − fD

L

) dhL (x)

dx+(f ⋆R − fD

R

) dhR (x)

dx

] ∣∣∣∣x=xj

, (2.32)

where xj is the location of solution point j in reference space. Equation (2.32) can be

evaluated at all j = 1, · · · , Np solution points in order to obtain Np equations for the


Np unknowns

[∂(uD)

1

∂t, . . . ,

∂(uD)Np

∂t

]in each element. Then, the resulting equations

can be marched forward in time using an explicit or implicit time-stepping scheme.

The behavior of the energy stable FR schemes in 1D is determined by the following

factors:

• The locations of the solution points xi.

• The methodology for computing the common solution values u⋆L and u⋆R.

• The methodology for computing the common flux values f ⋆L and f ⋆

R.

• The methodology for computing the energy stable FR solution correction func-

tions gL (and thus gR).

• The methodology for computing the energy stable FR flux correction functions

hL (and thus hR).

Vincent et al. [49] and Castonguay et al. [51] discuss how these factors effect the

energy stable FR schemes in 1D. In the following chapters, the 1D schemes will be

extended to higher dimensions (2D and 3D) and the stability of the resulting schemes

will be shown for linear advection-diffusion problems.

Chapter 3


for Advection-Diffusion Problems on

Triangles

32

CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 33

3.1 Preamble

For the first time, this chapter constructs energy stable FR schemes (VCJH schemes)

for advection-diffusion problems on triangles, and proves the stability of these schemes

for linear advection-diffusion problems, for all orders of accuracy. Note that this chap-

ter is adapted from the article “Energy Stable Flux Reconstruction for Advection-

Diffusion Problems on Triangles” by D. M. Williams, P. Castonguay, P. E. Vin-

cent, and A. Jameson, which has been submitted to the Journal of Computational

Physics. The main results and over 95% of the text for this article were contributed

by D. M. Williams.

Due to the size of this chapter, it is useful to briefly examine its format. Section 2 of

this chapter presents a FR approach for solving advection-diffusion problems on trian-

gles, sections 3 and 4 define the VCJH correction fields on triangles, section 5 presents

plots of the VCJH correction fields, section 6 develops a range of VCJH schemes for

linear advection-diffusion problems on triangles, and proves that these schemes are

stable, and finally, section 7 presents results of linear numerical experiments involving

the newly proposed schemes.

3.2 Flux Reconstruction Approach for Advection-

Diffusion Problems on Triangles

In what follows, the FR schemes for advection problems on triangles, developed by

Castonguay et al. in [52], are extended to advection-diffusion problems. For readers

unfamiliar with the FR approach, the authors recommend a review of the 1D descrip-

tions of FR for advection in [23] and [49], or advection-diffusion in [51] and Chapter 2

(of this thesis) before proceeding further.

3.2.1 Preliminaries

Consider the 2D conservation law

∂u

∂t+∇·f(u,∇u) = 0, (3.1)


where u is a scalar variable and f is a vector-valued flux. Equation (3.1) can be

rewritten as a first-order system,

∂u

∂t+∇·f(u,q) = 0, (3.2)

q−∇u = 0. (3.3)

The solution u = u(x, y, t) to the system evolves inside the 2D domainΩ (with bound-

ary Γ), where x and y are spatial coordinates, q = (qx, qy) is the auxiliary variable

with components qx = qx(u,∇u) and qy = qy(u,∇u), and f = (f, g) has components

f = f(u,q) and g = g(u,q). Assume that the domain Ω can be discretized into N

non-overlapping, conforming, straight-sided triangular elements Ωk such that

Ω =

N⋃

k=1

Ωk, (3.4)

Ωi ∩Ωj = ∅ ∀i 6= j. (3.5)

Approximations to equations (3.2) and (3.3) can be constructed within each element

Ωk. In equation (3.2), the exact solution u can be replaced by an approximate

solution uDk = uDk (x, y, t), which is a two-dimensional polynomial of degree p within

Ωk and is identically zero outside the element. In general, the sum of uDk and uDk+1 is

discontinuous at the interface between neighboring elements Ωk and Ωk+1, and as a

result each quantity is designated with a superscript D.

The flux f in equation (3.2) can be approximated by a function fk = (fk, gk) =

fk(x, y, t), which is a polynomial of degree p + 1 within Ωk and is identically zero

outside Ωk. The normal components of fk and fk+1 are required to be equivalent to

one another on the boundary between Ωk and Ωk+1.

Analogous approximations can be introduced into equation (3.3). Here, the auxiliary

variable q can be replaced by a function qDk = (qDx , q

Dy )k = qD

k (x, y, t) which is a

polynomial of degree p within Ωk and is identically zero outside. In general, the sum

of qDk and qD

k+1 is discontinuous at the interface between Ωk and Ωk+1. In addition,

the exact solution u can be replaced by an approximate solution uk = uk(x, y, t),

which is a polynomial of degree p + 1 within Ωk and is identically zero outside (and

where it is important to note that uk 6= uDk ). The approximate solutions uk and uk+1


are required to be equivalent to one another on the boundary between Ωk and Ωk+1.

Based on these definitions, the approximate first-order system within each element

Ωk becomes

∂uDk∂t

+∇·fk = 0, (3.6)

qDk −∇uk = 0. (3.7)

Next, in order to facilitate implementation, each element Ωk in physical space is

mapped to a reference right triangle ΩS. The mapping Θk for a linear triangular

element is

x = Θk(x) = −(x+ y)

2v1,k +

(x+ 1)

2v2,k +

(y + 1)

2v3,k, (3.8)

where x = (x, y) are the physical coordinates, x = (x, y) are the reference coordinates,

v1,k, v2,k, and v3,k are the vertex coordinates of the element Ωk in physical space,

and (−1,−1), (1,−1), and (−1, 1) are the vertex coordinates of the element ΩS in

reference space. Once a mapping is established, the physical quantities uk, uDk , fk, and

qDk can be reformulated as transformed quantities u, uD, f , and qD in the reference

domain using the following transformations due to Viviand [54] and Vinokur [55]

u = Jkuk(Θk(x), t), uD = JkuDk (Θk(x), t), (3.9)

f =

[fg

]= Jk J

−1k fk, qD =

[qDxqDy

]= ∇u = Jk J

Tk qD

k , (3.10)

where

fk =

[fkgk

], qD

k =

[qDxk

qDyk

]= ∇uk, (3.11)

and

Jk =

[∂x∂x

∂x∂y

∂y∂x

∂y∂y

], Jk = det(Jk). (3.12)

In equation (3.11), the operator ∇ is defined as the gradient in the reference domain

taken with respect to x and y (where thus far ∇ has been the gradient in the physical

domain taken with respect to x and y).


The following identities arise from equations (3.9)-(3.12)

∇·fk =1

Jk

(∇·f), (3.13)

∇uk · fk =1

J2k

(∇u · f

), (3.14)

∫

Ωk

Jkuk (∇·fk) dΩk =

∫

ΩS

u(∇·f)dΩS, (3.15)

∫

Ωk

Jk (∇uk · fk) dΩk =

∫

ΩS

∇u · fdΩS, (3.16)

∫

Γk

Jkuk (fk · n) dΓk =

∫

ΓS

u(f · n

)dΓS. (3.17)

These identities are used extensively in the proof of stability of the VCJH schemes

for the linear advection-diffusion problem on triangles (cf. section 3.6).

Upon utilizing equations (3.9)-(3.12) to reformulate the physical quantities in equa-

tions (3.6) and (3.7), one obtains

∂uD

∂t+ ∇·f = 0, (3.18)

qD − ∇u = 0. (3.19)

The next section will discuss the FR procedure for solving equations (3.18) and (3.19).

3.2.2 Procedure

The FR approach for advection-diffusion problems on triangles consists of seven

stages. In practice, when implementing the method, several of these stages can be

combined, however, in what follows they will be presented as separate stages for the

sake of clarity.

In the first stage of the FR approach on triangles, the approximate transformed

solution uD within the reference element ΩS is defined using a two-dimensional poly-

nomial of degree p. The polynomial is constructed from values of the approximate

transformed solution at Np =12(p+ 1)(p+ 2) solution points. (Figure (3.1) shows an

example of the locations of the solution points for the case of p = 2).


Figure 3.1: Example of solution point locations (denoted by circles) in the reference elementfor the case of p = 2.

The resulting approximate transformed solution takes the form

uD(x, t) =

Np∑

i=1

uDi ℓi(x), (3.20)

where uDi = Jk(xi) · uDk (Θk(xi), t) is the value of uD at solution point i, xi is the

location of solution point i, and ℓi(x) is the nodal basis function which takes on

the value of 1 at solution point i and the value of 0 at all other solution points.

The approximate transformed solution uD lies in the space Pp(ΩS), where Pp(ΩS)

defines the polynomial space of degree ≤ p on ΩS. Consequently, uD also lies in

the polynomial space Rp(ΓS) =φ|φ ∈ L2(ΓS), φ|Γf

∈ Pp(Γf), ∀Γf

which contains

polynomials of degree ≤ p defined on each side Γf of ΩS.

The second stage involves calculating common values for the transformed solution u⋆

at a set of flux points along the edges of ΩS. In what follows, a quantity labeled

with the indices f , j corresponds to a quantity at the flux point j of face f , where

1 ≤ f ≤ 3 and 1 ≤ j ≤ Nfp. Here, Nfp is the number of flux points on each face,

and is equal to p + 1. The location of flux point j on face f is denoted xf,j . The

convention used to number faces and flux points is illustrated in Figure (3.2).


j = 1 j = 2 j = Nfp

= 3

f = 1

f = 2

f = 3

Figure 3.2: Example of the numbering convention for the faces and flux points on thereference element for the case of p = 2. The flux points are denoted by squares and the fluxpoint index increases counterclockwise along an edge.

Common values of the transformed solution at each flux point (u⋆f,j) are computed

by first evaluating the multiply defined values of the physical solution uDf,j at each

flux point. At each flux point, uD(f,j)− is defined as the value of uDf,j computed using

the information local to the current element and uD(f,j)+ as the value of uDf,j computed

using information from the adjoining element. For a given element, the physical

solution at each flux point is obtained by evaluating the transformed solution (uD

in equation (3.20)) at each flux point, and converting the result to physical space

using equation (3.9). Once both physical solution values (uD(f,j)− and uD(f,j)+) are ob-

tained, an approach such as the Central Flux (CF) [13], Local Discontinuous Galerkin

(LDG) [53], Compact Discontinuous Galerkin (CDG) [56], Internal Penalty (IP) [57],

Bassi Rebay 1 (BR1) [58], or Bassi Rebay 2 (BR2) approach [59] can be used to

compute a common solution value u⋆f,j. The LDG approach is of particular interest

because it is identical to the CDG approach in 1D, and recovers the BR1 and CF ap-

proaches in 1D, 2D, and 3D. If one elects to employ the LDG approach, the common


solution value u⋆f,j is computed as follows

u⋆f,j = uDf,j − β · [[uDf,j ]], β = (βx, βy) , (3.21)

where β is a directional parameter which allows u⋆f,j to assume a value that is biased

in either the upwind or downwind direction, and · and [[·]] are ‘average’ and ‘jump’

operators, respectively which are defined such that

uDf,j =uD(f,j)− + uD(f,j)+

2, [[uDf,j ]] = uD(f,j)−n− + uD(f,j)+n+, (3.22)

where n− and n+ denote the outward and inward pointing unit normals, respectively.

Note that choosing β = ±0.5n− promotes compactness of the FR scheme in multiple

dimensions, but does not ensure it for certain cases on general grids [13, 56]. Alter-

native approaches (such as CDG or BR2) can be employed to ensure compactness in

multiple dimensions. These compact approaches are essential for cases in which the

full matrix is to be formed, i.e. when a spatial discretization is paired with an im-

plicit time-stepping approach. However, this work is concerned with demonstrating

favorable aspects of pairing FR discretizations with explicit time-stepping approaches.

In this context, compactness is less important, and the LDG approach is frequently

sufficient.

After the common solution u⋆f,j is computed using the LDG approach or an alternative

approach, the transformed common solution denoted by u⋆f,j can be obtained from

equation (3.9).

Next, in preparation for the third stage, one may compute uCf,j the transformed solu-

tion correction at each flux point as follows

uCf,j = u⋆f,j − uDf,j. (3.23)

Note that the superscript ‘C’ on uCf,j stands for ‘correction.’

The third stage involves constructing the transformed auxiliary variable qD from the

transformed solution correction uCf,j at each flux point. Towards this end, one may


expand qD (in equation (3.19)) in order to obtain

qD = ∇u = ∇uD + ∇uC , (3.24)

where ∇uD is computed by applying the transformed gradient operator ∇ to uD in

equation (3.20), and ∇uC is computed by ‘lifting’ values of uCf,j from the element

boundary as follows

∇uC(x) =3∑

f=1

Nfp∑

j=1

uCf,j ψf,j(x) nf,j. (3.25)

The function ψf,j is a ‘correction field’ or ‘lifting operator’ which transforms uC defined

on ΓS into ∇uC defined throughout ΩS. Details regarding the construction of ψf,j

are given in section 3.4.

On rewriting equation (3.24) in terms of uD from equation (3.20) and ∇uC from

equation (3.25), one obtains the following expression for qD

qD(x) = ∇uD(x) + ∇uC(x) =Np∑

i=1

uDi ∇ℓi(x) +3∑

f=1

Nfp∑

j=1

uCf,j ψf,j(x) nf,j. (3.26)

In the fourth stage, the transformed discontinuous flux fD can be computed using

uD from equation (3.20) and qD from equation (3.26). Each component of the flux

fD = (fD, gD) can be expressed using a degree p polynomial as follows

fD(x) =

Np∑

i=1

fDi ℓi(x), gD(x) =

Np∑

i=1

gDi ℓi(x), (3.27)

where the quantities fDi and gDi are the components of the transformed flux at solution

point i, and are evaluated using uDi and qDi .

The fifth stage involves calculating transformed numerical fluxes f⋆ at flux points

along the edges of ΩS. At each flux point, the transformed flux f⋆f,j is calculated

based on the physical flux f⋆f,j which is composed from advective and diffusive parts

f⋆f,j = f⋆(f,j) adv+f⋆(f,j) dif . In turn, f⋆(f,j) adv and f⋆(f,j) dif are constructed from the multiply


defined discontinuous fluxes at each flux point denoted by

fD(f,j) adv− = f(uD(f,j)−

), fD(f,j) adv+ = f

(uD(f,j)+

), (3.28)

fD(f,j) dif− = f(uD(f,j)−,q

D(f,j)−

), fD(f,j) dif+ = f

(uD(f,j)+,q

D(f,j)+

). (3.29)

The solution values uD(f,j)− and uD(f,j)+ were obtained in stage 2, and the auxiliary vari-

able values qD(f,j)− and qD

(f,j)+ are obtained by evaluating qD at each flux point using

equation (3.26), and converting the result to physical space using equation (3.10).

Once fD(f,j) adv−, fD(f,j) adv+, f

D(f,j) dif−, and fD(f,j) dif+ are obtained, the numerical advec-

tive flux f⋆(f,j) adv and the numerical diffusive flux f⋆(f,j) dif can be computed. For linear

advection-diffusion equations, f⋆(f,j) adv can be computed using the Lax-Friedrichs ap-

proach (as defined in [13]), and for the nonlinear Navier-Stokes equations, f⋆(f,j) advcan be computed using a Roe [27] or Rusanov [60] approach. For both linear and

nonlinear equations, the numerical diffusive flux f⋆(f,j) dif can be computed using one

of the aforementioned CF, LDG, CDG, IP, BR1, or BR2 approaches. In particular,

if one elects to employ the LDG approach, the common numerical diffusive flux is

computed as follows

f⋆(f,j) dif = fD(f,j) dif+ τ [[uDf,j ]] + β[[fD(f,j) dif ]], (3.30)

where τ is a penalty parameter controlling the jump in the solution, β is the directional

parameter (defined previously), and · and [[·]] are the average and jump operators

for the flux which are defined such that

fD(f,j) dif =fD(f,j) dif− + fD(f,j) dif+

2, (3.31)

[[fD(f,j) dif ]] = fD(f,j) dif− · n− + fD(f,j) dif+ · n+. (3.32)

Note that the parameter β in equation (3.30) is preceded by a ‘+’ sign and in equa-

tion (3.21) it is preceded by a ‘−’ sign. Opposite signs help ensure symmetry of the

diffusive process in the following sense: if the common solution u⋆f,j is upwind biased

then the numerical flux f⋆(f,j) dif is downwind biased or vice versa.

Once the total numerical flux f⋆f,j = f⋆(f,j) adv + f⋆(f,j) dif is computed, the transformed

numerical flux denoted by f⋆f,j can be obtained from equation (3.10).


The sixth stage involves forming the transformed correction flux fC which will be

added to the transformed discontinuous flux fD to form the transformed continuous

flux f (i.e. f = fD + fC). fC is constructed using vector correction functions hf,j (x)

which are restricted to lie in the Raviart-Thomas space [61] of order p, denoted by

RTp(ΩS). Note that RTp(ΩS) = (Pp (ΩS))2 + xPp (ΩS), where (Pp (ΩS))

2 is the two-

dimensional vector space with components with degree ≤ p. As a consequence, the

vector correction functions possess the following attributes

φf,j(x) ∈ Pp(ΩS), hf,j(x) · n|ΓS∈ Rp(ΓS), (3.33)

where

φf,j(x) ≡ ∇·hf,j(x) (3.34)

is referred to as the ‘correction field’, and hf,j(x) · n|ΓSis referred to as the ‘normal

trace.’

Furthermore, if k represents a particular face (1 ≤ k ≤ 3) and ℓ represents a particular

flux point (1 ≤ ℓ ≤ Nfp), each of the correction functions hf,j is required to satisfy

hf,j(xk,ℓ) · nk,ℓ =

1 if f = k and j = ℓ0 if f 6= k or j 6= ℓ

. (3.35)

Equations (3.33) and (3.35) ensure that the resulting FR schemes are conservative

[52]. Figure (3.3) shows an example of a vector correction function hf,j which satisfies

equations (3.33) and (3.35) for the case of p = 2.


Figure 3.3: Example of a vector correction function hf,j associated with flux pointf = 2, j = 2 for the case of p = 2.

Now, having defined the vector correction functions hf,j , one may define the trans-

formed correction flux fC as follows

fC(x) =

3∑

f=1

Nfp∑

j=1

[(f⋆f,j − fDf,j

)· nf,j

]hf,j(x) =

3∑

f=1

Nfp∑

j=1

∆f,jhf,j(x). (3.36)

Here, ∆f,j is defined as the difference between the normal transformed numerical flux

and the normal transformed discontinuous flux at the flux point f ,j.

The final stage involves calculating the divergence of the continuous transformed flux

f . The result is used to update the transformed solution at the solution point xi as

follows

duDidt

= −(∇·f) ∣∣∣∣

xi

= −(∇·fD

) ∣∣∣∣xi

−(∇·fC

) ∣∣∣∣xi

= −Np∑

n=1

(fDn

) ∂ℓn∂x

∣∣∣∣xi

−Np∑

n=1

(gDn) ∂ℓn∂y

∣∣∣∣xi

−3∑

f=1

Nfp∑

j=1

∆f,jφf,j(xi). (3.37)

For triangles, the overall behavior of the FR scheme depends on six factors, namely:

1. The location of the solution points xi.


2. The location of the flux points xf,j .

3. The methodology for calculating the transformed common solution values u⋆f,j.

4. The methodology for calculating the transformed numerical flux values f⋆f,j.

5. The form of the solution correction field ψf,j .

6. The form of the flux correction field φf,j.

The FR schemes can be simplified by choosing the same solution and flux correction

fields (choosing ψf,j = φf,j). Nevertheless, in an effort to generalize the subsequent

discussions, the solution and flux correction fields are assumed to be distinct unless

otherwise indicated.

Finally, note that the computational effort associated with the FR approach is iden-

tical to that associated with a collocation-based nodal DG approach. When the cor-

rections fields are chosen appropriately, the FR approach recovers a collocation-based

nodal DG scheme (as mentioned previously).

3.3 VCJH Flux Correction Fields on Triangles

For linear advection problems on triangles, Castonguay et al. [52] have identified a

range of correction fields φf,j for correcting the flux, which lead to energy stable FR

schemes. The correction fields φf,j are polynomials of degree p, and can therefore be

expressed in terms of an orthonormal basis Ln of degree p as follows

φf,j =

Np∑

n=1

σnLn(x), (3.38)

where the coefficients σn are constants. Here each Ln is a member of the 2D Dubiner

basis [62] defined on the reference right triangle ΩS. This basis is given by

Ln(x) =√2Qv(a)Q

(2v+1,0)w (b)(1− b)v, (3.39)

k = w + (p + 1)v + 1− v

2(v − 1), (v, w) ≥ 0, v + w ≤ p,


where

a = 2(1 + x)

(1− y) − 1, b = y, (3.40)

and Q(α,β)m is the mth order Jacobi polynomial (normalized as described in [13]).

Castonguay et al. [52] showed that stability for linear advection problems can be

achieved if the coefficients σn are computed by solving the following system of equa-

tions

Np∑

n=1

σn

p+1∑

v=1

cv

(D(p,v)Li

)(D(p,v)Ln

)= −σi +

∫

ΓS

(hf,j · n)Li dΓS for 1 ≤ i ≤ Np,

(3.41)

where each dot product (hf,j · n) is defined via equation (3.35), each derivative oper-

ator D(p,v) is of the form

D(p,v) =∂p

∂x(p−v+1)∂y(v−1), 1 ≤ v ≤ p+ 1, (3.42)

and each constant cv is of the form

cv = c

(p

v − 1

)= c

[p!

(v − 1)!(p− v + 1)!

]. (3.43)

In equation (3.43), c is a scalar, where 0 ≤ c ≤ ∞. The correction fields φf,j obtained

from the solution of equation (3.41) lead to VCJH schemes on triangles which are an

infinite family of FR schemes parameterized by c. For linear advection problems on

triangles, if c = cdg = 0, a collocation-based nodal DG scheme is recovered [52].

The stability of VCJH schemes for linear advection problems on triangles (for which

periodic boundary conditions can be imposed) is ensured because it can be shown

that a Sobolev-type norm of the solution is non-increasing, i.e.

d

dt‖UD‖2p,c ≤ 0, (3.44)

where the Sobolev-type norm is defined as

‖UD‖p,c =

N∑

k=1

∫

Ωk

[(uDk )

2 +1

AS

p+1∑

v=1

cv

(D(p,v) uDk

)2]dΩk

1/2

, (3.45)


UD is the total (domain-wide) solution defined as

UD =

N∑

k=1

uDk , (3.46)

and AS denotes the area of the reference element ΩS. Note that, in equation (3.45),

the derivative operator D(p,v) in ΩS has been applied to uDk in the physical domain

Ωk using the chain rule.

3.4 VCJH Solution Correction Fields on Triangles

Consider the solution correction function gf,j and solution correction field (‘lifting

operator’) ψf,j , where ψf,j = ∇·gf,j. According to the previous section, the correction

field ψf,j can be classified as a ‘VCJH correction field’ if it takes the following form

ψf,j =

Np∑

n=1

ξnLn(x), (3.47)

where each coefficient ξn is obtained from the following system of equations

Np∑

n=1

ξn

p+1∑

v=1

κv

(D(p,v)Li

)(D(p,v)Ln

)= −ξi +

∫

ΓS

(gf,j · n)Li dΓS for 1 ≤ i ≤ Np,

(3.48)

where each D(p,v) is a derivative operator of degree p (defined by equation (3.42) in

section 3.3).

In equation (3.48), each constant κv is defined as

κv = κ

(p

v − 1

)= κ

[p!

(v − 1)!(p− v + 1)!

], 0 ≤ κ ≤ ∞, (3.49)

and hence the correction fields ψf,j are parameterized by the scalar κ.

As currently constructed, the correction fields ψf,j possess an important ability to act

as ‘lifting operators’, which transform uC defined on the element boundary ΓS into


∇uC defined within the element interior ΩS, according to the following identity

∫

ΩS

∇uC dΩS =

∫

ΓS

uCn dΓS. (3.50)

In what follows, the derivation of this identity will be examined.

First, recall that ψf,j = ∇ · gf,j, and thus according to the divergence theorem, gf,j

and ψf,j are related as follows

∫

ΩS

ψf,j dΩS =

∫

ΓS

gf,j · n dΓS. (3.51)

The quantity gf,j ·n (which satisfies equation (3.35) with gf,j in place of hf,j) vanishes

on all faces except for face f . As a result, equation (3.51) becomes

∫

ΩS

ψf,j dΩS =

∫

Γf

gf,j · nf,j dΓf . (3.52)

Multiplying equation (3.52) by uCf,j and nf,j , and summing over f and j yields

∫

ΩS

3∑

f=1

Nfp∑

j=1

uCf,jψf,jnf,j dΩS =

3∑

f=1

∫

Γf

Nfp∑

j=1

uCf,j (gf,j · nf,j) nf,j dΓf

. (3.53)

Substituting equation (3.25) into this equation, and replacing the term gf,j · nf,j with

the 1D Lagrange polynomial ℓ 1Df,j (because gf,j · nf,j satisfies equation (3.35) and

gf,j ∈ RTp(ΩS)), one obtains

∫

ΩS

∇uC dΩS =

3∑

f=1

∫

Γf

Nfp∑

j=1

uCf,j ℓ1Df,j nf,j dΓf

. (3.54)

Note that the 1D Lagrange polynomial ℓ 1Df,j is a degree p polynomial on face f , which

takes on the value of 1 at flux point j, and the value of 0 at all other flux points on

the face.


A corollary to the divergence theorem requires that

∫

ΩS

∇uC dΩS =

∫

ΓS

uCn dΓS =3∑

f=1

[∫

Γf

uC∣∣Γf

nf dΓf

]. (3.55)

From equations (3.54) and (3.55), one obtains

3∑

f=1

[∫

Γf

uC∣∣Γf

nf dΓf

]=

3∑

f=1

∫

Γf

Nfp∑

j=1

uCf,j ℓ1Df,j nf,j dΓf

. (3.56)

Each domain Γf is arbitrary (as the standard element and its associated faces can

be chosen arbitrarily) and thus the integrands in equation (3.56) must be equal.

Consequently, uC on ΓS takes the form

uC∣∣∣∣Γf

=

Nfp∑

j=1

uCf,j ℓ1Df,j uC

∣∣∣∣ΓS

=

uC∣∣Γ1

on Γ1

uC∣∣Γ2

on Γ2

uC∣∣Γ3

on Γ3

. (3.57)

In conclusion, the VCJH correction fields ψf,j serve as ‘lifting operators’, because

they provide a relationship (given by equation (3.55)) between uC defined on ΓS

(equation (3.57)) and ∇uC defined within ΩS (equation (3.25)).

3.5 Visualization of VCJH Correction Fields

The VCJH flux and solution correction fields φf,j and ψf,j assume the same analytical

form (with the caveat that the parameterizing coefficients c and κ need not be the

same), and therefore, for the sake of brevity this section will only present plots of

the flux correction fields φf,j. In the following figures, contours of the fields φf,j are

plotted for the case of p = 2 and c = c+ = 3.13 × 10−2. It should be noted that the

scheme with c = c+ is optimal in the following sense: for linear advection problems,

the combination of the VCJH scheme with the value of c = c+ and the 5-stage, 4th

order, Runge-Kutta time-stepping scheme (derived in [16]) yielded a maximal explicit

time-step limit relative to all other schemes tested by Castonguay et al. in [52].


f=1, j=1 f=1, j=2

f=1, j=3

Figure 3.4: Plots of the VCJH correction fields φf,j for the case of p = 2, f = 1, andc = c+ = 3.13 × 10−2.


f=2, j=1 f=2, j=2

f=2, j=3



f=3, j=1 f=3, j=2

f=3, j=3


From Figures (3.4) – (3.6), it is clear that the VCJH correction fields behave in accor-

dance with physical intuition, as the symmetries of the correction fields are consistent

with the underlying symmetries of the right triangle. This is expected, as Castonguay

et al. [52] demonstrated (via numerical experiments) that if the VCJH correction fields

are parameterized according to equations (3.43) and (3.49), the resulting fields are

symmetric under rotations and reflections of the triangle unto itself.


3.6 Proof of Stability of VCJH Schemes for the

Linear Advection-Diffusion Equation on Tri-

angles

In this section, it is shown that if the correction fields ψf,j and φf,j are chosen to be

VCJH correction fields, (i.e. ψf,j = ∇ ·gf,j and φf,j = ∇ ·hf,j where hf,j and gf,j are

VCJH vector correction functions), and the transformed common solution values u⋆f,jand numerical flux values f⋆f,j are obtained appropriately, the resulting FR schemes

are stable for the linear advection-diffusion equation on straight-sided triangles.

3.6.1 Preliminaries

The linear advection-diffusion equation in 2D can be expressed as the following first-

order system

∂u

∂t+∇· (au− bq) = 0, (3.58)

q−∇u = 0, (3.59)

where a = (ax, ay) is a velocity vector (with constant components) and b is a dif-

fusivity coefficient (with constant value b ≥ 0). Equations (3.58) and (3.59) can be

rewritten in terms of physical coordinates in the kth element (Ωk), yielding equa-

tions (3.6) and (3.7), where fk = fadv,k + fdif,k, and where fadv,k = auDk and fdif,k =

−bqDk .

3.6.2 The Stability of VCJH Schemes

The stability of VCJH schemes for linear advection-diffusion problems on triangles

can be established by examining the evolution in time of the Sobolev-type norm of

the solution given in equation (3.45). An expression for the time rate of change of

this norm will be constructed using contributions from the time rate of change of the

solution itself∂uD

k

∂t(equation (3.6)) and the auxiliary variable qD

k (equation (3.7)).

The following lemmas will manipulate and combine equations (3.6) and (3.7) in order

to establish an upper bound for the time rate of change of the norm.


Lemma 3.6.1. For all FR schemes, equation (3.6) holds, and therefore the following

results also hold

Jk2

d

dt

∫

Ωk

(uDk)2dΩk = −

∫

ΩS

uD(∇·fD + ∇·fC

)dΩS (3.60)

and

Jk2AS

d

dt

∫

Ωk

(D(p,v) uDk

)2dΩk = −

(D(p,v) uD

)(D(p,v)

(∇·fC

)). (3.61)

Proof. The following relation holds for all FR schemes: ∇·fk = ∇·fDk +∇·fCk . Upon

combining this relation and equation (3.6) and rearranging the result, one obtains

∂uDk∂t

= −∇·fDk −∇·fCk . (3.62)

Multiplying this equation by JkuDk and integrating over the element domain Ωk yields

the following

Jk2

d

dt

∫

Ωk

(uDk)2dΩk = −

∫

Ωk

JkuDk

(∇·fDk

)dΩk −

∫

Ωk

JkuDk

(∇·fCk

)dΩk. (3.63)

Upon replacing the integrals over the physical domain Ωk on the RHS of equa-

tion (3.63) with integrals over the reference domain ΩS (using the identity from

equation (3.15)), one obtains equation (3.60), the first result of Lemma 3.6.1.

The second result of Lemma 3.6.1 can be derived as follows. Consider applying the

operator D(p,v) to both sides of equation (3.62) in order to obtain

∂

∂t

(D(p,v) uDk

)= −D(p,v)

(∇·fDk

)− D(p,v)

(∇·fCk

)= −D(p,v)

(∇·fCk

). (3.64)

Note that D(p,v)(∇·fDk

)= 0 because ∇·fDk is at most degree (p− 1).

If both sides of equation (3.64) are multiplied by (D(p,v) uDk ) and integrated over Ωk,

one obtains

1

2

d

dt

∫

Ωk

(D(p,v) uDk

)2dΩk = −

∫

Ωk

(D(p,v) uDk

)(D(p,v)

(∇·fCk

))dΩk. (3.65)


Substituting equations (3.9) and (3.13) into equation (3.65) yields

1

2

d

dt

∫

Ωk

(D(p,v) uDk

)2dΩk = −

∫

ΩS

1

Jk

(D(p,v) uD

)(D(p,v)

(∇·fC

))dΩS

= −AS

Jk

(D(p,v) uD

)(D(p,v)

(∇·fC

)). (3.66)

Upon rearranging equation (3.66) one obtains equation (3.61). This completes the

proof of Lemma 3.6.1.

Lemma 3.6.2. For all FR schemes, equation (3.7) holds, and therefore the following

results also hold

Jk

∫

Ωk

fDdif,k · qDk dΩk =

∫

ΩS

fDdif ·(∇uD + ∇uC

)dΩS (3.67)

and

JkAS

∫

Ωk

(D(p,v) fDdif,k

)·(D(p,v) qD

k

)dΩk =

(D(p,v) fDdif

)·(D(p,v)

(∇uC

)). (3.68)

Proof. The following relation holds for all FR schemes: ∇uk = ∇uDk + ∇uCk . Upon

combining this relation and equation (3.7) and rearranging the result, one obtains

qDk = ∇uDk +∇uCk . (3.69)

Taking the dot product of equation (3.69) with Jk fDdif,k and integrating over the

element domain Ωk yields

Jk

∫

Ωk

fDdif,k · qDk dΩk = Jk

∫

Ωk

(fDdif,k · ∇uDk

)dΩk + Jk

∫

Ωk

(fDdif,k · ∇uCk

)dΩk. (3.70)

Upon transforming the RHS of equation (3.70) to the reference domain using equa-

tion (3.16) (with fdif,k and fdif in place of fk and f), one obtains equation (3.67), the

first result of Lemma 3.6.2.

The second result of Lemma 3.6.2 can be derived as follows. Consider applying the

operator D(p,v) to both sides of equation (3.69) in order to obtain

D(p,v) qDk = D(p,v)

(∇uDk

)+ D(p,v)

(∇uCk

)= D(p,v)

(∇uCk

). (3.71)


The term D(p,v)(∇uDk

)vanishes because ∇uDk is of degree (p − 1). Taking the dot

product of equation (3.71) with(D(p,v) fDdif,k

)and integrating over Ωk yields

∫

Ωk

(D(p,v) fDdif,k

)·(D(p,v) qD

k

)dΩk =

∫

Ωk

(D(p,v) fDdif,k

)·(D(p,v)

(∇uCk

))dΩk.

(3.72)

The RHS of equation (3.72) can be simplified via the identities in equations (3.10)

and (3.11). On substituting fDdif,k, fDdif , u

Ck , and uC in place of fk, f , uk, and u in

equations (3.10) and (3.11), and rearranging the result, one obtains

fDdifJk

= J−1k fDdif,k, (3.73)

∇uCJk

= JTk ∇uCk . (3.74)

Upon applying the derivative operator D(p,v) to equations (3.73) and (3.74), one

obtains

1

JkD(p,v)

(fDdif

)= D(p,v)

(J−1k fDdif,k

), (3.75)

1

JkD(p,v)

(∇uC

)= D(p,v)

(JTk ∇uCk

). (3.76)

Next, consider taking the dot product of equations (3.75) and (3.76) as follows

D(p,v)(J−1k fDdif,k

)· D(p,v)

(JTk ∇uCk

)=

1

J2k

(D(p,v) fDdif

)·(D(p,v)

(∇uC

)). (3.77)

Now, one should note that, for straight-sided triangles Jk is a matrix of constant

values (independent of x and y) and therefore the following identities hold

D(p,v)(J−1k fDdif,k

)= J−1

k D(p,v)(fDdif,k

), (3.78)

D(p,v)(JTk ∇uCk

)= JT

k D(p,v)

(∇uCk

). (3.79)

On substituting equations (3.78) and (3.79) into equation (3.77) and simplifying the


result, one obtains

(D(p,v) fDdif,k

)·(D(p,v)

(∇uCk

))=

1

J2k

(D(p,v) fDdif

)·(D(p,v)

(∇uC

)). (3.80)

Upon substituting equation (3.80) into the RHS of equation (3.72), one obtains

∫

Ωk

(D(p,v) fDdif,k

)·(D(p,v) qD

k

)dΩk =

∫

Ωk

1

J2k

(D(p,v) fDdif

)·(D(p,v)

(∇uC

))dΩk

=AS

Jk

(D(p,v) fDdif

)·(D(p,v)

(∇uC

)). (3.81)

On rearranging equation (3.81) one obtains equation (3.68). This completes the proof

of Lemma 3.6.2.

Lemma 3.6.3. If φf,j and ψf,j are the VCJH correction fields, the following identity

holds

Jk2

d

dt

∫

Ωk

(uDk)2dΩk +

Jk2AS

d

dt

∫

Ωk

p+1∑

v=1

cv

(D(p,v) uDk

)2dΩk

− Jk∫

Ωk

fDdif,k · qDk dΩk −

JkAS

∫

Ωk

p+1∑

v=1

κv

(D(p,v) fDdif,k

)·(D(p,v) qD

k

)dΩk

= −∫

ΩS

uD(∇·fD

)dΩS −

∫

ΓS

uD(fC · n

)dΓS

−∫

ΩS

fDdif ·(∇uD

)dΩS −

∫

ΓS

uC(fDdif · n

)dΓS, (3.82)

where constants cv and κv are related via equations (3.43) and (3.49) to constants

c and κ, respectively, which parameterize the VCJH correction fields φf,j and ψf,j,

respectively.

Proof. Consider subtracting equation (3.67) from equation (3.60) in order to obtain

Jk2

d

dt

∫

Ωk

(uDk)2dΩk − Jk

∫

Ωk

fDdif,k · qDk dΩk

= −∫

ΩS


)dΩS −

∫

ΩS


)dΩS. (3.83)

Next, consider multiplying equation (3.61) by cv and equation (3.68) by−κv, summing


each equation over v, and adding the results to equation (3.83) in order to obtain

Jk2

d

dt

∫

Ωk

(uDk)2dΩk +

Jk2AS

d

dt

∫

Ωk

p+1∑

v=1

cv

(D(p,v) uDk

)2dΩk

− Jk∫

Ωk

fDdif,k · qDk dΩk −

JkAS

∫

Ωk

p+1∑

v=1

κv

(D(p,v) fDdif,k

)·(D(p,v) qD

k

)dΩk

= −∫

ΩS


)dΩS −

p+1∑

v=1

cv

(D(p,v) uD

)(D(p,v)

(∇·fC

))

−∫

ΩS


)dΩS −

p+1∑

v=1

κv

(D(p,v) fDdif

)·(D(p,v)

(∇uC

)). (3.84)

Suppose that ∇·fC and ∇uC are constructed based on correction fields φf,j = ∇ ·hf,j

and ψf,j = ∇·gf,j, respectively, which satisfy the following identities (that are satisfied

if hf,j and gf,j are the VCJH correction functions as shown in [52])

∫

ΩS

hf,j · ∇Li dΩS −p+1∑

v=1

cv

(D(p,v) Li

)(D(p,v) φf,j

)= 0, (3.85)

∫

ΩS

gf,j · ∇Li dΩS −p+1∑

v=1

κv

(D(p,v) Li

)(D(p,v) ψf,j

)= 0. (3.86)

One may manipulate equations (3.85) and (3.86) in order to obtain two additional

equations that can be utilized to simplify equation (3.84). Towards this end, one may

perform integration by parts on equation (3.85) and rearrange the result, in order to

obtain

∫

ΓS

(hf,j · n)Li dΓS =

∫

ΩS

φf,jLi dΩS +

p+1∑

v=1

cv

(D(p,v)Li

)(D(p,v)φf,j

). (3.87)

Note that uD ∈ Pp (ΩS) can be written as a linear combination of the orthonormal

polynomials Li. As a result, equation (3.87) can be rewritten in terms of uD as follows

∫

ΓS

(hf,j · n) uD dΓS =

∫

ΩS

φf,j uD dΩS +

p+1∑

v=1

cv

(D(p,v)uD

)(D(p,v)φf,j

). (3.88)


On multiplying equation (3.88) by ∆f,j and summing over f and j, one obtains

∫

ΓS

3∑

f=1

Nfp∑

j=1

∆f,j hf,j · n

uD dΓS

=

∫

ΩS

3∑

f=1

Nfp∑

j=1

∆f,j φf,j

uD dΩS +

p+1∑

v=1

cv

(D(p,v)uD

)D(p,v)

3∑

f=1

Nfp∑

j=1

∆f,j φf,j

.

(3.89)

Upon substituting the definition of fC (equation (3.36)) into equation (3.89) and

rearranging the result, one obtains

p+1∑

v=1

cv

(D(p,v) uD

)(D(p,v)

(∇·fC

))=

∫

ΓS

uD(fC · n

)dΓS −

∫

ΩS

uD(∇ · fC

)dΩS.

(3.90)

Setting equation (3.90) aside for the moment, consider performing integration by

parts on equation (3.86) and rearranging the result, in order to obtain

∫

ΓS

(gf,j · n)Li dΓS =

∫

ΩS

ψf,jLi dΩS +

p+1∑

v=1

κv

(D(p,v)Li

)(D(p,v)ψf,j

). (3.91)

On the LHS, gf,j · n vanishes on all faces except for face f (as required by equa-

tion (3.35)). As a result, equation (3.91) becomes

∫

Γf

(gf,j · nf,j)Li dΓf =

∫

ΩS

ψf,jLi dΩS +

p+1∑

v=1

κv

(D(p,v)Li

)(D(p,v)ψf,j

). (3.92)

Because (fDdif · nf,j) ∈ Pp (ΩS), it can be expressed as a linear combination of the

orthonormal polynomials Li and therefore, equation (3.92) can be rewritten with

(fDdif · nf,j) in place of Li as follows

∫

Γf

(gf,j · nf,j)(fDdif · nf,j

)dΓf

=

∫

ΩS

ψf,j

(fDdif · nf,j

)dΩS +

p+1∑

v=1

κv

(D(p,v)

(fDdif · nf,j

))(D(p,v)ψf,j

). (3.93)


On multiplying equation (3.93) by uCf,j and summing over f and j, one obtains

3∑

f=1

∫

Γf

Nfp∑

j=1

uCf,j (gf,j · nf,j)(fDdif · nf,j

)dΓf

=

∫

ΩS

fDdif ·

3∑

f=1

Nfp∑

j=1

uCf,j ψf,jnf,j

dΩS +

p+1∑

v=1

κv D(p,v)(fDdif ) · D(p,v)

3∑

f=1

Nfp∑

j=1

uCf,j ψf,jnf,j

.

(3.94)

Upon combining equations (3.25), (3.57), and (3.94) and rearranging the result, one

obtains

p+1∑

v=1

κv

(D(p,v) fDdif

)·(D(p,v)

(∇uC

))=

∫

ΓS

uC(fDdif · n

)dΓS−

∫

ΩS

(fDdif · ∇uC

)dΩS.

(3.95)

Upon substituting equations (3.90) and (3.95) into the RHS of equation (3.84), one

obtains equation (3.82). This completes the proof of Lemma 3.6.3.

Theorem 3.6.1. If the VCJH schemes on triangles (for which Lemmas 3.6.1–3.6.3

hold) are employed in conjunction with the Lax-Friedrichs formulation for the advec-

tive numerical flux f⋆adv

f⋆adv = fDadv+λ

2

(max

u∈[uD−,uD

+ ]

∣∣∣∣∂fadv∂u· n∣∣∣∣

)[[uD]], (3.96)

with 0 ≤ λ ≤ 1, and the LDG formulation for the common solution u⋆ and diffusive

numerical flux f⋆dif ,

u⋆ = uD − β · [[uD]], (3.97)

f⋆dif = fDdif+ τ [[uD]] + β[[fDdif ]], (3.98)

with β = (βx, βy) and τ ≥ 0, then it can be shown that the following result holds

d

dt‖UD‖2p,c ≤ 0. (3.99)

Proof. The fluxes fC and fD in equation (3.82) (from Lemma 3.6.3) can be ex-

pressed in terms of advective and diffusive parts: in particular fC = fCadv + fCdif and


fD = fDadv + fDdif . In addition, the diffusive flux fdif,k on the LHS of equation (3.82)

is defined such that fdif,k = −bqDk (as mentioned previously). As a result, equa-

tion (3.82) becomes

Jk2

d

dt

∫

Ωk

[(uDk)2

+1

AS

p+1∑

v=1

cv

(D(p,v) uDk

)2]dΩk

+ b Jk

∫

Ωk

[qDk · qD

k +1

AS

p+1∑

v=1

κv

(D(p,v) qD

k

)·(D(p,v) qD

k

)]dΩk

= −∫

ΩS

uD(∇·fDadv

)dΩS −

∫

ΓS

uD(fCadv · n

)dΓS −

∫

ΩS

uD(∇·fDdif

)dΩS

−∫

ΩS

(∇uD · fDdif

)dΩS −

∫

ΓS

uD(fCdif · n

)dΓS −

∫

ΓS

uC(fDdif · n

)dΓS. (3.100)

Setting equation (3.100) aside for the moment, one may consider the following identity

uD(∇·fDadv) =1

2∇ · (uD fDadv), (3.101)

which holds because fDadv = J−1k auD, J−1

k is a constant matrix, and a is a constant

vector. On substituting equation (3.101) into the RHS of equation (3.100), and

employing the divergence theorem and integration by parts, one obtains

Jk2

d

dt

∫

Ωk

[(uDk)2

+1

AS

p+1∑

v=1

cv

(D(p,v) uDk

)2]dΩk

+ b

∫

Ωk

[qDk · qD

k +1

AS

p+1∑

v=1

κv

(D(p,v) qD

k

)·(D(p,v) qD

k

)]dΩk

= −12

∫

ΓS

uD(fDadv · n

)dΓS −

∫

ΓS

uD(fCadv · n

)dΓS

−∫

ΓS

uD(fDdif · n

)dΓS −

∫

ΓS

uD(fCdif · n

)dΓS −

∫

ΓS

uC(fDdif · n

)dΓS. (3.102)

Consider the following identities

u

∣∣∣∣ΓS

=(uD + uC

) ∣∣∣∣ΓS

= u⋆, f · n∣∣∣∣ΓS

=(fD + fC

)· n∣∣∣∣ΓS

= f⋆ · n, (3.103)


where (on each face Γf)

u⋆ =

Nfp∑

j=1

u⋆f,j ℓ1Df,j , f⋆ =

Nfp∑

j=1

f⋆f,j ℓ1Df,j . (3.104)

One may use equation (3.103) to eliminate fC and uC from equation (3.102), and

thereafter one may use equation (3.17) to transform the integrals on the RHS in

order to obtain the following expression

1

2

d

dt

∫

Ωk

[(uDk)2

+1

AS

p+1∑

v=1

cv

(D(p,v) uDk

)2]dΩk

+ b

∫

Ωk

[qDk · qD

k +1

AS

p+1∑

v=1

κv

(D(p,v) qD

k

)·(D(p,v) qD

k

)]dΩk

=

∫

Γk

[1

2uDk(fDadv,k · n

)− uDk (f⋆adv · n)

]dΓk

+

∫

Γk

[uDk(fDdif,k · n

)− uDk

(f⋆dif · n

)− u⋆

(fDdif,k · n

)]dΓk, (3.105)

where the quantities u⋆, f⋆adv, and f⋆dif are defined (on each face Γf) as

u⋆ =

Nfp∑

j=1

u⋆f,j ℓ1Df,j , f⋆adv =

Nfp∑

j=1

f⋆(f,j) adv ℓ1Df,j , f⋆dif =

Nfp∑

j=1

f⋆(f,j) dif ℓ1Df,j . (3.106)

To obtain a description of the solution behavior within the entire domain, one must

sum over all the elements in the mesh (summing over k on equation (3.105)) as follows

1

2

d

dt

N∑

k=1

∫

Ωk

[(uDk)2

+1

AS

p+1∑

v=1

cv

(D(p,v) uDk

)2]dΩk

+ bN∑

k=1

∫

Ωk

[qDk · qD

k +1

AS

p+1∑

v=1

κv

(D(p,v) qD

k

)·(D(p,v) qD

k

)]dΩk

=

N∑

k=1

∫

Γk

[1

2uDk(fDadv,k · n


]dΓk

+N∑

k=1

∫

Γk

[uDk(fDdif,k · n

)− uDk

(f⋆dif · n

)− u⋆

(fDdif,k · n

)]dΓk

. (3.107)


For constants cv and κv that satisfy 0 ≤ cv <∞ and 0 ≤ κv <∞, the expressions

‖UD‖p,c =

N∑

k=1

∫

Ωk

[(uDk)2

+1

AS

p+1∑

v=1

cv

(D(p,v) uDk

)2]dΩk

1/2

(3.108)

and

‖QD‖p,κ =

N∑

k=1

∫

Ωk

[qDk · qD

k +1

AS

p+1∑

v=1

κv

(D(p,v) qD

k

)·(D(p,v) qD

k

)]dΩk

1/2

(3.109)

are broken Sobolev-type norms of the solution UD and the auxiliary variable QD,

respectively. The expressions

Θadv =

N∑

k=1

∫

Γk

[1

2uDk(fDadv,k · n


]dΓk

(3.110)

and

Θdif =N∑

k=1

∫

Γk

[uDk(fDdif,k · n

)− uDk

(f⋆dif · n

)− u⋆

(fDdif,k · n

)]dΓk

(3.111)

represent contributions from the advective and diffusive fluxes at the element bound-

aries. Using equations (3.108) - (3.111), equation (3.107) can be rewritten as

1

2

d

dt‖UD‖2p,c = −b ‖QD‖2p,κ +Θadv +Θdif . (3.112)

On the RHS of equation (3.112), the term −b ‖QD‖2p,κ is guaranteed to be non-

positive for b ≥ 0, and the terms Θadv and Θdif are guaranteed to be non-positive

for appropriate choices of the common solution u⋆, and the common numerical fluxes

f⋆adv and f⋆dif .

The Lax-Friedrichs formulation provides a suitable expression for f⋆adv (equation (3.96)).

Upon substituting fadv = au into equation (3.96) and taking the dot product with n,


one obtains

f⋆adv · n =

(auD+ λ

2|a · n| [[uD]]

)· n

=1

2

(uD+ + uD−

)(axnx + ayny) +

λ

2|axnx + ayny|

(uD− − uD+

), (3.113)


uD+ =

Nfp∑

j=1

uD(f,j)+ ℓ1Df,j , uD− =

Nfp∑

j=1

uD(f,j)− ℓ1Df,j , (3.114)

and where λ is an upwinding parameter in the sense that λ > 0 results in an upwind

biased flux and λ = 0 results in a central flux. Castonguay et al. [52] demonstrated

that this choice for f⋆adv ensures Θadv ≤ 0.

The LDG formulation provides suitable expressions for u⋆ and f⋆dif (equations (3.97)

and (3.98)). Upon expanding equation (3.97), one obtains

u⋆ = uD − β · [[uD]]

=1

2

(uD− + uD+

)−(uD− − uD+

)(βxnx + βyny) , (3.115)

and upon substituting fdif = −bq into equation (3.98) and taking the dot product

with n, one obtains

f⋆dif · n =(fDdif+ τ [[uD]] + β[[fDdif ]]

)· n

=− b[1

2

(qDx− + qDx+

)nx +

1

2

(qDy− + qDy+

)ny

+((qDx− − qDx+

)nx +

(qDy− − qDy+

)ny

)(βxnx + βyny)

]+ τ

(uD− − uD+

),

(3.116)


[qDx+qDy+

]= qD

+ =

Nfp∑

j=1

qD(f,j)+ ℓ

1Df,j ,

[qDx−qDy−

]= qD

− =

Nfp∑

j=1

qD(f,j)− ℓ

1Df,j . (3.117)


In what follows, it will be shown that this choice for u⋆ and f⋆dif ensures Θdif ≤ 0.

On substituting equations (3.115) and (3.116) into the combination of equations (3.111)

and (3.112), and replacing uDk and qDk with uD− and qD

− = (qDx−, qDy−), respectively, one

obtains

1

2

d

dt‖UD‖2p,c

= −b ‖QD‖2p,κ +Θadv +

N∑

k=1

∫

Γk

[b

(uD−2

(qDx+nx + qDy+ny

)+uD+2

(qDx−nx + qDy−ny

)

+(uD+(qDx−nx + qDy−ny

)− uD−

(qDx+nx + qDy+ny

))(βxnx + βyny)

)+ τ

(uD−u

D+ − (uD−)

2)]dΓk

.

(3.118)

The last term on the RHS of equation (3.118) can be rewritten as a sum over edges

instead of a sum over elements. Assuming the domain is such that periodic bound-

ary conditions can be imposed, each edge receives contributions from two adjacent

elements. Using the notation uDe,+, qDxe,+

, qDye,+ and uDe,−, qDxe,−

, qDye,− to define the solu-

tion and auxiliary variable from the elements on the right and left sides of an edge,

respectively, equation (3.118) becomes

1

2

d

dt‖UD‖2p,c

= −b ‖QD‖2p,κ +Θadv +

Ne∑

e=1

∫

Γe

[b

(uDe,−2

(qDxe,+

nx + qDye,+ ny

)+uDe,+2

(qDxe,−

nx + qDye,− ny

)

− uDe,+2

(qDxe,−

nx + qDye,− ny

)− uDe,−

2

(qDxe,+

nx + qDye,+ ny

)

+(uDe,−

(qDxe,+

nx + qDye,+ ny

)− uDe,+

(qDxe,−

nx + qDye,− ny

))(βxnx + βyny)

+(uDe,+

(qDxe,−

nx + qDye,− ny

)− uDe,−

(qDxe,+

nx + qDye,+ ny

))(βxnx + βyny)

)

+ τ(uDe,−u

De,+ − (uDe,−)

2)+ τ

(uDe,+u

De,− − (uDe,+)

2)]dΓe

= −b ‖QD‖2p,κ +Θadv − τNe∑

e=1

∫

Γe

(uDe,+ − uDe,−

)2dΓe

. (3.119)

In the equations above, Γe represents edge e. Equation (3.99) follows immediately

from equation (3.119) because b ≥ 0, τ ≥ 0, and Θadv ≤ 0 from [52]. This completes


the proof of Theorem 3.6.1

Remark. Theorem 3.6.1 guarantees the stability of VCJH schemes for linear advection-

diffusion problems on triangles. In summary, this theorem was obtained assuming

that:

1. The approximate solution UD is computed on a domain of straight-sided trian-

gles, each with a constant value of Jk.

2. The correction functions (gf,j and hf,j) and fields (ψf,j and φf,j) are the VCJH

correction functions and fields defined in sections 3.3 and 3.4. This ensures that

equations (3.90) and (3.95) are satisfied, constants c and κ are non-negative,

and thus ‖UD‖p,c and ‖QD‖p,κ are norms.

3. The advective numerical flux is computed using the Lax-Friedrichs approach

(equation (3.113)).

4. The common solution and diffusive numerical flux are computed using the LDG

approach (equations (3.115) and (3.116)).

Note that, the stability proof has been constructed for the specific case in which the

Lax-Friedrichs approach is used for the advective numerical flux and the LDG ap-

proach is used for the common solution and diffusive numerical flux. However, the

proof still has broad applicability because the Lax-Friedrichs approach recovers the

central and upwind approaches, and the LDG approach recovers the BR1 and CF

approaches. Furthermore, it appears that the proof can be extended to alternative

flux formulations, in particular to compact approaches such as the CDG, BR2, or IP

approaches. However, such extensions are beyond the scope of this work because, as

discussed previously, this work is concerned with demonstrating the favorable per-

formance of VCJH discretizations paired with explicit time-stepping approaches, for

which compactness is not necessarily essential.

3.7 Linear Numerical Experiments

This section will investigate the orders of accuracy and explicit time-step limits asso-

ciated with the VCJH schemes on triangles by performing numerical experiments to

solve the linear advection-diffusion equation.


Consider the 2D, time-dependent, linear advection-diffusion of a scalar u = u(x, t)

governed by equations (3.58) and (3.59) in the square domain [−1, 1]×[−1, 1], subjectto a sinusoidal initial condition u(x, 0) = sin(πx) sin(πy), and periodic boundary

conditions. The exact solution takes the form

ue = exp(−2bπ2t

)sin (π (x− axt)) sin (π (y − ayt)) , (3.120)

where ax = a cos θ and ay = a sin θ are the wave speeds in the x and y directions, and

b is the diffusivity coefficient. Numerical experiments were performed on two variants

of this problem: a diffusion problem with a = 0 and b = 0.1, and an advection-

diffusion problem with a = 1, b = 0.1, and θ = π/6. These problems were solved

using the VCJH schemes in conjunction with a Lax-Friedrichs formulation for the

advective numerical flux and a LDG formulation for the common solution and diffusive

numerical flux. The schemes were marched forward in time from t = 0 to t = 1 using

an explicit, low-storage, 5 stage, 4th order Runge-Kutta scheme for time advancement

(denoted RK54) [16]. For the order of accuracy analysis, the time-step was chosen

sufficiently small to ensure that temporal errors were negligible relative to spatial

errors.

In the experiments, VCJH schemes parameterized by c and κ were paired with Lax-

Friedrichs fluxes parameterized by λ and LDG fluxes parameterized by β and τ .

Values of c, κ, λ, β, and τ were selected as follows:

• Choosing c and κ: Four values of c, (namely, cdg, csd, chu, and c+), have been

shown to produce favorable results for advection problems on triangles [63]. For

advection problems on triangles, Castonguay et al. [63, 52] demonstrated that

cdg recovers a collocation-based nodal DG scheme (as mentioned previously),

csd and chu recover schemes with properties similar to the one-dimensional SD

scheme [48] and Huynh’s g2 scheme [23], and finally c+ recovers a scheme which

yields a maximum explicit time-step limit (as mentioned previously). In the

experiments documented in this work, these four values of c were paired with

four equivalent values of κ, namely, κ = κdg = cdg, κ = κsd = csd, κ = κhu = chu,

and κ = κ+ = c+. Table (3.1) shows the numerical values of c and κ for

polynomial orders p = 2 and p = 3, for several different time-stepping schemes,

including the RK54 time-stepping scheme.


p=2 p=3RK33 cdg = κdg 0. 0.

csd = κsd 3.75e-03 1.23e-04chu = κhu 8.45e-03 2.19e-04c+ = κ+ 2.61e-02 4.92e-04

RK44 cdg = κdg 0. 0.csd = κsd 3.51e-03 1.13e-04chu = κhu 7.90e-03 2.01e-04c+ = κ+ 2.44e-02 4.50e-04

RK54 cdg = κdg 0. 0.csd = κsd 4.50e-03 1.17e-04chu = κhu 1.01e-02 2.08e-04c+ = κ+ 3.13e-02 4.67e-04

Table 3.1: Reference values of c and κ for p = 2 and p = 3, for the 3-stage, 3rd-orderRunge-Kutta scheme (RK33), the 4-stage, 4th-order, Runge-Kutta scheme (RK44), and a5-stage, 4th-order, Runge-Kutta scheme (RK54) [16]. These values were computed usingvon Neumann analysis (similar to the analysis performed in [52]) of the VCJH schemes onuniform grids of right triangular elements.

Note that the experiments did not utilize values of c and κ that were larger

than c+ and κ+, as it has been shown that choosing c≫ c+ and κ≫ κ+ results

in a significant reduction in the order of accuracy of VCJH schemes for 1D

advection-diffusion problems [51].

• Choosing λ: All of the numerical experiments were performed with λ = 1.

This value of λ ensures that the advective numerical flux is computed using

information exclusively from the upwind direction.

• Choosing β: All of the numerical experiments were performed with β = ±0.5n−.

This value of β promotes compactness of the scheme (as mentioned previ-

ously) [53, 56].

• Choosing τ : All numerical experiments were performed with τ = 0.1. This

value of τ has been shown to yield favorable results for linear advection-diffusion

problems in 1D [51].

3.7.1 Orders of Accuracy and Explicit Time Step Limits

The orders of accuracy were evaluated on regular triangular meshes created by di-

viding cartesian N × N quadrilateral meshes into meshes with N = 2N2 triangular


elements. Triangular meshes with N = 16, 24, 32, 48, 64, 96, 128, and 192 were cre-

ated. On each mesh, errors in the solution and the solution gradient were measured

using the following L2 norm and seminorm

E(L2) =

√√√√N∑

k=1

∫

Ωk

(ue − uDk )2 dΩk, (3.121)

E(L2s) =

√√√√N∑

k=1

∫

Ωk

[(∂ue

∂x− ∂uDk

∂x

)2

+

(∂ue

∂y− ∂uDk

∂y

)2]dΩk. (3.122)

In equations (3.121) and (3.122), the integrals over each element domainΩk were com-

puted using a quadrature rule of sufficient strength. The expected order of accuracy

for E(L2) was p+ 1, and the expected order for E(L2s) was p.

In addition to orders of accuracy, explicit time-step limits were obtained for the VCJH

schemes. The explicit time-step limit for each scheme was determined by using an

iterative method to find the largest time-step that allowed the solution to remain

bounded at t = 1.

For each of the schemes, Tables (3.2) to (3.4) contain values of the time-step limit

and the absolute error obtained on the grid with N = 32, along with the order of

accuracy obtained on the sequence of grids described above.


c κ L2 err. O(L2) L2s err. O(L2s) ∆tmax

cdg κdg 1.26e-05 3.00 1.29e-03 2.00 3.18e-04κsd 1.27e-05 3.00 1.22e-03 2.00 3.77e-04κhu 1.33e-05 3.00 1.19e-03 2.00 4.05e-04κ+ 1.72e-05 2.99 1.17e-03 2.00 4.51e-04

csd κdg 1.25e-05 3.00 1.29e-03 2.00 3.75e-04κsd 1.27e-05 3.00 1.22e-03 2.00 4.37e-04κhu 1.33e-05 3.00 1.19e-03 2.00 4.76e-04κ+ 1.72e-05 2.99 1.17e-03 2.00 5.42e-04

chu κdg 1.25e-05 3.00 1.30e-03 2.00 4.02e-04κsd 1.27e-05 3.00 1.22e-03 2.00 4.73e-04κhu 1.33e-05 3.00 1.19e-03 2.00 5.22e-04κ+ 1.72e-05 2.99 1.17e-03 2.00 6.04e-04

c+ κdg 1.25e-05 3.00 1.30e-03 2.00 4.42e-04κsd 1.27e-05 3.00 1.22e-03 2.00 5.33e-04κhu 1.34e-05 3.00 1.19e-03 2.00 5.97e-04κ+ 1.74e-05 3.00 1.18e-03 2.00 7.07e-04

Table 3.2: VCJH scheme accuracy properties and explicit time-step limits for the modellinear advection-diffusion problem on triangles with a = 0, b = 0.1, and p = 2. Values ofλ = 1, β = ±0.5n−, and τ = 0.1 were used in the experiments. The time-step limit (∆tmax)and absolute errors (L2 and L2s err.) were obtained on the grid with N = 32.














In particular, Tables (3.2) and (3.3) show the results for the diffusion problem, where

the order of accuracy was obtained on grids with N = 24, 32, 48, 64, 96, 128, and 192

for p = 2 and N = 16, 24, 32, 48, 64, 96, and 128 for p = 3. Table (3.4) shows the

results for the advection-diffusion problem, where the order of accuracy was obtained

on grids with N = 16, 24, 32, 48, 64, 96, and 128 for p = 3. Note that the results

for the diffusion and the advection-diffusion problems were similar, and thus limited

results (only the results for p = 3) are shown for the advection-diffusion problem.

The data in Tables (3.2) to (3.4) demonstrates that each scheme obtains the expected

order of accuracy. In addition, the data demonstrates that schemes with larger values

of c and κ have larger maximum time-steps. The best time-step improvements are

shown in Table (3.2) for p = 2, c = c+, and κ = κ+, where the maximum time-step is

approximately 2.2 times larger than that of the collocation-based nodal DG scheme.

Chapter 4


for Advection-Diffusion Problems on

Tetrahedra

71

CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 72

4.1 Preamble

For the first time, this chapter constructs energy stable FR schemes (VCJH schemes)

for advection-diffusion problems on tetrahedra, and proves the stability of these

schemes for linear advection-diffusion problems for all orders of accuracy. Note

that this chapter is adapted from the article “Energy Stable Flux Reconstruction

for Advection-Diffusion Problems on Tetrahedra” by D. M. Williams and A. Jame-

son, which has been submitted to the Journal of Scientific Computing. The main

results and over 95% of the text for this article were contributed by D. M. Williams.

Due to the size of this chapter, it is useful to briefly examine its format. Section 2

of this chapter describes the general FR approach for treating tetrahedral elements,

section 3 introduces the energy stable FR approach (i.e. the VCJH approach) for

tetrahedral elements and proves the stability of this approach for linear advection-

diffusion problems, section 4 describes how to construct the VCJH correction fields,

section 5 identifies values of the coefficients c and κ that preserve the spatial symmetry

of the schemes, section 6 presents plots of the symmetric VCJH correction fields,

section 7 proves that the resulting class of VCJH schemes is equivalent to a class of

filtered DG schemes, section 8 describes how to construct the filtering matrices for

the filtered DG schemes, section 9 identifies the VCJH schemes which have explicit

time-step limits that are maximal (for linear advection problems), and finally, in an

effort to further assess the capabilities of the VCJH schemes, section 10 presents the

results of numerical experiments on a canonical linear advection-diffusion problem.

4.2 Flux Reconstruction for Advection-Diffusion

Problems on Tetrahedral Elements

In what follows, the general FR methodology for advection-diffusion problems on

triangular elements, described by Castonguay et al. ([51] and [52]) and Williams et

al. ([64] and Chapter 3 of this thesis), is extended to tetrahedral elements.


4.2.1 Preliminaries

Consider the 3D advection-diffusion equation

∂u

∂t+∇ · f = 0, (4.1)

where u is a scalar solution, t is time, ∇ is the gradient operator defined such that∇ ≡(∂∂x, ∂∂y, ∂∂z

), x = (x, y, z) = (x1, x2, x3) are spatial coordinates, and f = f (u,∇u) is

an advective-diffusive flux. In keeping with the standard approach (from [58]), one

may eliminate second derivative terms from Equation (4.1) by rewriting it as the

following first-order system

∂u

∂t+∇ · f = 0, (4.2)

q−∇u = 0. (4.3)

This system depends on u and a new variable q that is commonly referred to as the

‘auxiliary variable.’ One seeks a solution to this system within the 3D domain Ω with

boundary Γ. Towards this end, one may assume that the domain can be divided into

N conforming, non-overlapping, straight-sided tetrahedral elements Ωk as follows

Ω =

N⋃

k=1

Ωk, (4.4)

Ωi ∩Ωj = ∅ ∀i 6= j. (4.5)

Within the kth element Ωk, the solution u can be approximated by a function uDkwhich is continuously defined within Ωk and which vanishes outside of the element.

The function uDk is labeled with a superscript D to indicate that, in general, it is

discontinuous at the boundary between neighboring elements Ωk and Ωk+1. In ac-

cordance with the traditional, discontinuous Finite Element approach (as described

in [13]), uDk can be approximated using a polynomial of degree p which takes the

following form

uDk =

Np∑

i=1

(uDk )i ℓi (x) , (4.6)


where (uDk )i is the value of the solution at solution point i within Ωk and ℓi (x) is

the multi-dimensional nodal basis function which assumes the value of 1 at solution

point i and the value of 0 at all other solution points. Figure (4.1) shows an example

of the Np = 10 solution points which can be used to define a degree p = 2 polynomial

approximation of the solution on the tetrahedron.

(−1,1,−1)

(1,−1,−1)

(−1,−1,−1)

(−1,−1,1)

Figure 4.1: Example of the Np = 10 solution point locations (denoted by spheres) in thereference element for the case of p = 2.

In general, note that Np = (p+1)(p+2)(p+3)6

solution points are required to define a

degree p polynomial approximation of the solution on the tetrahedron.

In a similar fashion, the auxiliary variable q and the flux f within Ωk can be approx-

imated by vector-valued functions qDk and fDk (respectively) whose components are


polynomials of degree p which are defined as follows

qDk =

(qDxk, qDyk , q

Dzk

)=(qD1k , q

D2k, qD3k

),

∀ m = 1, 2, 3, qDmk=

Np∑

i=1

(qDmk)i ℓi (x) , (4.7)

fDk =(fDxk, fD

yk, fD

zk

)=(fD1k, fD

2k, fD

3k

),

∀ m, fDmk

=

Np∑

i=1

(fDmk

)i ℓi (x) , (4.8)

where (qDmk)i is the value of the mth component of the auxiliary variable at solution

point i, and (fDmk

)i is the value of the mth component of the flux at solution point i.

Note that, in general, the flux (fDmk

)i is a nonlinear function of both the solution (uDk )i

and the auxiliary variable (qDk )

i.

Upon substituting uDk , qDk , and fDk in place of u, q, and f in equations (4.2) and (4.3),

one obtains

∂uDk∂t

+∇ · fDk = 0, (4.9)

qDk −∇uDk = 0. (4.10)

As currently constructed, equations (4.9) and (4.10) involve only information that

is local to the kth element, as they relate the solution uDk in Ωk to the flux fDk and

auxiliary variable qDk in Ωk. In order to construct a valid numerical scheme, it is

necessary to incorporate information from neighboring elements. Towards this end,

the FR approach replaces the discontinuous flux fDk in equation (4.9) with a ‘recon-

structed flux’ fk of degree p + 1 which is required to be continuous in the following

sense: the normal components of fk and fk+1 are required to be equivalent to one

another on the boundary between neighboring elements Ωk and Ωk+1. In addition,

the FR approach replaces the discontinuous solution uDk in equation (4.10) with a

reconstructed solution uk of degree p + 1 which is required to be continuous in the

sense that uk and uk+1 are required to be equivalent to one another on the boundary

between Ωk and Ωk+1. Upon replacing fDk and uDk with fk and uk in equations (4.9)


and (4.10), one obtains

∂uDk∂t

+∇ · fk = 0, (4.11)

qDk −∇uk = 0. (4.12)

Evidently, equations (4.11) and (4.12) represent a valid numerical scheme in the sense

that they utilize information from both the kth element and its neighbors.

Prior to solving equations (4.11) and (4.12), it is convenient to first transform these

equations from the domain of the physical element Ωk to the domain of a ‘reference

element’ ΩS. Towards this end, suppose that the element Ωk has vertices v1.k, v2,k,

v3,k, and v4,k, and the reference tetrahedronΩS has vertices (−1,−1,−1), (1,−1,−1),(−1, 1,−1), and (−1,−1, 1). One may define a mapping Θk between the physical

coordinates x in Ωk and the reference coordinates x = (x, y, z) in ΩS, as follows

x = Θk (x) = −x+ y + z + 1

2v1,k +

x+ 1

2v2,k +

y + 1

2v3,k +

z + 1

2v4,k. (4.13)

Using the mapping in equation (4.13), one may construct the Jacobian matrix Jk =

∇Θk (where ∇ =(

∂∂x, ∂∂y, ∂∂z

)) and the Jacobian determinant Jk = det(Jk). Next,

one may use Jk and Jk in order to transform the physical quantities uk, uDk , fk, and

qDk defined on Ωk into reference quantities u, uD, f , and qD defined on ΩS as follows

u = Jk uk (Θk (x) , t) , uD = Jk uDk (Θk (x) , t) , (4.14)

f = Jk J−1k fk, qD = Jk J

Tk q

Dk = ∇u = Jk J

Tk∇uk. (4.15)

Note that, as a result of equations (4.14) and (4.15), the following equations also hold

∇ · fk = J−1k

(∇ · f

), (4.16)

∇uk · fk = J−2k

(∇u · f

), (4.17)

∫

Ωk

uk (∇ · fk) dΩk = J−1k

∫

ΩS

u(∇ · f

)dΩS, (4.18)

∫

Ωk

∇uk · fk dΩk = J−1k

∫

ΩS

∇u · f dΩS, (4.19)

∫

Γk

uk (fk · n) dΓk = J−1k

∫

ΓS

u(f · n

)dΓS. (4.20)


Equations (4.14) - (4.20) will be used frequently in subsequent discussions. For now,

consider substituting equations (4.14) - (4.16) into equations (4.11) and (4.12), in

order to obtain the following

∂uD

∂t+ ∇ · f = 0, (4.21)

qD − ∇u = 0. (4.22)

Observe that equations (4.21) and (4.22) are defined exclusively on the reference

domain ΩS, as they have been obtained by transforming equations (4.11) and (4.12)

from the physical domain Ωk to the reference domain ΩS. In what follows, the FR

procedure for solving equations (4.21) and (4.22) will be discussed.

4.2.2 The FR Procedure

The FR procedure for solving equations (4.21) and (4.22) involves obtaining the

unknown quantities qD, the auxiliary variable in reference space, and ∂uD

∂t, the time

rate of change of the solution in reference space, from computations of (respectively)

∇u, the gradient of the reconstructed solution in reference space, and ∇ · f , the

divergence of the reconstructed flux in reference space.

4.2.2.1 Computing ∇u, the Gradient of the Reconstructed Solution in

Reference Space

In order to facilitate the computation of ∇u, one must first formulate a more precise

definition for u. Towards this end, the FR approach requires u to take the following

form on the element boundary ΓS

u

∣∣∣∣ΓS

=(uD + uC

) ∣∣∣∣ΓS

, (4.23)

where uC is a ‘solution correction’ that corrects uD. On the element boundary, the

sum of uD and uC is required to assume the value of u⋆, the value of the common

numerical solution in reference space. This is equivalent to requiring that the following

expression holds

u

∣∣∣∣ΓS

= u⋆∣∣∣∣ΓS

=(uD + uC

) ∣∣∣∣ΓS

. (4.24)


Equation (4.24) ensures that u (and thus uk) is continuous at the boundary between

neighboring elements Ωk and Ωk+1. In practice, equation (4.24) is enforced pointwise

at the l = 1, . . . , Nfp ‘flux points’ on the f = 1, . . . , Nef faces of the element, as

follows

uf,l = u⋆f,l = uDf,l + uCf,l ∀f, l (4.25)

where (for example) uDf,l is the value of uD at flux point l on face f .

For reference, the f = 1, . . . , 4 faces of the tetrahedron are shown in Figure (4.2), and

the l = 1, . . . , 6 flux points for the case of f = 1, p = 2 are shown in Figure (4.3).

f = 2 (back)

f = 1 (front)

f = 4 (bottom)

f = 3 (back)

Figure 4.2: Example of the numbering convention for the faces on the reference element.


l = 3

l = 5

l = 2

l = 6

l = 4

l = 1

Figure 4.3: Example of the numbering convention for the flux points on the reference elementfor the case of p = 2. The flux points (denoted by squares) are shown for the face f = 1.

Note that, in general Nfp = (p+ 1)(p+ 2)/2 and Nef = 4 for a tetrahedron.

Now, having formulated u in terms of uD and uC via equations (4.23)-(4.25), one may

formulate ∇u in a similar fashion. In particular, one may express ∇u as follows

∇u = ∇uD + ∇uC , (4.26)

where ∇uD can be computed by applying the gradient operator ∇ to uD, and ∇uCcan be computed by applying the ‘lifting operators’ (or ‘correction fields’) ψf,l to

values of uCf,l as follows

∇uC (x) =

Nfe∑

f=1

Nfp∑

l=1

uCf,lnf,l ψf,l (x)

=

Nfe∑

f=1

Nfp∑

l=1

[u⋆f,l − uDf,l

]nf,l ψf,l (x) . (4.27)

The lifting operators ψf,l(x) are designed to ‘lift’ (or transform) values of uC defined

on ΓS into values of ∇uC defined onΩS. In order to ensure that the operators perform

this task, they are defined such that ψf,l ≡ ∇ · gf,l, where each gf,l is a vector-valued

‘correction function’ associated with flux point l on face f . In addition, the normal


component of the correction function (gf,l · n) is required to assume the value of 1 at

flux point l on face f and to assume the value of 0 at all neighboring flux points as

follows

gf,l (xv,w) · nv,w =

1 if v = f and w = l

0 if v 6= f or w 6= l. (4.28)

Furthermore, gf,l (x) is required to belong to the Raviart-Thomas space of degree p

(defined in [61]) in order to ensure that the following holds

ψf,l ≡ ∇ · gf,l ∈ Pp (ΩS) , gf,l · n ∈ Rp (ΓS) , (4.29)

where Pp (ΩS) and Rp (ΓS) are spaces which contain the polynomials of degree ≤ p on

ΩS and ΓS, respectively. In [64] (and Chapter 3 of this thesis), Williams et al. showed

that equations (4.28) and (4.29) ensure that ψf,l serves as a lifting operator that

transforms uC defined on ΓS into ∇uC defined on ΩS.

Figures (4.4) and (4.5) show an example of a vector correction function gf,l and an

associated correction field (lifting operator) ψf,l, for the case of p = 2.

Figure 4.4: Example of a vector correction function gf,l associated with flux pointf = 1, l = 2 for the case of p = 2.


Figure 4.5: Contours of a correction field ψf,l (where ψf,l ≡ ∇ · gf,l) associated with fluxpoint f = 1, l = 2 for the case of p = 2.

4.2.2.2 Computing ∇ · f , the Divergence of the Reconstructed Flux in

Reference Space

One must now define a precise form for f in order to facilitate the computation of

∇ · f . Towards this end, f is required to take the following form in reference space

f = fD + fC , (4.30)

where fD is the discontinuous flux (fDk ) in reference space, and fC is a ‘corrective flux’

which corrects fD. On each face of the element, the sum of the normal components

of fD and fC is required to equal the normal component of f⋆, the common numerical

flux in reference space. Equivalently, the following expression is required to hold

f · n∣∣∣∣ΓS

= f⋆ · n∣∣∣∣ΓS

=(fD + fC

)· n∣∣∣∣ΓS

. (4.31)

Equation (4.31) ensures that the normal component of f (and thus fk) is continuous at

the boundary between neighboring elements. In practice, equation (4.31) is enforced


pointwise at the flux points on the element faces as follows

ff,l · nf,l = f⋆f,l · nf,l =(fDf,l + fCf,l

)· nf,l ∀f, l. (4.32)

Now, having obtained a definition of f in terms of fD and fC , one can obtain a similar

definition of ∇ · f . Towards this end, one may express ∇ · f as follows

∇ · f = ∇ · fD + ∇ · fC , (4.33)

where ∇ · fD is obtained by applying the divergence operator ∇· to fD, and ∇ · fCis obtained by applying lifting operators (correction fields) φf,l to values of fCf,l, as

follows

∇ · fC =

Nfe∑

f=1

Nfp∑

l=1

[fCf,l · nf,l

]φf,l

=

Nfe∑

f=1

Nfp∑

l=1

[(f⋆f,l − fDf,l

)· nf,l

]φf,l. (4.34)

In equation (4.34), each correction field φf,l is defined as the divergence of a vector

correction function hf,l. The functions hf,l are required to possess the same properties

as the functions gf,l (which were defined previously). This ensures that the resulting

FR schemes are conservative as shown by Castonguay et al. [52].

4.2.2.3 Obtaining a Final System of Equations

Upon substituting the expressions for ∇ · f (equations (4.33) and (4.34)) and ∇u(equations (4.26) and (4.27)) into equations (4.21) and (4.22), respectively, one ob-

tains a complete description of the FR approach on the reference element ΩS, as

follows

∂uD

∂t+ ∇ · fD + ∇ · fC

=∂uD

∂t+ ∇ · fD +

Nfe∑

f=1

Nfp∑

l=1

[(f⋆f,l − fDf,l

)· nf,l

]φf,l = 0, (4.35)


and

qD − ∇uD − ∇uC

= qD − ∇uD −Nfe∑

f=1

Nfp∑

l=1

[u⋆f,l − uDf,l

]nf,l ψf,l = 0. (4.36)

Equations (4.35) and (4.36) can be evaluated at theNp solution points withinΩS in or-

der to yield 4Np equations for the 4Np unknowns (∂uD

∂t)1, . . . , (∂u

D

∂t)Np and (qD)1, . . . , (qD)Np.

The behavior of the FR scheme defined by equations (4.35) and (4.36) is determined

by six factors:

1. The locations of the solution points xi within the reference element.

2. The locations of the flux points xf,l on the boundary of the reference element.

3. The procedure for computing the common numerical solution values u⋆f,l in

reference space.

4. The procedure for computing the common numerical flux values f⋆f,l in reference

space.

5. The procedure for computing the solution correction fields ψf,l.

6. The procedure for computing the flux correction fields φf,l.

If appropriate procedures are chosen for computing the common numerical solution

values u⋆f,l in reference space, the common numerical flux values f⋆f,l in reference

space, the solution correction fields ψf,l, and the flux correction fields φf,l, the FR

schemes can be proven stable for linear advection-diffusion problems, independent of

the locations of the solution points xi and flux points xf,l. This will be shown in the

following section.


4.3 Proof of Stability of VCJH Schemes for Linear


In this section, it will be shown that if the solution and flux correction fields ψf,l

and φf,l are chosen to be the ‘VCJH correction fields’, and if the common numerical

solution values u⋆f,l and common numerical flux values f⋆f,l are chosen appropriately,

the ensuing FR schemes (the VCJH schemes) are stable for linear advection-diffusion

problems.

4.3.1 Preliminaries

In order to examine the stability of the FR approach, it is useful to reformulate the

approach on the physical element Ωk. On Ωk, equations (4.35) and (4.36) of the FR

approach can be expressed succinctly as follows

∂uDk∂t

+∇ · fDk +∇ · fCk = 0, (4.37)

qDk −∇uDk −∇uCk = 0, (4.38)

where the reference quantities (defined on ΩS) in equations (4.35) and (4.36) were

converted into physical quantities (defined on Ωk) in equations (4.37) and (4.38) via

the transformations in equations (4.14) - (4.16).

The FR approach for solving equations (4.37) and (4.38) is considered to be ‘energy

stable’ ifN∑

k=1

(d

dt‖uDk ‖2

)≤ 0, (4.39)

or equivalentlyN∑

k=1

(d

dt‖Uk‖2

M

)≤ 0, (4.40)

where Uk =[(uDk )

1 . . . (uDk )Np]T

is a vector containing the solution values,

‖Uk‖M = UTk Mk Uk (4.41)

is a matrix-based norm, and Mk is a symmetric positive-definite matrix. Note that


the precise definition of Mk will be given later on in this section.

In equation (4.40), the squared norm of the solution ‖Uk‖2M

characterizes the en-

ergy of the solution. Therefore, equation (4.40) is a condition that ensures ‘energy

stability’, because it insists that the time rate of change of the solution energy is

non-positive.

It will be shown that equation (4.40) holds for a particular class of FR schemes,

referred to as the VCJH schemes. The proof of stability of the VCJH schemes will

consist of lemmas and a final theorem proving the stability of the schemes. In par-

ticular, the lemmas will summarize intermediate results which will be obtained from

manipulating equations (4.37) and (4.38), and the theorem will combine these results

in order to prove that equation (4.40) holds.

Lemma 4.3.1. Given that equation (4.37) holds for all FR schemes and provided

that the flux correction functions and fields (hf,l and φf,l) are chosen to be the VCJH

correction functions and fields, the following result holds

∫

Ωk

∂uDk∂t

ℓj dΩk +1

VS

p+1∑

v=1

v∑

w=1

cvw

∫

Ωk

∂

∂t

(D(p,v,w)

(uDk))D(p,v,w) (ℓj) dΩk

+

∫

Ωk

(∇ · fDk

)ℓj dΩk = −

∫

Γk

(fCk · n

)ℓj dΓk, (4.42)

where VS is the volume of the reference element ΩS, each cvw is a constant which

parameterizes φf,l (and thus hf,l), and each D(p,v,w) is a derivative operator which

will be subsequently defined.

Proof. Consider defining a derivative operator D(p,v,w) of degree p as follows

D(p,v,w) (·) = ∂p (·)∂x(p−v+1)∂y(v−w)∂z(w−1)

, (4.43)

where v = 1, . . . , p and w = 1, . . . , v. Note that, upon substituting all possible

combinations of v and w into equation (4.43), one recovers the (p + 1)(p + 2)/2

distinct derivatives of degree p in 3D.


One may apply D(p,v,w) to equation (4.37) as follows

∂

∂t

(D(p,v,w)

(uDk))

+ D(p,v,w)(∇ · fDk

)+ D(p,v,w)

(∇ · fCk

)

=∂

∂t

(D(p,v,w)

(uDk))

+ D(p,v,w)(∇ · fCk

)= 0, (4.44)

where terms involving derivatives of the physical quantities w.r.t the reference coor-

dinates (i.e. terms such as D(p,v,w)(uDk)) are computed via the chain rule, and where

D(p,v,w)(∇ · fDk

)= 0 because ∇ · fDk is a degree p− 1 polynomial.

On multiplying equation (4.44) by D(p,v,w) (ℓj) and integrating over Ωk, one obtains

the following

∫

Ωk

∂

∂t

(D(p,v,w)

(uDk))D(p,v,w) (ℓj) dΩk +

∫

Ωk

D(p,v,w)(∇ · fCk

)D(p,v,w) (ℓj) dΩk = 0.

(4.45)

Upon substituting equation (4.16) (with fC in place of f) into equation (4.45) and

defining ℓj ≡ Jk ℓj, one obtains

∫

Ωk

∂

∂t

(D(p,v,w)

(uDk))D(p,v,w) (ℓj) dΩk+

1

Jk

∫

Ωk

D(p,v,w)(∇ · fC

)D(p,v,w)

(ℓj

)dΩS = 0.

(4.46)

On noting that D(p,v,w)(∇ · fC

)and D(p,v,w)

(ℓj

)are constants because ∇ · fC and

ℓj are degree p polynomials, one obtains

∫

Ωk

∂

∂t

(D(p,v,w)

(uDk))D(p,v,w) (ℓj) dΩk +

VSJkD(p,v,w)

(∇ · fC

)D(p,v,w)

(ℓj

)= 0.

(4.47)

Consider multiplying equation (4.47) by constant coefficients cvw, and thereafter sum-

ming over v and w as follows

p+1∑

v=1

v∑

w=1

cvw

∫

Ωk

∂

∂t

(D(p,v,w)


+VSJk

p+1∑

v=1

v∑

w=1

cvwD(p,v,w)

(∇ · fC

)D(p,v,w)

(ℓj

)= 0. (4.48)

The double summation in equation (4.48) contains (p+1)(p+2)/2 terms, one for each

of the distinct (p+1)(p+2)/2 derivative operators of degree p that can be formed in


3D. As a result, the term of the form∑p+1

v=1

∑vw=1 cvwD

(p,v,w) (·) is a general, weighted

sum of all distinct derivatives of degree p.

Next, it should be noted that if fC and ∇ · fC are constructed using the VCJH

correction functions hf,l (which were discussed in section 4.2.2.1) and the VCJH

fields φf,l (which have yet to be precisely defined), the following identity holds

p+1∑

v=1

v∑

w=1

cvw D(p,v,w)

(∇ · fC

)D(p,v,w)

(ℓj

)=

∫

ΓS

(fC · n

)ℓj dΓS−

∫

ΩS

(∇ · fC

)ℓj dΩS.

(4.49)

Section 4.4 will discuss a procedure for constructing the fields φf,l so as to ensure that

equation (4.49) holds.

Upon substituting equation (4.49) into equation (4.48) and rearranging the result,

one obtains

1

VS

p+1∑

v=1

v∑

w=1

cvw

∫

Ωk

∂

∂t

(D(p,v,w)


+1

Jk

[∫

ΓS

(fC · n

)ℓj dΓS −

∫

ΩS

(∇ · fC

)ℓj dΩS

]= 0. (4.50)

On substituting equations (4.18) and (4.20) with fC , fCk , ℓj, and ℓj in place of f , fk,

u, and uk into equation (4.50), one obtains

1

VS

p+1∑

v=1

v∑

w=1

cvw

∫

Ωk

∂

∂t

(D(p,v,w)


+

∫

Γk

(fCk · n

)ℓj dΓk −

∫

Ωk

(∇ · fCk

)ℓj dΩk = 0. (4.51)

Setting equation (4.51) aside for the moment, consider multiplying equation (4.37)

by the test function ℓj and integrating over Ωk, as follows

∫

Ωk

∂uDk∂t

ℓj dΩk +

∫

Ωk

(∇ · fDk

)ℓj dΩk +

∫

Ωk

(∇ · fCk

)ℓj dΩk = 0. (4.52)

Upon summing equations (4.52) and (4.51), and rearranging the result, one obtains

equation (4.42). This completes the proof of Lemma 4.3.1.


Lemma 4.3.2. Given that equation (4.38) holds for all FR schemes and provided

that the solution correction functions and fields (gf,l and ψf,l) are chosen to be the

VCJH correction functions and fields, the following result holds

∫

Ωk

qDk · Lm,j dΩk +

1

VS

p+1∑

v=1

v∑

w=1

κvw

∫

Ωk

D(p,v,w)(qDk

)· D(p,v,w) (Lm,j) dΩk

−∫

Ωk

∇uDk · Lm,j dΩk =

∫

Γk

uCk (Lm,j · n) dΓk, (4.53)

where each κvw is a constant which parameterizes ψf,l (and thus gf,l), and each Lm,j

is a vectorial generalization of the nodal basis function ℓj, i.e. L1,j = Lx,j ≡ (ℓj, 0, 0),

L2,j = Ly,j ≡ (0, ℓj, 0), and L3,j = Lz,j ≡ (0, 0, ℓj).

Proof. Consider applying the derivative operator D(p,v,w) to both sides of equation (4.38)

as follows

D(p,v,w)(qDk

)− D(p,v,w)

(∇uDk

)− D(p,v,w)

(∇uCk

)

= D(p,v,w)(qDk

)− D(p,v,w)

(∇uCk

)= 0, (4.54)

where D(p,v,w)(∇uDk

)= 0 because ∇uDk is at most a degree p− 1 polynomial.

Upon taking the dot product of equation (4.54) with D(p,v,w) (Lm,j) and integrating

over Ωk, one obtains

∫

Ωk

D(p,v,w)(qDk

)· D(p,v,w) (Lm,j) dΩk −

∫

Ωk

D(p,v,w)(∇uCk

)· D(p,v,w) (Lm,j) dΩk = 0.

(4.55)

Next, one may reformulate equation (4.55) by utilizing an expression which can be

derived from equation (4.15). Consider defining Lm,j ≡ Jk J−1k Lm,j (where m = 1, 2, 3,

L1,j = Lx,j, L2,j = Ly,j, and L3,j = Lz,j), and substituting uCk , Lm,j , uC , and Lm,j in

place of uk, fk, u, and f in equation (4.15) in order to obtain

J−1k Lm,j =

(J−1k

)Lm,j, (4.56)

JTk ∇uCk =

(J−1k

)∇uC . (4.57)

Upon applying the derivative operator D(p,v,w) to equations (4.56) and (4.57), one


obtains

J−1k D(p,v,w) (Lm,j) =

(J−1k

)D(p,v,w)

(Lm,j

), (4.58)

JTk D

(p,v,w)(∇uCk

)=(J−1k

)D(p,v,w)

(∇uC

), (4.59)

where it should be noted that the following identities have been used to simplify

equations (4.58) and (4.59)

D(p,v,w)(J−1k Lm,j

)= J−1

k D(p,v,w) (Lm,j) , (4.60)

D(p,v,w)(JTk ∇uCk

)= JT

k D(p,v,w)

(∇uCk

). (4.61)

Equations (4.60) and (4.61) hold because Ωk is a straight-sided tetrahedron element

and thus Jk is a matrix with constant entries, independent of x, y, and z.

On taking the dot product of equations (4.58) and (4.59) and simplifying the results,

one obtains

D(p,v,w)(∇uCk

)· D(p,v,w) (Lm,j) =

(J−2k

)D(p,v,w)

(∇uC

)· D(p,v,w)

(Lm,j

). (4.62)

On substituting equation (4.62) into equation (4.55), one obtains

∫

Ωk

D(p,v,w)(qDk

)·D(p,v,w) (Lm,j) dΩk−

1

Jk

∫

ΩS

D(p,v,w)(∇uC

)·D(p,v,w)

(Lm,j

)dΩS = 0.

(4.63)

Upon multiplying equation (4.63) by constant coefficients κvw, summing over v and

w, and noting that D(p,v,w)(∇uC

)and D(p,v,w)

(Lm,j

)are constant vectors because

the components of ∇uC and Lm,j are at most degree p polynomials, one obtains the

following

p+1∑

v=1

v∑

w=1

κvw

∫

Ωk

D(p,v,w)(qDk


− VSJk

p+1∑

v=1

v∑

w=1

κvw D(p,v,w)

(∇uC

)· D(p,v,w)

(Lm,j

)= 0. (4.64)

Next, it should be noted that if ∇uC is constructed using VCJH correction fields ψf,l

(where ψf,l ≡ ∇ · gf,l and each gf,l is a VCJH correction function), the following


identity holds

p+1∑

v=1

v∑

w=1

κvw D(p,v,w)

(∇uC

)·D(p,v,w)

(Lm,j

)=

∫

ΓS

uC(Lm,j · n

)dΓS−

∫

ΩS

∇uC·Lm,j dΩS.

(4.65)

On substituting equation (4.65) into equation (4.64) and manipulating the result, one

obtains

1

VS

p+1∑

v=1

v∑

w=1

κvw

∫

Ωk

D(p,v,w)(qDk


− 1

Jk

[∫

ΓS

uC(Lm,j · n

)dΓS −

∫

ΩS

∇uC · Lm,j dΩS

]= 0. (4.66)

Upon substituting equations (4.19) and (4.20) with uCk , Lm,j , uC , and Lm,j in place

of uk, fk, u, and f into equation (4.66), one obtains

1

VS

p+1∑

v=1

v∑

w=1

κvw

∫

Ωk

D(p,v,w)(qDk


−∫

Γk

uCk (Lm,j · n) dΓk +

∫

Ωk

∇uCk · Lm,j dΩk = 0. (4.67)

Setting this equation aside for the moment, consider taking the dot product of equa-

tion (4.38) with Lm,j and integrating over Ωk, in order to obtain

∫

Ωk

qDk · Lm,j dΩk −

∫

Ωk

∇uDk · Lm,j dΩk −∫

Ωk

∇uCk · Lm,j dΩk = 0. (4.68)

On summing equations (4.68) and (4.67) and manipulating the result, one obtains

equation (4.53). This completes the proof of Lemma 4.3.2.

Lemma 4.3.3. Suppose that the VCJH schemes (for which Lemmas 4.3.1 and 4.3.2

hold) are employed to solve the linear advection-diffusion equation for which the flux

f takes the following form

f = fadv + fdif = (au)− (bq) , (4.69)

where a is a constant vector of wave speeds a = (ax, ay, az) = (a1, a2, a3) and b ≥ 0

is a constant coefficient of diffusivity. Then the following reformulation of the VCJH


schemes (in terms of matrices) holds

Mk d

dtUk + SkAUk − bSkQk

=

∫

Γk

[((auDk − bqD

k

)− (au− bq)⋆

)· n]L dΓk, (4.70)

Mk Qk − SkUk = −∫

Γk

(uDk − u⋆

)NL dΓk, (4.71)

where Uk is a vector of solution values, Qk is a vector of auxiliary variable values,

Mk and Mk are positive-definite mass matrices (provided that cvw ≥ 0 and κvw ≥ 0),

Sk and Sk are stiffness matrices, A is a matrix of wave speeds, N is a matrix of face

normals, and L is a vector of nodal basis functions. Note that these matrix and vector

quantities will be defined more precisely in what follows.

Proof. Upon substituting expressions for the discontinuous flux fDk =(auDk − bqD

k

)

and the flux correction fCk = f⋆ − fDk = (au− bq)⋆ −(auDk − bqD

k

)into equa-

tion (4.42), and insisting that the result holds for all j, one obtains

∀j∫

Ωk

∂uDk∂t

ℓj dΩk +1

VS

p+1∑

v=1

v∑

w=1

cvw

∫

Ωk

∂

∂t

(D(p,v,w)


+

∫

Ωk

∇ ·(auDk − bqD

k

)ℓj dΩk =

∫

Γk

((auDk − bqD

k

)− (au− bq)⋆

)· n ℓj dΓk.

(4.72)

Also, on substituting an expression for the solution correction uCk =(u⋆ − uDk

)into

equation (4.53), and insisting that the result hold for all j and m, one obtains

∀j,m∫

Ωk

qDk · Lm,j dΩk +

1

VS

p+1∑

v=1

v∑

w=1

κvw

∫

Ωk

D(p,v,w)(qDk


−∫

Ωk

∇uDk · Lm,j dΩk = −∫

Γk

(uDk − u⋆

)n · Lm,j dΓk. (4.73)

Equations (4.72) and (4.73) can be manipulated and recast into matrix form. Towards

this end, consider modifying equation (4.72) by transforming the second integral on


the left hand side (LHS) from an integral over Ωk into an integral over ΩS, as follows

∀j∫

Ωk

∂uDk∂t

ℓj dΩk +JkVS

p+1∑

v=1

v∑

w=1

cvw

∫

ΩS

∂

∂t

(D(p,v,w)

(uDk (x)

))D(p,v,w) (ℓj (x)) dΩS

+

∫

Ωk

∇ ·(auDk − bqD

k

)ℓj dΩk =

∫

Γk

((auDk − bqD

k

)− (au− bq)⋆

)· n ℓj dΓk,

(4.74)

where the quantities uDk (x) and ℓj (x) are defined via the mapping Θk (x) in equa-

tion (4.13) as follows

uDk (x) ≡ uDk (Θk (x)) = uDk (x) ≡ uDk , (4.75)

ℓj (x) ≡ ℓj (Θk (x)) = ℓj (x) ≡ ℓj. (4.76)

Upon applying the same procedure to the second integral on the LHS of equa-

tion (4.73), one obtains

∀j,m∫

Ωk

qDk · Lm,j dΩk +

JkVS

p+1∑

v=1

v∑

w=1

κvw

∫

ΩS

D(p,v,w)(qDk (x)

)· D(p,v,w) (Lm,j (x)) dΩS

−∫

Ωk

∇uDk · Lm,j dΩk = −∫

Γk

(uDk − u⋆

)n · Lm,j dΓk, (4.77)

where

qDk (x) ≡ qD

k (Θk (x)) = qDk (x) ≡ qD

k , (4.78)

Lm,j (x) ≡ Lm,j (Θk (x)) = Lm,j (x) ≡ Lm,j. (4.79)


Now, equations (4.74) and (4.77) can be recast in matrix form as follows

(Mk +

JkNp

p+1∑

v=1

v∑

w=1

cvw

(D(p,v,w)

)TD(p,v,w)

)d

dtUk +

3∑

m=1

(amS

kmUk − bSk

mQmk

)

=

∫

Γk

[((auDk − bqD

k

)− (au− bq)⋆

)· n]L dΓk, (4.80)

∀m,(Mk +

JkNp

p+1∑

v=1

v∑

w=1

κvw

(D(p,v,w)

)TD(p,v,w)

)Qmk

− SkmUk

= −∫

Γk

(uDk − u⋆

)nm L dΓk, (4.81)

where Mk ∈ RNp×Np is the local mass matrix with entries

[Mk]ij=

∫

Ωk

ℓi (x) ℓj (x) dΩk =

∫

Ωk

ℓi ℓj dΩk, (4.82)

Skm ∈ R

Np×Np is the local stiffness matrix with entries

[Skm

]ij=

∫

Ωk

ℓi (x)∂ℓj (x)

∂xmdΩk =

∫

Ωk

ℓi∂ℓj∂xm

dΩk, (4.83)

D(p,v,w) is the matrix form of the derivative operator D(p,v,w) defined such that

[D(p,v,w)Uk

]i≡ D(p,v,w)

(uDk (x)

) ∣∣∣∣xi

=

[Np∑

=1

(uDk)D(p,v,w) (ℓ (x))

]

xi

, (4.84)

Uk =[(uDk )

1 . . . (uDk )Np]T

is a vector containing the solution values (as mentioned

previously), Qmk=[(qDmk

)1 . . . (qDmk)Np]T

are vectors containing the auxiliary variable

values, and L = L (x) =[ℓ1 (x) . . . ℓNp (x)

]Tis a vector containing the nodal basis

functions.

Equations (4.80) and (4.81) can be simplified by introducing the following block

matrix definitions (in terms of cartesian coordinates x, y, and z)

Mk =

Mk 0 00 Mk 00 0 Mk

, (4.85)


Sk =[Skx Sk

y Skz

], Sk =

Skx

Sky

Skz

, (4.86)

Qk =

Qxk

Qyk

Qzk

, A =

axIayIazI

, N =

nxInyInzI

, (4.87)

where I ∈ RNp×Np. In addition, one may define the following matrices in order to

further simply the notation

Kk =JkNp

p+1∑

v=1

v∑

w=1

cvw

(D(p,v,w)

)TD(p,v,w), (4.88)

Kk =JkNp

p+1∑

v=1

v∑

w=1

κvw

(D(p,v,w)

)TD(p,v,w) 0 0

0(D(p,v,w)

)TD(p,v,w) 0

0 0(D(p,v,w)

)TD(p,v,w)

.

(4.89)

Upon substituting equations (4.85) - (4.89) into equations (4.80) and (4.81), one

obtains

(Mk +Kk

) d


=

∫

Γk

[((auDk − bqD

k

)− (au− bq)⋆

)· n]L dΓk, (4.90)

(Mk +Kk

)Qk − SkUk = −

∫

Γk

(uDk − u⋆

)NL dΓk. (4.91)

One may now define ‘modified mass matrices’ as follows

Mk ≡(Mk +Kk

), (4.92)

Mk ≡(Mk +Kk

), (4.93)

where Mk and Mk are guaranteed to be symmetric positive-definite provided that Kk

and Kk are symmetric positive-semidefinite. Note that if one requires that cvw ≥ 0

and κvw ≥ 0, then Kk and Kk are (in fact) symmetric positive-semidefinite.

On substituting equations (4.92) and (4.93) into equations (4.90) and (4.91), one


obtains equations (4.70) and (4.71). This completes the proof of Lemma 4.3.3.

Theorem 4.3.1. If the VCJH schemes (for which Lemmas 4.3.1 – 4.3.3 hold) are

employed in conjunction with the Lax-Friedrichs formulation [65, 66] for the advective

numerical flux f⋆adv

f⋆adv = (au)⋆ = auD+ λ

2|a · n| [[uD]], (4.94)

for which 0 ≤ λ ≤ 1 is an upwinding parameter, · is an averaging operator, [[·]]is a differencing (jump) operator, and the LDG formulation [53] for the common

numerical solution u⋆ and diffusive numerical flux f⋆dif

u⋆ = uD − β · [[uD]], (4.95)

f⋆dif = (bq)⋆ = bqD+ τ [[uD]] + β b[[qD]], (4.96)

for which β = (βx, βy, βz) = (β1, β2, β3) is a directional parameter and τ ≥ 0 is a

penalty parameter, then it can be shown that the following result holds

N∑

k=1

(d

dt‖Uk‖2

M

)≤ 0. (4.97)

Proof. Consider multiplying equation (4.70) on the left by UTk and equation (4.71)

on the left by QTk in order to obtain

UTk Mk d

dtUk +UT

k SkAUk − bUTk SkQk

=

∫

Γk

[((auDk − bqD

k

)− (au− bq)⋆

)· n]uDk dΓk, (4.98)

QTk MkQk −QT

k SkUk = −∫

Γk

(uDk − u⋆

) (qDk · n

)dΓk, (4.99)

where the fact that uDk = UTk L and qD

k ·n = QTk NL has been used. Equations (4.98)

and (4.99) can be simplified by introducing the following identities

UTk Mk d

dtUk =

1

2

d

dt

(UT

k MkUk

)=

1

2

d

dt‖Uk‖2

M, (4.100)

QTk MkQk = ‖Qk‖2M, (4.101)


where, because Mk and Mk are symmetric positive-definite matrices, ‖Uk‖M and

‖Qk‖M are norms. Upon using equations (4.100) and (4.101) to simplify equa-

tions (4.98) and (4.99), one obtains

1

2

d

dt‖Uk‖2

M+UT

k SkAUk − bUTk SkQk

=

∫

Γk

[((auDk − bqD

k

)− (au− bq)⋆

)· n]uDk dΓk, (4.102)

‖Qk‖2M −QTk SkUk = −

∫

Γk

(uDk − u⋆

) (qDk · n

)dΓk. (4.103)

One may now multiply equation (4.103) by b and add the result to equation (4.102)

in order to obtain

1

2

d

dt‖Uk‖2

M+ b ‖Qk‖2M +UT

k SkAUk − b(UT

k SkQk +QTk SkUk

)

=

∫

Γk

[((auDk − bqD

k

)− (au− bq)⋆

)· n]uDk dΓk − b

∫

Γk

(uDk − u⋆

) (qDk · n

)dΓk.

(4.104)

Equation (4.104) can be simplified by noting that

UTk SkAUk

=

∫

Ωk

uDk ∇ ·(auDk

)dΩk =

1

2

∫

Ωk

∇ ·(a(uDk)2)

dΩk =1

2

∫

Γk

(uDk)2

(a · n) dΓk,

(4.105)

and that

UTk SkQk +QT

k SkUk

=

∫

Ωk

(uDk(∇ · qD

k

)+ qD

k · ∇uDk)dΩk =

∫

Γk

uDk(qDk · n

)dΓk. (4.106)

Upon substituting equations (4.105) and (4.106) into equation (4.104) and rearranging


the result, one obtains

1

2

d

dt‖Uk‖2

M+ b ‖Qk‖2M

=

∫

Γk

[uDk

((auDk2− bqD

k

)− (au− bq)⋆

)+ b u⋆qD

k

]· n dΓk

=

∫

Γk

[uDk

(auDk2− (au)⋆

)− uDk

(bqD

k − (bq)⋆)+ b u⋆qD

k

]· n dΓk, (4.107)

where the fact that (au− bq)⋆ = (au)⋆− (bq)⋆ has been used. Next, upon summing

equation (4.107) over all elements in the mesh, one obtains

1

2

N∑

k=1

(d

dt‖Uk‖2

M

)

=− bN∑

k=1

‖Qk‖2M +

N∑

k=1

∫

Γk

[uDk

(auDk2− (au)⋆

)− uDk

(bqD

k − (bq)⋆)+ b u⋆qD

k

]· n dΓk

.

(4.108)

In order to demonstrate that the VCJH schemes are stable (in accordance with equa-

tion (4.40)), one must show that the right hand side (RHS) of equation (4.108) is

non-positive. The first term on the RHS of equation (4.108) is clearly non-positive,

however, the second term on the RHS has an ambiguous sign, and is only assured to

be non-positive for appropriate formulations of the numerical fluxes (au)⋆ and (bq)⋆,

and the common numerical solution u⋆. The Lax-Friedrichs approach [65, 66] is a

well-known approach for treating the advective numerical flux (au)⋆ and the Cen-

tral Flux (CF) [13, 58], Bassi Rebay 1 (BR1) [58], Bassi Rebay 2 (BR2) [59], Local

Discontinuous Galerkin (LDG) [53], Compact Discontinuous Galerkin (CDG) [56],

and Internal Penalty (IP) [57] approaches are well-known approaches for treating

the diffusive numerical flux (bq)⋆ and the common numerical solution u⋆. For the

sake of brevity, the remainder of this discussion will not consider all the different

possible combinations of advective and diffusive flux formulations. Rather, for illus-

trative purposes, it will be shown that a combination of the Lax-Friedrichs and LDG

flux formulations ensures that the second term on the RHS of equation (4.108) is

non-positive.

Consider the normal component of the Lax-Friedrichs advective numerical flux which


takes the following form

(au)⋆ · n =

(auD+ λ

2|a · n| [[uD]]

)· n, (4.109)

where λ is a directional parameter which yields an upwind biased flux for λ > 0, and

· and [[·]] are average and jump operators defined such that

auD = uDa =1

2

(uDk− + uDk+

)a, [[uD]] = uDk−n− + uDk+n+, (4.110)

where uDk− and n− represent the solution and normal vector which belong to the kth

element and uDk+ and n+ represent the solutions and normal vectors which belong

to adjoining elements. Upon substituting equation (4.110) into equation (4.109) and

noting that n ≡ n−, one obtains

(au)⋆ · n =1

2

(uDk− + uDk+

)(a · n) + λ

2|a · n|

(uDk− − uDk+

). (4.111)

Next, one may consider the LDG common numerical solution which takes the follow-

ing form

u⋆ = uD − β · [[uD]]

=1

2

(uDk− + uDk+

)−(uDk− − uDk+

)(β · n) , (4.112)

and the normal component of the LDG diffusive numerical flux which takes the fol-

lowing form

(bq)⋆ · n =(bqD+ τ [[uD]] + β b[[qD]]

)· n

= b

(1

2

(qDk− + qD

k+

)+(qDk− − qD

k+

)(β · n)

)· n+ τ

(uDk− − uDk+

), (4.113)

where bqD and [[qD]] have been defined such that

bqD = qDb = 1

2

(qDk−

+ qDk+

)b, [[qD]] = qD

k−− qD

k+. (4.114)

In equations (4.112) and (4.113), β is a directional parameter which biases the solution

and the auxiliary variable in opposite directions (if one is biased upwind the other


is biased downwind and vice versa), τ is a penalty parameter which controls jumps

in the solution, qDk−

denotes the auxiliary variable which belongs to the kth element,

and qDk+

denotes the auxiliary variables which belong to adjoining elements.

Upon substituting equations (4.111), (4.112), and (4.113) into equation (4.108) and

replacing uDk and qDk with uDk− and qD

k−, respectively, one obtains an expression of the

form

1

2

N∑

k=1

(d

dt‖Uk‖2

M

)

=− bN∑

k=1

‖Qk‖2M +

N∑

k=1

∫

Γk

[uDk−

(uDk+2

(a · n)− λ

2|a · n|

(uDk− − uDk+

))

− uDk−(bqD

k− · n− b(1

2

(qDk− + qD

k+

)+(qDk− − qD

k+

)(β · n)

)· n− τ

(uDk− − uDk+

))

+ b

(1

2

(uDk− + uDk+

)−(uDk− − uDk+

)(β · n)

)qDk− · n

]dΓk

. (4.115)

One may simplify equation (4.115) by replacing the summation over elements in the

mesh (applied to the second term on the RHS) with a summation over faces (f) in

the mesh. Thereafter, upon assuming periodic boundary conditions and canceling

like terms, one obtains

1

2

N∑

k=1

(d

dt‖Uk‖2

M

)

=− bN∑

k=1

‖Qk‖2M −Nf∑

f=1

∫

Γf

[(λ

2|a · n|+ τ

)(uDf− − uDf+

)2]dΓf

, (4.116)

where Nf is the total number of faces in the mesh and uDf− and uDf+ are the solutions

on opposite sides of face f . Finally, because b ≥ 0, λ ≥ 0, and τ ≥ 0, the terms on

the RHS of equation (4.116) are non-positive and it follows that

1

2

N∑

k=1

(d

dt‖Uk‖2

M

)≤ 0. (4.117)

This completes the proof of Theorem 4.3.1.


Remark. In summary, the stability of a class of FR schemes (the VCJH schemes)

has been proven under the following assumptions:

• The flux is a linear advective-diffusive flux.

• The advection-diffusion problem is discretized on straight-sided tetrahedra Ωk,

for which Jk = const, Jk > 0, and equations (4.14) - (4.20) hold.

• The divergence of the flux correction ∇ · fC in reference space (equation (4.34))

is defined using a VCJH correction field φf,l ≡ ∇·hf,l parameterized by constant

coefficients cvw, such that equation (4.49) holds.

• The gradient of the solution correction ∇uC in reference space (equation (4.27))

is defined using a VCJH correction field ψf,l ≡ ∇·gf,l parameterized by constant

coefficients κvw, such that equation (4.65) holds.

• The coefficients of parameterization are non-negative (i.e. cvw ≥ 0 and κvw ≥ 0)

ensuring that the modified mass matrices Mk and Mk are symmetric positive-

definite and that ‖Uk‖M and ‖Qk‖M are norms.

• The advective numerical flux is computed using the Lax-Friedrichs approach

(equation (4.111)).

• The diffusive numerical flux and common numerical solution are computed using

the LDG approach (equations (4.112) and (4.113)).

Finally, note that stability has been proven for all orders of accuracy p ≥ 0, indepen-

dent of the locations of solution points xi and flux points xf,l.

4.4 Constructing the Energy Stable (VCJH) Cor-

rection Fields

In this section, a procedure will be presented for constructing energy stable (VCJH)

correction fields φf,l and ψf,l that satisfy equations (4.49) and (4.65), respectively.


4.4.1 Preliminaries

Recall that φf,l ≡ ∇ · hf,l and ψf,l ≡ ∇ · gf,l. In accordance with these definitions, it

may appear natural to first construct precise expressions for the correction functions

hf,l and gf,l, and thereafter apply the divergence operator ∇· to these expressions

in order to obtain φf,l and ψf,l. However, this is not the best approach, as hf,l and

gf,l may not be unique. In particular, for p > 1 there are an unlimited number of

correction functions which have the same divergence (or effectively, the same cor-

rection field). In addition, the implementation of the VCJH approach only requires

the precise formulation of the correction fields φf,l and ψf,l, and not of the correction

functions themselves. Specifically, the VCJH approach only requires definitions of the

normal components of the correction functions (hf,l · n and gf,l · n) which are given by

equations (4.28) and (4.29). In what follows, it will be shown that equations (4.28),

(4.29), (4.49), and (4.65) are sufficient for constructing precise definitions of φf,l and

ψf,l.

4.4.2 Constructing the Correction Fields φf,l

One may arrive at a procedure for constructing the correction fields φf,l by manip-

ulating equation (4.49) and utilizing the definition of hf,l · n (equations (4.28) and

(4.29)). Recall from section 4.3, Lemma 4.3.1 that the correction functions hf,l and

correction fields φf,l are required to satisfy equation (4.49), which can be expanded

as follows

p+1∑

v=1

v∑

w=1

cvw D(p,v,w)

(∇ · fC

)D(p,v,w)

(ℓj

)=

∫

ΓS

(fC · n

)ℓj dΓS −

∫

ΩS

(∇ · fC

)ℓj dΩS

=

p+1∑

v=1

v∑

w=1

cvw D(p,v,w)

Nfe∑

f=1

Nfp∑

l=1

[(f⋆f,l − fDf,l

)· nf,l

]φf,l

D(p,v,w)

(ℓj

)

=

∫

ΓS

Nfe∑

f=1

Nfp∑

l=1

[(f⋆f,l − fDf,l

)· nf,l

]hf,l · n

ℓj dΓS

−∫

ΩS

Nfe∑

f=1

Nfp∑

l=1

[(f⋆f,l − fDf,l

)· nf,l

]φf,l

ℓj dΩS. (4.118)


Upon rearranging and simplifying equation (4.118), one obtains the following

p+1∑

v=1

v∑

w=1

cvw D(p,v,w) (φf,l) D

(p,v,w)(ℓj

)=

∫

ΓS

(hf,l · n) ℓj dΓS −∫

ΩS

(φf,l) ℓj dΩS.

(4.119)

Here, each basis function ℓj is equal to the product of Jk and ℓj , each normal com-

ponent of a correction function (hf,l · n) is a polynomial that is uniquely defined by

equations (4.28) and (4.29) (with hf,l in place of gf,l), and each correction field φf,l

remains to be determined. In order to begin the process of computing φf,l, note that

it is a degree p polynomial function which can be expressed as follows

φf,l =

Np∑

=1

σ L (x) , (4.120)

where each σ is a constant coefficient (yet to be computed) and each basis function

L (x) is a member of an orthonormal basis of degree p on the reference element ΩS

defined as follows

L (x) =√8Pu (a)P

(2u+1,0)v

(b)(

1− b)uP (2u+2v+2,0)w (c) (1− c)u+v , (4.121)

= 1 +(11 + 12p+ 3p2) u

6+

(2p+ 3) v

2+ w − (2 + p) u2

2− uv − v2

2+u3

6,

0 ≤ u, 0 ≤ v, 0 ≤ w, u+ v + w ≤ p,

where P(α,β)n is the normalized nth order Jacobi polynomial (as defined in [13]), and

where

a = −2(1 + x)

y + z− 1, b = 2

(1 + y)

1− z − 1, c = z. (4.122)

In addition, note that each function ℓj in equation (4.119) is a degree p polynomial

which can be expressed as follows

ℓj =

Np∑

ı=1

γıLı (x) , (4.123)

where each constant coefficient γı is a known quantity because each function ℓj is a

known quantity.


On substituting equations (4.120) and (4.123) into equation (4.119), one obtains

p+1∑

v=1

v∑

w=1

cvw D(p,v,w)

(Np∑

=1

σ L

)D(p,v,w)

(Np∑

ı=1

γı Lı

)

=

∫

ΓS

(hf,l · n)(

Np∑

ı=1

γıLı

)dΓS −

∫

ΩS

(Np∑

=1

σ L

)(Np∑

ı=1

γı Lı

)dΩS, (4.124)

or equivalently

Np∑

=1

σ

p+1∑

v=1

v∑

w=1

cvw D(p,v,w) (L) D

(p,v,w) (Lı)

=

∫

ΓS

(hf,l · n)Lı dΓS −Np∑

=1

σ

∫

ΩS

L Lı dΩS. (4.125)

Upon noting that Lı and L are orthonormal polynomials, one may derive the follow-

ing expression from equation (4.125)

Np∑

=1

σ

p+1∑

v=1

v∑

w=1

cvw D(p,v,w) (L) D

(p,v,w) (Lı) = −σı +∫

ΓS

(hf,l · n)Lı dΓS. (4.126)

Equation (4.126) holds for ı = 1, . . . , Np and provides a system of Np equations for

the Np unknown coefficients σ. Together, equations (4.126) and (4.120) completely

define φf,l.

4.4.3 Constructing the Correction Fields ψf,l

In following the approach of the previous section, one may arrive at a procedure

for constructing the correction fields ψf,l by manipulating equation (4.65) and utiliz-

ing the definition of gf,l · n (equations (4.28) and (4.29)). Recall from section 4.3,

Lemma 4.3.2 that the correction functions gf,l and correction fields ψf,l are required


to satisfy equation (4.65), which can be expanded as follows

p+1∑

v=1

v∑

w=1

κvw D(p,v,w)

(∇uC

)· D(p,v,w)

(Lm,j

)=

∫

ΓS

uC(Lm,j · n

)dΓS −

∫

ΩS

∇uC · Lm,j dΩS

=

p+1∑

v=1

v∑

w=1

κvw D(p,v,w)

Nfe∑

f=1

Nfp∑

l=1

[u⋆f,l − uDf,l

]nf,l ψf,l

· D(p,v,w)

(Lm,j

)

=

∫

ΓS

Nfe∑

f=1

Nfp∑

l=1

[u⋆f,l − uDf,l

](gf,l · nf,l)

(Lm,j · n

)dΓS

−∫

ΩS

Nfe∑

f=1

Nfp∑

l=1

[u⋆f,l − uDf,l

]nf,l ψf,l

· Lm,j dΩS, (4.127)

where the fact that

uC∣∣∣∣ΓS

=

Nfe∑

f=1

Nfp∑

l=1

[u⋆f,l − uDf,l

](gf,l · nf,l) , (4.128)

has been used.

Upon simplifying equation (4.127), one obtains

p+1∑

v=1

v∑

w=1

κvw D(p,v,w) (ψf,l) · D(p,v,w)

(Lm,j · nf,l

)

=

∫

ΓS

(gf,l · nf,l)(Lm,j · n

)dΓS −

∫

ΩS

ψf,l

(Lm,j · nf,l

)dΩS. (4.129)

Both ψf,l and(Lm,j · nf,l

)are degree p polynomial functions which can be expressed

as follows

ψf,l =

Np∑

=1

ξ L (x) , (4.130)

(Lm,j · nf,l

)=

Np∑

ı=1

ζı Lı (x) . (4.131)

Here the coefficients ξ are unknown because ψf,l is unknown, while the coefficients


ζı are known because each Lm,j is defined in terms of a nodal basis function ℓj.

On substituting equations (4.130) and (4.131) into equation (4.129) and simplifying

the result, one obtains

Np∑

=1

ξ

p+1∑

v=1

v∑

w=1

κvw D(p,v,w) (L) D

(p,v,w) (Lı) = −ξı +∫

ΓS

(gf,l · n)Lı dΓS. (4.132)

Equation (4.132) provides a system of Np equations for the Np unknown coefficients

ξ. Together, equations (4.132) and (4.130) completely define ψf,l.

4.5 Choosing the Parameterizing Coefficients for

the VCJH Schemes

Thus far, there has not been a discussion of the precise approach for selecting the

coefficients cvw and κvw for parameterizing the VCJH correction functions and fields.

In particular, section 4.3 demonstrated that only loose constraints on cvw and κvw are

required to ensure stability, as it was shown that choosing cvw ≥ 0 and κvw ≥ 0 results

in a class of energy stable FR schemes (the VCJH schemes). This section will propose

an additional set of constraints on cvw and κvw based on symmetry considerations,

and thereby provide a more explicit approach for selecting the coefficients.

4.5.1 Symmetry Considerations Pertaining to Derivative Op-

erators

It is important to ensure that the derivative operators used in the formulation of

the VCJH schemes are symmetric. In particular, it is necessary to ensure that the

operators are directionally unbiased so as to avoid introducing artificial asymmetries

into simulations of (potentially) symmetric physical phenomena. Towards this end,

one may impose constraints on the coefficients cvw and κvw in order to ensure that

the operators are symmetric.


4.5.1.1 Preliminaries

In order to begin the process of identifying constraints on the coefficients, one must

first examine the precise form of the derivative operators used in the formulation of

the VCJH schemes. Towards this end, consider the ‘VCJH derivative operators’ of

degree p which take the following form

p+1∑

v=1

v∑

w=1

cvw D(p,v,w) (f) D(p,v,w) (g) , (4.133)

andp+1∑

v=1

v∑

w=1

κvw D(p,v,w) (F) · D(p,v,w) (G) , (4.134)

where f and g are arbitrary, continuous, p times differentiable, scalar-valued functions,

and F and G are arbitrary, continuous, p times differentiable, vector-valued functions.

The derivative operators in equations (4.133) and (4.134) were used in the formulation

of the VCJH schemes, and (more precisely) in the context of equations (4.49) and

(4.65), respectively. In particular, equation (4.49) contains the derivative operator

defined in equation (4.133), with functions f and g replaced by functions ∇· fC and ℓj ,

respectively. Furthermore, equation (4.65) contains the derivative operator defined

in equation (4.134), with functions F and G replaced by functions ∇uC and Lm,j,

respectively.

Next, in order to evaluate the symmetry of the VCJH derivative operators, it is useful

to rewrite them in quadratic form. The quadratic form of a generic derivative operator

D (f) D (g) can be obtained by substituting f in place of g in order to obtain

D (f) D (f) =(D (f)

)2. (4.135)

Similarly, the quadratic form of a generic derivative operator D (F) · D (G) can be

obtained by substituting F in place of G in order to obtain

D (F) · D (F) . (4.136)

In following this procedure, one may substitute f in place of g in equation (4.133)

and F in place of G in equation (4.134), in order to obtain the following quadratic


forms of the VCJH derivative operators

p+1∑

v=1

v∑

w=1

cvw

(D(p,v,w) (f)

)2, (4.137)

andp+1∑

v=1

v∑

w=1

κvw D(p,v,w) (F) · D(p,v,w) (F) . (4.138)

The VCJH operators in equations (4.137) and (4.138) are now in a form that can be

conveniently analyzed. In particular, well-known ‘operator theory’, (a good review

of which appears in [67], [68], and chapter 6 of [69]), can now be used to determine

constraints on the VCJH operators that will ensure their symmetry.

In what follows, operator theory will be used to construct a general, symmetric op-

erator of degree p and thereafter this symmetric operator will be compared to the

operators in equations (4.137) and (4.138) in order to arrive at a set of constraints

on cvw and κvw.

4.5.1.2 Theory for Constructing a Symmetric Derivative Operator

In order to construct a general symmetric derivative operator of degree p, one may

first consider the general quadratic form for a derivative operator of degree p

(∆f)T C (∆f) , (4.139)

where ∆f ∈ Rdp is a vector containing the pth degree derivatives of the function f , d is

the number of spatial dimensions, and C ∈ Rdp×dp is a symmetric matrix containing

at most dp(dp + 1)/2 unique entries. For d = 3 and p = 2, ∆f takes the following

form

∆f =

[∂2f

∂x2,∂2f

∂y2,∂2f

∂z2,∂2f

∂x∂y,∂2f

∂y∂x,∂2f

∂x∂z,∂2f

∂z∂x,∂2f

∂y∂z,∂2f

∂z∂y

]T, (4.140)

where it is important to note that although derivatives such as ∂2f∂x∂y

and ∂2f∂y∂x

are

identical for continuous functions, for now they will be treated distinctly in order to

simplify the analysis (as recommended in [67]).


According to operator theory, a derivative operator is defined to be ‘symmetric’ if it is

invariant under orthogonal transformations of the space, i.e. rotations and reflections

of the coordinate system [69, 67]. The operator in equation (4.139) is invariant under

orthogonal transformations if and only if it satisfies the following condition

(R∆f)T C (R∆f) = (∆f)T C (∆f) , (4.141)

where R ∈ Rdp×dp is an arbitrary orthogonal matrix. Upon manipulating equa-

tion (4.141) and noting that RT = R−1, one obtains the following equivalent condi-

tion

RTC = CRT . (4.142)

An obvious choice of C which satisfies equation (4.142) is C = cI where c is an

arbitrary scalar, and I is the identity matrix in Rdp×dp. Upon substituting this choice

of C into equation (4.139), one obtains the following derivative operator

c (∆f)T (∆f) . (4.143)

Next, one may reformulate equation (4.143) by expanding the inner product (∆f)T (∆f)

and then combining all resulting non-distinct derivative terms (such as the terms

c(

∂2f∂x∂y

)2and c

(∂2f∂y∂x

)2which arise in equation (4.143) when d = 3 and p = 2). For

the case of d = 3 and arbitrary p, there are a total of

(p

v − 1

)(v − 1

w − 1

), (4.144)

non-distinct terms associated with the pth degree derivative term that takes the form

c

(∂pf

∂x(p−v+1)∂y(v−w)∂z(w−1)

)2

, (4.145)

where v = 1, . . . , p and w = 1, . . . , v. It is interesting to note that equation (4.144)

contains an expression for the pth degree trinomial coefficients (the coefficients on the

pth degree terms in the expansion of (x+ y + z)p).

On combining all non-distinct terms for the operator in equation (4.143), for the case


of d = 3, one obtains

c (∆f)T (∆f) = c

p+1∑

v=1

v∑

w=1

(p

v − 1

)(v − 1

w − 1

)(∂pf

∂x(p−v+1)∂y(v−w)∂z(w−1)

)2

. (4.146)


c

p+1∑

v=1

v∑

w=1

(p

v − 1

)(v − 1

w − 1

)(D(p,v,w) (f)

)2. (4.147)

Equation (4.147) contains a final, convenient formulation for a pth degree derivative

operator that is invariant under orthogonal transformations in 3D.

4.5.2 Defining the Coefficients

In order to obtain appropriate definitions for the coefficients cvw and κvw, one may

compare the derivative operator in equation (4.147) to the derivative operators in

equations (4.137) and (4.138). Upon comparing equations (4.137) and (4.147), one

obtains the following expression for each coefficient cvw

cvw = c

(p

v − 1

)(v − 1

w − 1

), (4.148)

where in order to ensure that cvw ≥ 0, one requires that c ≥ 0.

Next, in order to define each coefficient κvw, one must first substitute the component-

wise definition of F (i.e. F = (Fx,Fy,Fz) = (F1,F2,F3)) into equation (4.138), as

follows

p+1∑

v=1

v∑

w=1

κvwD(p,v,w) (F) · D(p,v,w) (F)

=

p+1∑

v=1

v∑

w=1

κvw

[(D(p,v,w) (F1)

)2+(D(p,v,w) (F2)

)2+(D(p,v,w) (F3)

)2]. (4.149)

Next, setting equation (4.149) aside for the moment, consider modifying equation (4.147)

by replacing c with an arbitrary coefficient κ and replacing f with the mth component


of F as follows

κ

p+1∑

v=1

v∑

w=1

(p

v − 1

)(v − 1

w − 1

)(D(p,v,w) (Fm)

)2, m = 1, 2, 3. (4.150)

On comparing equation (4.150) to equation (4.149), one obtains the following expres-

sion for each coefficient κvw

κvw = κ

(p

v − 1

)(v − 1

w − 1

), (4.151)

where in order to ensure that κvw ≥ 0, one requires that κ ≥ 0.

In summary, if the coefficients cvw and κvw are chosen in accordance with equa-

tions (4.148) and (4.151), one obtains derivative operators for each scheme which

preserve physical symmetry in the sense that they are invariant under orthogonal

transformations of the coordinate system. Furthermore, the resulting ‘symmetry-

preserving’ schemes are straightforwardly defined in terms of two parameters, c and

κ, and are guaranteed to be stable for all choices of c ≥ 0 and κ ≥ 0.

4.6 Visualizing the Symmetric VCJH Correction

Fields

The symmetry of the VCJH derivative operators ensures the symmetry of the associ-

ated VCJH correction fields φf,l and ψf,l. Observe that the VCJH derivative operators

in equations (4.133) and (4.134) appear in the equations for the coefficients of the

VCJH correction fields (equations (4.126) and (4.132)). As a result, if the VCJH

operators (and hence the VCJH correction fields) are parameterized in accordance

with equations (4.148) and (4.151), the VCJH correction fields are symmetric in the

following sense: the VCJH correction fields are invariant under affine transformations

of the tetrahedron unto itself. This can be clearly demonstrated via an examination

of the plots of the correction fields φf,l (in Figures (4.6) – (4.8)) for the case in which

p = 2, cvw is given by equation (4.148), and c = c+ = 3.07× 10−2. It should be noted

that, for a particular time-stepping scheme, the value of c+ is an ‘optimal’ value of c,

the significance of which will be discussed in section 4.9.


f=1, l=1 f=1, l=2

Figure 4.6: Plots of the VCJH correction fields φf,l for the case of p = 2, f = 1, andc = c+ = 3.07 × 10−2.

f=1, l=3 f=1, l=4



f=1, l=5 f=1, l=6


Figures (4.6) – (4.8) clearly show that the symmetries of the VCJH correction fields

are consistent with the inherent symmetries of the reference tetrahedron.

4.7 Equivalence of VCJH Schemes and Certain Fil-

tered DG Schemes

In this section, it will be shown that, for all choices of c ≥ 0 and κ ≥ 0, the class

of VCJH schemes is equivalent to a class of filtered, collocation-based, nodal DG

schemes. The equivalence of VCJH schemes with certain filtered DG schemes has

been shown for 1D linear advection problems by Allaneau and Jameson [70]. In what

follows, this result will be extended to the case of 3D advection-diffusion problems on

tetrahedral elements.

4.7.1 Preliminaries

In order to begin demonstrating the equivalence of the VCJH schemes and certain

collocation-based, nodal DG schemes, one must first examine the formulation of the

unfiltered, collocation-based, nodal DG scheme. Towards this end, consider substitut-


ing the definition of the advective-diffusive flux fDk = auDk − bqDk into the standard

first-order system of equations (equations (4.9) and (4.10)) as follows

∂uDk∂t

+∇ ·(auDk

)−∇ ·

(bqD

k

)= 0, (4.152)

qDk −∇uDk = 0. (4.153)

Next, in keeping with the DG approach, the left hand sides of equations (4.152) and

(4.153) are required to be orthogonal to the test functions ℓj and Lm,j as follows

∀j∫

Ωk

(∂uDk∂t

+∇ ·(auDk

)−∇ ·

(bqD

k

))ℓj dΩk = 0, (4.154)

∀j,m∫

Ωk

(qDk −∇uDk

)· Lm,j dΩk = 0. (4.155)

Upon employing integration-by-parts in equations (4.154) and (4.155) and rearranging

the results, one obtains

∀j∫

Ωk

(∂uDk∂t

ℓj −(auDk − bqD

k

)· ∇ℓj

)dΩk = −

∫

Γk

(auDk − bqD

k

)ℓj · n dΓk,

(4.156)

∀j,m∫

Ωk

(qDk · Lm,j + uDk (∇ · Lm,j)

)dΩk =

∫

Γk

uDk (Lm,j · n) dΓk.

(4.157)

Upon replacing the flux on the right hand side of equation (4.156) with the numerical

flux (au− bq)⋆, replacing the solution on the right hand side of equation (4.157)

with the numerical solution u⋆, employing integration by parts on the left hand sides

of equations (4.156) and (4.157), and rearranging the results, one obtains

∀j∫

Ωk

∂uDk∂t

ℓj dΩk +

∫

Ωk

∇ ·(auDk − bqD

k

)ℓj dΩk

=

∫

Γk

((auDk − bqD

k

)− (au− bq)⋆

)· n ℓj dΓk, (4.158)

∀j,m∫

Ωk

qDk · Lm,j dΩk −

∫

Ωk

∇uDk · Lm,j dΩk

= −∫

Γk

(uDk − u⋆

)n · Lm,j dΓk, (4.159)


Equations (4.158) and (4.159) represent the so-called ‘strong’ formulation of the DG

method. The strong formulation can be recast in matrix form using the matrices

defined by equations (4.82), (4.83), and (4.85) - (4.87) as follows

Mk d


=

∫

Γk

[((auDk − bqD

k

)− (au− bq)⋆

)· n]L dΓk, (4.160)


Γk

(uDk − u⋆

)NL dΓk. (4.161)

Now, consider simplifying equations (4.160) and (4.161) by introducing the following

abbreviations

RHSDG1 ≡∫

Γk

[((auDk − bqD

k

)− (au− bq)⋆

)· n]L dΓk, (4.162)

RHSDG2 ≡ −∫

Γk

(uDk − u⋆

)NL dΓk. (4.163)

Upon substituting equations (4.162) and (4.163) into equations (4.160) and (4.161)

and rearranging the result, one obtains

Mk d

dtUk = −SkAUk + bSkQk +RHSDG1, (4.164)

MkQk = SkUk +RHSDG2. (4.165)

On multiplying equation (4.164) by(Mk)−1

and equation (4.165) by(Mk

)−1, one

obtains the DG residual RDG and auxiliary variable QDG

RDG ≡d

dtUk =

(Mk)−1 (−SkAUk + bSkQk +RHSDG1

), (4.166)

QDG ≡ Qk =(Mk

)−1 (SkUk +RHSDG2

). (4.167)

Equations (4.166) and (4.167) will serve as a convenient, simplified formulation of

the collocation-based nodal DG scheme. In particular, the simplified DG formulation

in equations (4.166) and (4.167) will be compared to a simplified formulation of the

VCJH schemes (which will be defined in the next section), in order to provide insight

into the relationship between the schemes.


4.7.2 Simplified Formulation of VCJH Schemes and Recov-

ery of Filtered DG Schemes

In order to obtain a simplified formulation of the VCJH schemes, consider substituting

equations (4.162) and (4.163) into equations (4.70) and (4.71) and rearranging the

result in order to obtain

Mk d

dtUk = −SkAUk + bSkQk +RHSDG1, (4.168)

MkQk = SkUk +RHSDG2. (4.169)

On multiplying equation (4.168) by(Mk)−1

and equation (4.169) by(Mk

)−1

, one

obtains the VCJH residual RV CJH and auxiliary variable QV CJH

RV CJH ≡d

dtUk =

(Mk)−1 (

−SkAUk + bSkQk +RHSDG1

), (4.170)

QV CJH ≡ Qk =(Mk

)−1 (SkUk +RHSDG2

). (4.171)

Upon comparing RV CJH in equation (4.170) to RDG in equation (4.166) and QV CJH

in equation (4.171) to QDG in equation (4.167), one observes that the mass matrices

of the VCJH scheme and the unfiltered, collocation-based, nodal DG scheme appear

to be different. However, if RDG is multiplied by the invertible filter matrix F1 and

QDG is multiplied by the invertible filter matrix F2, one obtains the following filtered,

collocation-based, nodal DG approach

F1 RDG = F1

(Mk)−1 (−SkAUk + bSkQk +RHSDG1

), (4.172)

F2QDG = F2

(Mk

)−1 (SkUk +RHSDG2

), (4.173)

which is identical to the VCJH approach in equations (4.170) and (4.171) provided

that the following equations hold

F1

(Mk)−1

=(Mk)−1

=(Mk +Kk

)−1, (4.174)

F2

(Mk

)−1=(Mk

)−1

=(Mk +Kk

)−1, (4.175)


or (equivalently) the filters take the following form

F1 =(I+

(Mk)−1

Kk)−1

, F2 =(I +

(Mk

)−1Kk)−1

, (4.176)

where I is the identity matrix in R3Np×3Np. Note that the filters F1 and F2 exist and

are invertible as long as Kk and Kk are symmetric, positive-semidefinite matrices,

which is guaranteed to be the case for all VCJH schemes (for c ≥ 0 and κ ≥ 0). If

c 6= 0 or κ 6= 0, then Kk or Kk contains nontrivial entries, and one recovers a filtered

DG approach. Conversely, if c = 0 and κ = 0, then Kk and Kk vanish, and one

recovers the unfiltered DG approach.

The process of forming filters F1 and F2 is described in detail in the next section.

The general form of the filters is discussed, and it is shown that the filters act on the

highest order polynomial terms (pth degree terms) of the orthonormal basis function

expansions of the residual RDG and the auxiliary variable QDG. In section 4.9,

numerical experiments will show that these filters are effective in damping spurious

oscillations in the highest order polynomial terms, leading to larger explicit time-step

limits for certain VCJH schemes.

In summary, it has been shown that the VCJH schemes for linear advection-diffusion

problems on tetrahedra are equivalent to a class of collocation-based, nodal DG

schemes that impose filters F1 and F2 (equation (4.176)) on the residual RDG and

the auxiliary variable QDG. As the VCJH approach (from sections 4.2 – 4.5) and the

filtered DG approach (from this section) are mathematically identical, the reader can

implement whichever approach is more convenient for their particular application and

software architecture. In particular, one may easily implement the proposed filtering

approach within an existing DG implementation. This enables one to employ the

VCJH schemes without having to construct the associated correction fields (φf,l and

ψf,l).

4.8 Constructing the Energy Stable (VCJH) Filter

Matrices

This section discusses the procedure for forming the energy stable (VCJH) filter

matrices F1 and F2 and examines the resulting structure of the filter matrices.


4.8.1 Procedure for Forming the Filter Matrices

The filters F1 and F2 are obtained from the matrices Mk,Mk, Kk, and Kk via equa-

tion (4.176). The mass matrices Mk andMk can be constructed from inner products

of the nodal basis functions ℓj (x) in accordance with equations (4.82) and (4.85).

However, it is often difficult to compute the nodal basis functions ℓj (x) directly, and

one is frequently better served by using the orthonormal basis functions Lj (x) in

their place. In particular, one may use the orthonormal basis functions to define a

‘Vandermonde matrix’ Vk which has the following entries

[Vk]ij = Lj (xi) , (4.177)

and next, one may use Vk to compute the mass matrices as follows

Mk = Jk

(Vk(Vk)T)−1

, (4.178)

Mk = Jk

(Vk(Vk)T)−1

0 0

0(Vk(Vk)T)−1

0

0 0(Vk(Vk)T)−1

. (4.179)

Note that the derivation of equations (4.178) and (4.179) is discussed in detail in [13].

In a similar fashion, one may construct expressions for Kk and Kk in terms of the

orthonormal basis functions. Recall that equations (4.88) and (4.89) provide expres-

sions for Kk and Kk in terms of the matrices(D(p,v,w)

)TD(p,v,w), where each D(p,v,w)

is a matrix which has the following entries

[D(p,v,w)]ij = D(p,v,w) (ℓj (x))

∣∣∣∣xi

. (4.180)

One may use the orthonormal basis functions Lj (x) to construct similar matricesˆD(p,v,w) which have the following entries

[ˆD(p,v,w)]ij = D(p,v,w) (Lj (x))

∣∣∣∣xi

. (4.181)


One may then relate eachˆD(p,v,w) to each D(p,v,w) by using the following identity

D(p,v,w) =ˆD(p,v,w)

(Vk)−1

. (4.182)

Note that the derivation of equation (4.182) is discussed in detail in [13].

Upon substituting equation (4.182) into equations (4.88) and (4.89), one obtains the

following expressions for Kk and Kk

Kk =JkNp

p+1∑

v=1

v∑

w=1

cvw(Vk)−T

(ˆD(p,v,w)

)T ˆD(p,v,w)

(Vk)−1

, (4.183)

Kk =JkNp

p+1∑

v=1

v∑

w=1

κvw

(Vk)−T

(ˆD(p,v,w)

)T ˆD(p,v,w)

(Vk)−1

0 0

0(Vk)−T

(ˆD(p,v,w)

)T ˆD(p,v,w)

(Vk)−1

0

0 0(Vk)−T

(ˆD(p,v,w)

)T ˆD(p,v,w)

(Vk)−1

.

(4.184)

Next, on substituting equations (4.183), (4.184), (4.178), and (4.179) into equa-

tion (4.176), one obtains the following expressions for F1 and F2

F1 =

(I+

1

Np

p+1∑

v=1

v∑

w=1

cvw Vk(ˆD(p,v,w)

)T ˆD(p,v,w)

(Vk)−1

)−1

= Vk ˆF1

(Vk)−1

, (4.185)

where

ˆF1 =

(I+

1

Np

p+1∑

v=1

v∑

w=1

cvw

(ˆD(p,v,w)

)T ˆD(p,v,w)

)−1

, (4.186)

and

F2 =

I +

p+1∑

v=1

v∑

w=1

κvwNp

Vk(ˆD(p,v,w)

)T ˆD(p,v,w)

(Vk)−1

0 0

0 Vk(ˆD(p,v,w)

)T ˆD(p,v,w)

(Vk)−1

0

0 0 Vk(ˆD(p,v,w)

)T ˆD(p,v,w)

(Vk)−1

−1

= Vk ˆF2

(Vk)−1

, (4.187)


where

ˆF2 =

I + 1

Np

p+1∑

v=1

v∑

w=1

κvw

(ˆD(p,v,w)

)T ˆD(p,v,w) 0 0

0(ˆD(p,v,w)

)T ˆD(p,v,w) 0

0 0(ˆD(p,v,w)

)T ˆD(p,v,w)

−1

,

(4.188)

and

Vk =

Vk 0 00 Vk 00 0 Vk

. (4.189)

Note that equations (4.186) and (4.188) define new filtering matricesˆF1 and

ˆF2. These

matrices can be viewed as filters which act on the orthonormal basis (in contrast to

F1 and F2 which can be viewed as filters which act on the nodal basis). Conveniently,

the two sets of filters are related via left and right multiplication by the Vandermonde

matrix and its inverse (as shown in equations (4.185) and (4.187)).

Now, having established a method for constructing F1 and F2 using the orthonormal

basis (viaˆF1 and

ˆF2), one may obtain insights into the overall effects of the filtering

process by examining the sparsity patterns ofˆF1 and

ˆF2.

4.8.2 Sparsity Patterns of the Filter Matrices

On evaluating equation (4.186), one finds thatˆF1 has the following block structure

ˆF1 =

[IB1 00 FB

1

], (4.190)

where IB1 ∈ RNℓ

p×Nℓp is an identity matrix, FB

1 ∈ RNu

p ×Nup is a dense matrix of filtering

coefficients, N ℓp = Np − Nu

p is the number of orthonormal basis functions of degree

≤ (p− 1), and Nup = 1

2(p+ 1) (p+ 2) is the number of orthonormal basis functions

of degree p. The structure ofˆF1 ensures that only the degree p polynomial terms

(associated with the degree p orthonormal basis functions) are effected by the filtering

matrix. All terms of degree ≤ (p− 1) are multiplied by the identity matrix and remain

unaffected.


Similarly, on evaluating equation (4.188), one finds thatˆF2 has the following structure

ˆF2 =

IB2 0 · · · · · · · · · 0

0 FB2

. . ....

.... . . IB2 0

...... 0 FB

2. . .

......

. . . IB2 00 · · · · · · · · · 0 FB

2

, (4.191)

where IB2 ∈ RNℓ

p×Nℓp is an identity matrix and FB

2 ∈ RNu

p ×Nup is a dense matrix of

filtering coefficients. The structure ofˆF2 is similar to the structure of

ˆF1, as the

structure ofˆF2 also ensures that only the degree p polynomial terms are effected by

the filtering matrix.

In summary, the filters F1 and F2 are related (via the Vandermonde matrix) to the

filtersˆF1 and

ˆF2 which act on the orthonormal basis functions Lj(x), and effect only

the highest (degree p) polynomial terms of the residual and the auxiliary variable,

respectively.

4.9 Maximizing the Explicit Time-Step Limits of

the VCJH Schemes for Linear Advection Prob-

lems

The parameters c and κ have an effect on the explicit time-step limit and the accuracy

of a VCJH scheme. Williams et al. ([64] and Chapter 3 of this thesis) have shown

that choosing κ = c and 0 ≤ c ≤ c+ yields stable and accurate results for linear

advection-diffusion problems on triangles, where c+ is a value of c that maximizes

the explicit time-step limit for linear advection problems. Values of c > c+ were still

found to be stable, but resulted in a reduction in the explicit time-step limits and

the accuracy of the schemes. In light of these results, it is desirable to determine

values of c+ for linear advection problems on tetrahedra. In what follows, classical

von Neumann analysis (as outlined in [52, 71, 72, 73]) will be employed to determine

the value of c = c+ that maximizes the explicit time-step limit of the VCJH schemes

for a 3D, canonical, linear advection problem.


4.9.1 Preliminaries

Consider the following 3D linear advection equation

∂u

∂t+∇ · (au) = 0, (4.192)

where the velocity vector (a) of propagating waves has magnitude am and takes the

following form

a = am

cos ϑ sinϕsinϑ sinϕ

cosϕ

, (4.193)

and where ϑ ∈ [0, 2π] and ϕ ∈ [0, π] specify the orientation of propagating waves.

A solution to this equation can be obtained within a cubic domain Ω on which

periodic boundary conditions are imposed. In particular, under these conditions,

equation (4.192) has a solution of the form

u (x, t) = u exp [J (k · x− ωt)] , (4.194)

where u is an arbitrary constant, J =√−1, ω is a frequency, and k is a wave number

with magnitude km which takes the following form

k = km

cos ϑ sinϕsinϑ sinϕ

cosϕ

. (4.195)

Note that, according to the exact dispersion relation, ω ≡ a · k [73].

The solution in equation (4.194) can be partitioned into temporal and spatial parts

as follows

u (x, t) = u exp [−J ωt] exp [J k · x] = u (t) exp [J k · x] . (4.196)

One may now construct an approximation to the solution in equation (4.196). To-

wards this end, consider dividing the domain Ω into a regular grid of cubic elements,

each with edge length ∆X . Next, in order to obtain a regular grid of tetrahedral

elements, one may subdivide each cubic element into 6 tetrahedra. An example of

the tetrahedral division of the (i, j, k)th cubic element is shown in Figure (4.9).


(i,j,k) (i+1,j,k)(i-1,j,k)

(i,j,k-1)(i-1,j,k-1) (i+1,j,k-1)

(i,j,k+1)(i-1,j,k+1) (i+1,j,k+1)

(…,j+1,…)(…,j+1,...) (…,j+1,…)

(…,j+2,…)(…,j+2,…) (…,j+2,…)

(a) Regular cubic grid.

(i,j,k)

∆X

(b) Splitting of the (i, j, k)th cube into 6 tetrahedra.

Figure 4.9: Regular grid of tetrahedral elements used in the von Neumann analysis.

One may approximate the solution within the 6 tetrahedra within the (i, j, k)th cubic

element as follows

Ui,j,k = U exp [J km∆X (i cosϑ sinϕ+ j sinϑ sinϕ+ k cosϕ)] , (4.197)


where Ui,j,k ∈ R6Np and U = U (t) ∈ R

6Np.

The evolution in time of Ui,j,k is governed by the following equation

d

dt[Ui,j,k] =

am∆X [BUi,j,k +CUi−1,j,k +DUi+1,j,k

+ EUi,j−1,k + FUi,j+1,k +GUi,j,k−1 +HUi,j,k+1], (4.198)

where B, C, D, E, F, G, and H are matrices in R6Np×6Np which relate the time rate

of change of the solution in the (i, j, k)th element to itself and to the solutions in

neighboring elements. In general, the matrices are a function of the wave speed am,

the wave propagation angles ϑ and ϕ, the geometry of the tetrahedron elements (the

Jacobians Jk and face normal vectors n), the parameter λ in the Lax-Friedrichs flux

formulation, and the parameter c in the VCJH scheme. Note that the precise form

of the matrices is omitted for the sake of brevity.

Upon substituting equation (4.197) into equation (4.198) one obtains

d

dtU = WU, (4.199)

where

W =am∆X

(B+C exp [−J km∆X cosϑ sinϕ] +D exp [J km∆X cosϑ sinϕ]

+ E exp [−J km∆X sin ϑ sinϕ] + F exp [J km∆X sinϑ sinϕ]

+G exp [−J km∆X cosϕ] +H exp [J km∆X cosϕ]

). (4.200)

One may introduce non-dimensional quantities into equation (4.199) by multiplying

through by ∆X /am and defining k′m ≡ km∆X and t′ ≡ t∆X /am. Upon performing

these steps, one obtainsd

dt′U = W′U, (4.201)

where W′ is the non-dimensional form of W.

Upon applying an explicit time-stepping scheme (such as the classical 4 stage, 4th-

order Runge-Kutta scheme) to equation (4.201), one can obtain the following, fully


discrete equation

Un+1 = TUn, (4.202)

where Un+1 and Un are the solutions at time levels n + 1 and n, and where T is

a matrix that depends on the non-dimensional time-step ∆t′, the non-dimensional

wave number k′m, the wave propagation angles ϑ and ϕ, the parameter λ in the Lax-

Friedrichs flux formulation, and the parameter c in the VCJH scheme. According to

classical von Neumann stability analysis, equation (4.202) is stable provided that the

spectral radius ρ (T) ≤ 1. The largest time-step ∆t′ for which ρ (T) ≤ 1 is referred

to as the ‘explicit time-step limit’ ∆t′lim.

4.9.2 Results of Numerical Analysis

Numerical experiments were performed in order to determine the values of c+ (the val-

ues of c that coincided with the largest values of ∆t′lim) for different polynomial orders

and different time-stepping schemes. In particular, equation (4.202) was evaluated for

p = 2 to p = 5 for the classical 3 stage, 3rd-order Runge-Kutta scheme (RK33), the

classical 4 stage, 4th-order Runge-Kutta scheme (RK44), and the more recently devel-

oped, strong stability preserving, 5 stage, 4th-order Runge-Kutta scheme (RK54) [16].

For the RK33, RK44, and RK54 schemes, the matrix T in equation (4.202) took the

following forms

T = I+∆t′W′ +(∆t′)2

2(W′)

2+

(∆t′)3

6(W′)

3, (4.203)

T = I+∆t′W′ +(∆t′)2

2(W′)

2+

(∆t′)3

6(W′)

3+

(∆t′)4

24(W′)

4, (4.204)

T = I+∆t′W′ +(∆t′)2

2(W′)

2+

(∆t′)3

6(W′)

3+

(∆t′)4

24(W′)

4+

(∆t′)5

200(W′)

5.

(4.205)

For each explicit time-stepping scheme, time-step limits ∆t′lim were determined for

values of c ∈ [10−12, 102], for the Lax-Friedrichs flux with λ = 1. A conservative

estimate for each explicit time-step limit ∆t′lim was determined by finding the di-

mensionless wave number k′m ∈ [0, 2π] and wave orientation angles ϑ ∈ [0, 2π] and

ϕ ∈ [0, π] that resulted in the smallest value of ∆t′lim. In Figure (4.10), these values

of ∆t′lim are plotted vs. values of c for p = 2 to p = 5 and for the RK33, RK44, and


RK54 time-stepping schemes.

10−6

10−4

10−2

100

102

0.1

0.15

0.2

0.25

0.3

0.35

c

∆t’ lim

RK33RK44RK54

(a) p=2

10−8

10−6

10−4

10−2

100

0.07

0.1

0.13

0.16

0.19

0.22

c∆t

’ lim

RK33RK44RK54

(b) p=3

10−10

10−8

10−6

10−4

10−2

0.05

0.07

0.09

0.11

0.13

0.15

c

∆t’ lim

RK33RK44RK54

(c) p=4

10−12

10−10

10−8

10−6

10−4

0.04

0.055

0.07

0.085

0.1

0.115

c

∆t’ lim

RK33RK44RK54

(d) p=5

Figure 4.10: Plots of explicit time-step limits ∆t′lim vs. c obtained from the von Neumannanalysis.

The optimal values of c (the values of c+) for each polynomial order and explicit

time-stepping scheme are summarized in Table (4.1).


p=2 p=3 p=4 p=5

RK33 2.22e-02 6.53e-04 1.18e-05 1.25e-07RK44 2.99e-02 7.33e-04 1.23e-05 1.31e-07RK54 3.07e-02 5.45e-04 9.92e-06 1.10e-07

Table 4.1: Values of c+ for the 3-stage, 3rd-order RK scheme (RK33), the 4-stage, 4th-orderRK scheme (RK44), and a 5-stage, 4th-order RK scheme (RK54) [16].

The best results are obtained for the RK54 time-stepping scheme, for p = 2 and

c = c+ = 3.07×10−2. In this case, the time-step limit is 1.81x larger than that of the

collocation-based nodal DG scheme (for which c = 0).

4.10 Numerical Experiments on the Linear Advection-

Diffusion Equation

In an effort to further evaluate the performance of the VCJH schemes, they were

employed to solve a model linear advection-diffusion problem, and their orders of

accuracy and explicit time-step limits were computed.

The governing equation for the model linear advection-diffusion problem took the

following form∂u

∂t+∇ · (au) = b∆u, (4.206)

where the wave velocity vector a and the diffusion coefficient b were given the following

values: a = (ax, ay, az) =(1/2, 1/2,

√2/2)and b = 0.01. Approximate solutions to

equation (4.206) were sought within the domain Ω = [−1, 1]× [−1, 1]× [−1, 1], withperiodic boundary conditions imposed. At time t = 0, the solution was initialized to

u = sin (πx) sin (πy) sin (πz) throughout Ω, and thereafter, the solution was marched

forward in time to t = 1 using the explicit RK54 time-stepping scheme. At each

time-step, the advective numerical fluxes were computed using the Lax-Friedrichs

approach with λ = 1 and the common solutions and diffusive numerical fluxes were

computed using the LDG approach with τ = 1 and β = ±0.5n−.

The numerical experiments (described above) were used to evaluate the accuracy and

stability properties of two ‘representative’ VCJH schemes: the collocation-based nodal

DG scheme with c = 0 and κ = 0, and the scheme with c = c+ and κ = κ+ = c+.

The orders of accuracy and explicit time-step limits of these two VCJH schemes were


computed as follows:

1. The orders of accuracy were computed for p = 2 to p = 5 on a sequence of

regular tetrahedral grids. These grids were obtained using the approach from

section 4.9, i.e. by dividing the domain Ω into N3 cubic elements of equal size,

and then dividing each of these elements into 6 tetrahedral elements of equal

size, resulting in grids with a total of N = 6N3 tetrahedral elements. Grids with

N = 4, 6, 8, 12, 16, 24, 32, 48, and 64 were created, and on these grids, the

accuracy of the approximate solution was evaluated at t = 1 using the L2 norm

of the error between the approximate solution and the exact solution, where the

exact solution took the following form

uexact = exp(−3 b π2t

)sin (π (x− axt)) sin (π (y − ayt)) sin (π (z − azt)) .

The approximate solution was expected to converge towards the exact solution

at a rate of hp+1, where h was the mesh spacing.

In a similar fashion, the accuracy of the solution gradient was evaluated at t = 1

using the L2 norm of the error between the approximate solution gradient and

the exact solution gradient. In this case, the approximate solution gradient was

expected to converge towards the exact solution gradient at a rate of hp.

Note that, in each of the numerical experiments used to determine the rates of

convergence of the solution and its gradient, the time-steps were chosen small

enough to avoid effecting the spatial accuracy of the schemes.

Finally, note that due to the high accuracy of the schemes with p = 4 and

p = 5, reasonable results were obtained by computing the error on the coarser

meshes with N = 4, 6, 8, 12, 16, and 24 while neglecting the finest meshes with

N = 32, 48, and 64.

2. The explicit time-step limits were computed for p = 2 and p = 3 on the grid

with N = 64, and for p = 4 and p = 5 on the grid with N = 24. In each case,

the maximum time-step limit was taken to be the maximum time-step for which

the solution remained bounded at t = 1000∆t (i.e. boundedness was evaluated

after the first 1000 time-steps). Using this criterion, each maximum time-step

limit was determined via an iterative process, which computed the maximum


time-step to within an estimated tolerance of 1%.

The orders of accuracy and explicit time-step limits of the VCJH schemes are pre-

sented in Tables (4.2) and (4.3).

cdg, κdg c+, κ+

p N L2 u error O(L2) L2 ∇u error O(L2) L2 u error O(L2) L2 ∇u error O(L2)

2 4 6.29e-02 1.16e+00 2.96e-01 2.47e+006 1.84e-02 3.04 5.46e-01 1.86 9.95e-02 2.69 1.31e+00 1.588 7.92e-03 2.92 3.07e-01 2.00 4.18e-02 3.02 8.06e-01 1.6812 2.40e-03 2.94 1.31e-01 2.10 1.19e-02 3.10 3.88e-01 1.8016 1.00e-03 3.03 6.97e-02 2.20 4.79e-03 3.16 2.21e-01 1.9724 2.87e-04 3.09 2.83e-02 2.22 1.27e-03 3.27 9.01e-02 2.2132 1.18e-04 3.10 1.50e-02 2.20 4.78e-04 3.40 4.44e-02 2.4648 3.36e-05 3.09 6.27e-03 2.16 1.16e-04 3.50 1.53e-02 2.6364 1.39e-05 3.07 3.41e-03 2.11 4.20e-05 3.53 7.02e-03 2.70

3 4 1.72e-02 3.75e-01 5.97e-02 1.11e+006 3.18e-03 4.16 1.07e-01 3.09 1.55e-02 3.33 4.23e-01 2.378 9.24e-04 4.30 4.20e-02 3.25 5.21e-03 3.78 1.85e-01 2.8712 1.70e-04 4.18 1.13e-02 3.24 1.02e-03 4.02 5.24e-02 3.1216 5.22e-05 4.10 4.50e-03 3.21 3.10e-04 4.15 2.04e-02 3.2824 1.01e-05 4.06 1.24e-03 3.18 5.54e-05 4.25 5.11e-03 3.4132 3.14e-06 4.04 5.02e-04 3.14 1.59e-05 4.34 1.85e-03 3.5448 6.13e-07 4.03 1.43e-04 3.11 2.66e-06 4.41 4.20e-04 3.6564 1.93e-07 4.02 5.88e-05 3.08 7.40e-07 4.45 1.45e-04 3.71

4 4 2.59e-03 8.00e-02 1.76e-02 3.84e-016 3.55e-04 4.90 1.48e-02 4.17 2.99e-03 4.36 9.62e-02 3.428 8.65e-05 4.91 4.49e-03 4.14 7.91e-04 4.63 3.37e-02 3.6412 1.15e-05 4.98 8.33e-04 4.15 1.15e-04 4.76 7.11e-03 3.8416 2.70e-06 5.03 2.52e-04 4.15 2.82e-05 4.88 2.23e-03 4.0324 3.49e-07 5.05 4.73e-05 4.13 3.72e-06 5.00 4.04e-04 4.22

5 4 4.69e-04 1.51e-02 3.53e-03 1.01e-016 4.31e-05 5.88 1.91e-03 5.10 4.45e-04 5.11 1.72e-02 4.378 7.56e-06 6.05 4.30e-04 5.19 9.48e-05 5.38 4.65e-03 4.5612 6.44e-07 6.08 5.28e-05 5.17 9.78e-06 5.60 6.84e-04 4.7316 1.12e-07 6.06 1.22e-05 5.10 1.87e-06 5.75 1.68e-04 4.8924 9.68e-09 6.05 2.07e-06 4.37 1.74e-07 5.85 2.17e-05 5.04

Table 4.2: Accuracy of VCJH schemes for the advection-diffusion problem with a =(1/2, 1/2,

√2/2)and b = 0.01 on the grids with N = 4 to N = 64. The advective numerical

flux was computed using a Lax-Friedrichs flux with λ = 1 and the diffusive numerical fluxwas computed using a LDG flux with τ = 1 and β = ±0.5n−.


cdg, κdg c+, κ+

p N ∆t′lim ∆t′lim2 64 3.24e-04 6.66e-043 64 1.51e-04 2.45e-044 24 4.72e-04 6.94e-045 24 2.74e-04 3.76e-04

Table 4.3: Time-step limits of VCJH schemes for the advection-diffusion problem witha =

(1/2, 1/2,

√2/2)and b = 0.01 on the grids with N = 24 and 64. The advective

numerical flux was computed using a Lax-Friedrichs flux with λ = 1 and the diffusivenumerical flux was computed using a LDG flux with τ = 1 and β = ±0.5n−.

Table (4.2) demonstrates that the expected order of accuracy is obtained by both the

scheme with c = 0 and κ = 0 and the scheme with c = c+ and κ = κ+, for all values

of p, except for p = 5 for which roundoff errors appear to have effected the results.

In addition, note that for p = 2 and p = 3, the scheme with c = c+ and κ = κ+

appears to be superconvergent, in some cases achieving more than 1/2 an order of

accuracy above that which was expected. Table (4.3) demonstrates that the scheme

with c = c+ and κ = κ+ has the larger explicit time-step limits. In particular, for

p = 2, the scheme with c = c+ and κ = κ+ has an explicit time-step limit which is

2.06x larger than that of the collocation-based nodal DG scheme (with c = 0 and

κ = 0).

Chapter 5

Reformulation of Energy Stable Flux

Reconstruction on Triangles

130

CHAPTER 5. REFORMULATION OF ESFR ON TRIANGLES 131

5.1 Preamble

In this chapter, the energy stable FR schemes (VCJH schemes) on triangles, originally

presented in chapter 3, are reformulated to incorporate the advances in theory that

were presented in chapter 4. In particular, in the previous chapter it was shown that

a certain class of parameterizing coefficients cvw and κvw ensure symmetry of the

VCJH derivative operators on tetrahedra. Furthermore, it was shown that the VCJH

schemes on tetrahedra can be formulated as a class of stable, filtered DG schemes. In

what follows, these results will be extended to the VCJH schemes on triangles.

5.2 Choosing Parameterizing Coefficients on Tri-

angles

The VCJH schemes on triangles employ degree p derivative operators, and it is im-

portant (as discussed previously) to ensure that these operators are symmetric. For

triangles, one may ensure symmetry by enforcing constraints on the coefficients as-

sociated with the derivative operators. More precisely, one may ensure symmetry

by constraining the coefficients cv and κv that appear in the following derivative

operatorsp+1∑

v=1

cv D(p,v) (f) D(p,v) (g) , (5.1)

andp+1∑

v=1

κv D(p,v) (F) · D(p,v) (G) , (5.2)

where f and g are arbitrary, continuous, p times differentiable, 2D scalar-valued

functions, and F and G are arbitrary, continuous, p times differentiable, 2D vector-

valued functions. The operators in equations (5.1) and (5.2) can be rewritten in

quadratic form as followsp+1∑

v=1

cv

(D(p,v) (f)

)2, (5.3)

andp+1∑

v=1

κv D(p,v) (F) · D(p,v) (F) . (5.4)


Now, having obtained the quadratic formulation of the VCJH derivative operators,

one may follow the approach of section 4.5, and construct a general, symmetric,

degree p derivative operator with which to compare the VCJH derivative operators.

Recall, that a general, symmetric, derivative operator of degree p was defined in

equation (4.143). Evidently, for d = 2, this operator can be employed on triangles.

For example, when d = 2 and p = 2, ∆f in equation (4.143) can be defined as follows

∆f =

[∂2f

∂x2,∂2f

∂y2,∂2f

∂x∂y,∂2f

∂y∂x

]T. (5.5)

Upon expanding ∆fT (∆f) in equation (4.143), one obtains an expression which, for

p > 1, contains non-distinct derivative terms, e.g. for d = 2 and p = 2, one obtains

an expression with c(

∂2f∂x∂y

)2and c

(∂2f∂y∂x

)2. More generally, for d = 2 and p ≥ 1, the

expanded form of equation (4.143) contains a total of

(p

v − 1

)(5.6)

derivative terms of the form

c

(∂p (·)

∂x(p−v+1)∂y(v−1)

)2

. (5.7)

Here, it is interesting to note that equation (5.6) contains an expression for the pth

degree binomial coefficients.

Upon combining all non-distinct derivative terms, one obtains the following operator

with p+ 1 distinct derivative terms

c

p+1∑

v=1

(p

v − 1

)(∂p (·)

∂x(p−v+1)∂y(v−1)

)2

, (5.8)

or equivalently

c

p+1∑

v=1

(p

v − 1

)(D(p,v) (f)

)2. (5.9)

The symmetric derivative operator in equation (5.9) can now be compared to the


VCJH derivative operator in equation (5.3). From this comparison, it immediately

follows that in order for the VCJH operator to be symmetric, one must require the

coefficients cv to be defined as follows

cv = c

(p

v − 1

). (5.10)

In addition, upon substituting F = (F1,F2) into equation (5.4), substituting κ and

Fm (for m = 1, 2) into equation (5.9) (in place of c and f), and comparing the

resulting expressions, one finds that the coefficients κv must be defined as follows

κv = κ

(p

v − 1

). (5.11)

One may observe that the coefficients in equations (5.10) and (5.11) are the same that

were recommended for the VCJH schemes on triangles in sections 3.3 and 3.4. The

previous recommendation was based on the numerical experiments by Castonguay

et al. [52] which identified coefficients cv and κv that ensured the symmetry of the

VCJH operators for p = 1 to p = 6. In this section, the author has succeeded in

confirming the results of Castonguay et al. by presenting an analytical derivation of

the required coefficients. This derivation extends the original results, proving that

the desired form of the coefficients can be obtained for all p ≥ 1.

5.3 Proof of Stability and Formulation of a Class

of Filtered DG Schemes on Triangles

In section 3.6, a class of VCJH schemes on triangles was proven to be stable using

a vector-calculus-based approach; it turns out, that the same schemes can be proven

to be stable via the matrix-based approach that was utilized to prove the stability of

the VCJH schemes on tetrahedra in section 4.3. In fact, the matrix-based proof of

the stability of the VCJH schemes on triangles is virtually identical to the proof on

tetrahedra, with the exception that equations (4.49) and (4.65) now take the following


form

p+1∑

v=1

cv D(p,v)

(∇ · fC

)D(p,v)

(ℓj

)=

∫

ΓS

(fC · n

)ℓj dΓS −

∫

ΩS

(∇ · fC

)ℓj dΩS,

(5.12)

p+1∑

v=1

κv D(p,v)

(∇uC

)· D(p,v)

(Lm,j

)=

∫

ΓS

uC(Lm,j · n

)dΓS −

∫

ΩS

∇uC · Lm,j dΩS,

(5.13)

where ΩS denotes the domain of the reference right triangle.

In addition, it is useful to note that the VCJH schemes on triangles are equivalent to

a class of filtered, collocation-based, nodal DG schemes. For these schemes, the filter

matrices F1 and F2 are defined by equation (4.176), and the associated matrices Kk

and Kk are defined as follows

Kk =JkNp

p+1∑

v=1

cv

(D(p,v)

)TD(p,v), (5.14)

Kk =JkNp

p+1∑

v=1

κv

(D(p,v)

)TD(p,v) 0

0(D(p,v)

)TD(p,v)

, (5.15)

where

[D(p,v)]ij = D(p,v) (ℓj (x))

∣∣∣∣xi

. (5.16)

Part II

Stability Theory for Nonlinear


135

Chapter 6


for Nonlinear Advection-Diffusion

Problems on Tetrahedra

136

CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 137

6.1 Preamble

In this chapter, the stability of the ESFR (VCJH) schemes for nonlinear advection-

diffusion problems on tetrahedra is examined. It is shown that the stability of the

schemes is primarily effected by the following factors: the degree of nonlinearity of the

diffusive flux, the methods for computing the advective numerical flux, the common

solution, and the diffusive numerical flux, and the choices of locations for the solution

points and flux points. In particular, it is shown that the degree of nonlinearity

of the diffusion coefficient b has a critical effect on determining the boundedness

of contributions from the diffusive flux, that the methods for forming the common

solution and numerical fluxes have a critical effect on the signs of contributions from

the interfaces, and that the locations of the solution points and flux points have a

critical effect on so-called ‘aliasing’ errors that arise from the schemes’ attempts to

represent nonlinear advective-diffusive fluxes, which are generally infinite dimensional,

with finite dimensional polynomial bases.

6.2 Preliminaries

Consider the general 3D nonlinear advection-diffusion equation that takes the follow-

ing form∂u

∂t+∇ · (fadv (u) + fdif (u,∇u)) = 0. (6.1)

In order to facilitate the analysis, it is convenient to follow the approach of [53] and

to consider the following simplified version of equation (6.1)

∂u

∂t+∇ · (fadv (u)− b (u)∇u) = 0, (6.2)

where b (u) ≥ 0. Note that the non-negativity of b (u) is necessary to ensure that the

problem is well-posed.

In accordance with the standard approach, equation (6.2) can be rewritten as a first


order system in order to eliminate second derivative terms as follows

∂u

∂t+∇ · (fadv (u)− b (u)q) = 0, (6.3)

q−∇u = 0. (6.4)

In the following section, the stability of the VCJH schemes for solving equations (6.3)

and (6.4) will be examined.

6.3 Stability Analysis

In order to facilitate the analysis, the VCJH schemes for nonlinear advection-diffusion

can be cast in matrix form, as shown in the following lemma.

Lemma 6.3.1. Suppose that the VCJH schemes are employed to solve the nonlinear

advection-diffusion equation for which the flux f takes the following form

f = fadv + fdif = fadv (u)− b (u)q. (6.5)

Then the following formulation of the VCJH schemes (in terms of matrices) holds

Mk d

dtUk + Sk (Fk, adv + Fk, dif )

=

∫

Γk

[((fDk, adv + fDk, dif

)−(f⋆adv + f⋆dif

))· n]L dΓk, (6.6)


Γk

(uDk − u⋆

)NL dΓk, (6.7)

where Fk, adv and Fk, dif are matrices containing the x, y, and z components of the

advective and diffusive fluxes at the Np solution points, and Uk, Qk, Mk, Mk, Sk,

Sk, N, and L are matrices which were defined previously in Chapter 4.

Proof. Equation (6.7) was derived in Lemma 4.3.3, but the derivation of equation (6.6)

remains to be shown. In order to derive equation (6.6), consider substituting the

expressions for the discontinuous flux fDk = fDk, adv + fDk, dif and the flux correction

fCk = f⋆− fDk =(f⋆adv + f⋆dif

)−(fDk, adv + fDk, dif

)into equation (4.42), and insisting that


the result holds for all j in order to obtain

∀j∫

Ωk

∂uDk∂t

ℓj dΩk +1

VS

p+1∑

v=1

v∑

w=1

cvw

∫

Ωk

∂

∂t

(D(p,v,w)


+

∫

Ωk

(∇ ·(fDk, adv + fDk, dif

))ℓj dΩk =

∫

Γk

((fDk, adv + fDk, dif


))· n ℓj dΓk.

(6.8)

Next, the second integral on the RHS of equation (6.8) can be transformed from an

integral over Ωk into an integral over ΩS as follows

∀j∫

Ωk

∂uDk∂t

ℓj dΩk +JkVS

p+1∑

v=1

v∑

w=1

cvw

∫

ΩS

∂

∂t

(D(p,v,w)

(uDk (x)

))D(p,v,w) (ℓj (x)) dΩS

+

∫

Ωk

(∇ ·(fDk, adv + fDk, dif

))ℓj dΩk =

∫

Γk

((fDk, adv + fDk, dif


))· n ℓj dΓk,

(6.9)

where uDk (x) and ℓj (x) have been defined using the mapping Θk (x) (in accordance

with equations (4.75) and (4.76)). Equation (6.9) can now be rewritten in matrix

form as follows

Mk d

dtUk +

3∑

m=1

(SkmFmk , adv + Sk

mFmk , dif

)

=

∫

Γk



))· n]L dΓk, (6.10)

where Fmk , adv and Fmk , dif are vectors which contain the values of the advective and

diffusive fluxes at the Np solution points, i.e.

Fmk , adv =[(fDmk , adv

)1. . .(fDmk , adv

)Np]T, (6.11)

Fmk , dif =[(fDmk , dif

)1. . .(fDmk , dif

)Np]T

= −[b((uDk)1) (

qDmk

)1. . . b

((uDk)Np) (qDmk

)Np]T. (6.12)

Equation (6.10) can be simplified by introducing the following block matrix definitions


which are written in terms of cartesian coordinates x, y, and z

Fk, adv =

Fxk, adv

Fyk , adv

Fzk , adv

, Fk, dif =

Fxk, dif

Fyk, dif

Fzk, dif

, (6.13)

where one should note that (for example) Fxk, adv = F1k , adv and Fxk, dif = F1k , dif .

Upon substituting equations (4.86) and (6.13) into equation (6.10), one obtains equa-

tion (6.6). This completes the proof of Lemma 6.3.1.

Theorem 6.3.1. Suppose that in the limit of a weakly nonlinear flux or infinite mesh

resolution, the diffusion coefficients become element-wise constant, i.e. b(uDk)≈ bk

(to within machine precision), and therefore a matrix of the diffusion coefficients Bkcan be defined as follows

Bk = bkI. (6.14)

Then it can be shown that

N∑

k=1

(QT

kBkMk Qk

)≥ 0. (6.15)

Subject to these conditions, if the VCJH schemes (for which Lemma 6.3.1 holds) are

employed in conjunction with an ‘E-flux’ formulation of the advective numerical flux

f⋆adv for which the following holds

(uDf+ − uDf−

)(f⋆adv − fadv (ξ)) · n ≤ 0, (6.16)

where ξ ∈[min

(uDf−, u

Df+

),max

(uDf−, u

Df+

)], a formulation of the common numerical

solution u⋆ and diffusive numerical flux f⋆dif for which the following holds

[ (uDf+ − uDf−

)f⋆dif + u⋆

(fdif(uDf+,q

Df+

)− fdif

(uDf−,q

Df−

))

+ uDf−fdif(uDf−,q

Df−

)− uDf+fdif

(uDf+,q

Df+

) ]· n ≤ 0, (6.17)

and exact L2 projections of the nonlinear fluxes fDk, adv and fDk, dif for which all aliasing


errors vanish, then it can be shown that

N∑

k=1

(d

dt‖Uk‖2

M

)≤ 0. (6.18)

Proof. Consider defining the matrices Bk and Bk as follows

Bk =

b((uDk)1)

0 · · · · · · 0

0. . .

......

. . ....

.... . . 0

0 · · · · · · 0 b((uDk)Np)

, (6.19)

Bk =

Bk 0 00 Bk 00 0 Bk

. (6.20)

One may now define Fk, dif as

Fk, dif = −BkQk. (6.21)

This definition of Fk, dif can be used to rewrite equation (6.7). In particular, one may

multiply equation (6.7) on the left by QTkBk and substitute equation (6.21) into the

result in order to obtain

QTkBkMk Qk + FT

k, dif SkUk =

∫

Γk

(uDk − u⋆

) (fDk, dif · n

)dΓk, (6.22)

where the fact that fDk, dif · n = FTk, dif NL has been used.

Setting equation (6.22) aside for the moment, consider multiplying equation (6.6) on

the left by UTk and substituting equation (4.100) into the result in order to obtain

1

2

d

dt‖Uk‖2

M+UT

k Sk (Fk, adv + Fk, dif )

=

∫

Γk



))· n]uDk dΓk, (6.23)

where the fact that uDk = UTk L has been used. Upon summing equations (6.22) and


(6.23) and rearranging the result, one obtains

1

2

d

dt‖Uk‖2

M+QT

kBkMk Qk +UTk SkFk, adv +

(UT

k SkFk, dif + FTk, dif SkUk

)

=

∫

Γk



))· n]uDk dΓk +

∫

Γk

(uDk − u⋆

) (fDk, dif · n

)dΓk.

(6.24)

Equation (6.24) can be simplified by noting that

UTk SkFk, dif + FT

k, dif SkUk

=

∫

Ωk

(uDk(∇ · fDk, dif

)+ fDk, dif · ∇uDk

)dΩk =

∫

Γk

uDk(fDk, dif · n

)dΓk. (6.25)


1

2

d

dt‖Uk‖2

M+QT

kBkMk Qk +UTk SkFk, adv

=

∫

Γk

[(fDk, adv −

(f⋆adv + f⋆dif

))· n]uDk dΓk +

∫

Γk

(uDk − u⋆

) (fDk, dif · n

)dΓk. (6.26)

Now, one may simplify equation (6.26) by manipulating the third term on the LHS.

One may expand this term as follows

UTk SkFk, adv =

∫

Ωk

(∇ · fDk, adv

)uDk dΩk

=

∫

Γk

(fDk, adv · n

)uDk dΓk −

∫

Ωk

∇uDk · fDk, adv dΩk. (6.27)

Upon substituting equation (6.27) into equation (6.26) and rearranging the result,

one obtains

1

2

d

dt‖Uk‖2

M+QT

kBkMk Qk

=

∫

Ωk

∇uDk · fDk, adv dΩk −∫

Γk

(f⋆adv · n) uDk dΓk +

∫

Γk

[(uDk − u⋆

)fDk, dif − uDk f⋆dif

]· n dΓk.

(6.28)

Next, one may consider adding and subtracting the following expression from the


RHS of equation (6.28)

∫

Ωk

∇uDk · fadv(uDk)dΩk +

∫

Γk

[(uDk − u⋆

)fdif(uDk ,q

Dk

)]· n dΓk, (6.29)

in order to obtain

1

2

d

dt‖Uk‖2

M+QT

kBkMk Qk

=

∫

Ωk

∇uDk · fadv(uDk)dΩk −

∫

Γk

(f⋆adv · n) uDk dΓk

+

∫

Γk

[(uDk − u⋆

)fdif(uDk ,q

Dk

)− uDk f⋆dif

]· n dΓk + εΩk

+ εΓk, (6.30)

where εΩkand εΓk

are defined as follows

εΩk≡∫

Ωk

∇uDk ·(fDk, adv − fadv

(uDk))dΩk, (6.31)

εΓk≡∫

Γk

[(uDk − u⋆

) (fDk, dif − fdif

(uDk ,q

Dk

))]· n dΓk. (6.32)

Observe that εΩkand εΓk

are measures of the effective ‘distances’ between the exact

flux functions (fadv(uDk)and fdif

(uDk ,q

Dk

)) and the approximate fluxes (fDk, adv and

fDk, dif ).

Upon summing equation (6.30) over all N elements in the mesh, one obtains

1

2

N∑

k=1

(d

dt‖Uk‖2

M

)+

N∑

k=1

(QT

kBkMk Qk

)= Ξadv + Ξdif + Ξalias, (6.33)

where Ξadv and Ξdif denote contributions from the advective and diffusive fluxes that

are defined as follows

Ξadv ≡N∑

k=1

∫

Ωk

∇uDk · fadv(uDk)dΩk −

∫

Γk


, (6.34)

Ξdif ≡N∑

k=1

∫

Γk

[(uDk − u⋆

)fdif(uDk ,q

Dk

)− uDk f⋆dif

]· n dΓk

, (6.35)

and where Ξalias denotes the contribution from the aliasing error that is defined as


follows

Ξalias ≡N∑

k=1

(εΩk+ εΓk

) . (6.36)

Stability is ensured if the time rate of change of the norm on the LHS of equation (6.33)

is non-positive. In other words, stability is ensured if Ξadv ≤ 0, Ξdif ≤ 0, Ξalias ≤ 0,

andN∑

k=1

(QT

kBkMk Qk

)≥ 0. (6.37)

Unfortunately, the condition in equation (6.37) is difficult to satisfy, as it requires

BkMk to be a symmetric positive-definite matrix. In all cases, BkMk is at least

‘weakly’ positive-definite (according to [74]), as it can be written as the product of

‘strongly’ positive-definite matrices, where ‘strong positive-definiteness’ is equivalent

to the conventional notion of positive-definiteness. However, for a general b (u), BkMk

need not be symmetric or positive-definite. In fact, BkMk is only guaranteed to be

symmetric positive-definite if Bk and Mk commute, and if Bk and Mk are simulta-

neously diagonalizable (i.e. Bk and Mk possess the same eigenvectors). This occurs

when

Bk = bkI, (6.38)

where bk is a constant and (from before) I is the identity matrix in R3Np×3Np . In

practice, equation (6.38) can be satisfied (to within machine precision) if b(uDk)≈ bk.

This happens for all nonlinear functions b (u) in the limit of infinite mesh resolution,

and for some weakly nonlinear functions b (u) for coarser mesh resolutions.

Now, having determined the conditions under which equation (6.37) holds, one must

establish the conditions under which Ξadv ≤ 0, Ξdif ≤ 0, and Ξalias ≤ 0, in order to

complete the proof.

It turns out that if the advective numerical flux f⋆adv is chosen to be an ‘E-flux’ then

Ξadv ≤ 0. This can be seen as follows. Consider defining a vector-valued function

G (u) such that∂G

∂u(u) =

∂G

∂u= fadv. (6.39)


One may now substitute equation (6.39) into equation (6.34) in order to obtain

Ξadv =

N∑

k=1

∫

Ωk

∇uDk ·∂G

∂u

(uDk)dΩk −

∫

Γk

(f⋆adv · n)uDk dΓk

=N∑

k=1

∫

Ωk

∇ ·G(uDk)dΩk −

∫

Γk


. (6.40)

Upon applying the divergence theorem to the first term on the RHS of equation (6.40)

and rearranging the result, one obtains

Ξadv =

N∑

k=1

∫

Γk

[G(uDk)− uDk f⋆adv

]· n dΓk

. (6.41)

Assuming that the domain Ω is such that periodic boundary conditions can be im-

posed, one may rewrite equation (6.41) in terms of a summation over element faces

f as follows

Ξadv =

Nf∑

f=1

∫

Γf

[G(uDf−)−G

(uDf+)+(uDf+ − uDf−

)f⋆adv]· n dΓf

, (6.42)

where uDf− and uDf+ are values of the discontinuous solution in elements Ωk and Ωk+1

located on either side of face f , and n is the outward pointing unit normal vector

associated with element Ωk (i.e. n = n−).

Now, equation (6.42) can be simplified by manipulating the term[G(uDf−)−G

(uDf+)]·

n. In particular, one may rewrite this term as follows

[G(uDf−)−G

(uDf+)]· n =

∫ uDf−

uDf+

(∂G

∂u(u)

)· n du. (6.43)

In accordance with the mean value theorem and the definition of G (equation (6.39)),

one may reformulate equation (6.43) as follows

[G(uDf−)−G

(uDf+)]· n =

((∂G

∂u(u)

∣∣∣∣ξ

)· n)(uDf− − uDf+

)

= (fadv (ξ) · n)(uDf− − uDf+

), (6.44)


where ξ ∈[min

(uDf−, u

Df+

),max

(uDf−, u

Df+

)]. On substituting equation (6.44) into

equation (6.42), one obtains

Ξadv =

Nf∑

f=1

∫

Γf

[(uDf+ − uDf−

)(f⋆adv − fadv (ξ)) · n

]dΓf

. (6.45)

Now, if f⋆adv is chosen to be an E-flux as defined by Osher in [75], one is assured that

(uDf+ − uDf−

)(f⋆adv − fadv (ξ)) · n ≤ 0, (6.46)

and therefore that

Ξadv =

Nf∑

f=1

∫

Γf

[(uDf+ − uDf−


]dΓf

≤ 0. (6.47)

Note that a number of popular fluxes, including the aforementioned Lax-Friedrichs

advective numerical flux, can be categorized as E-fluxes. In order to demonstrate this

fact, consider defining the normal component of the Lax-Friedrichs flux as follows

f⋆adv · n = fadv · n+λ

2

(max

η∈[uDf−,uD

f+]

∣∣∣∣∂fadv∂u

(η) · n∣∣∣∣

)[[uD]] · n

=1

2

[(fadv

(uDf+)+ fadv

(uDf−))· n− λ

(uDf+ − uDf−

)(

maxη∈[uD

f−,uDf+]


(η) · n∣∣∣∣

)].

(6.48)

On subtracting 12fadv (ξ)·n from equation (6.48) and multiplying the result by

(uDf+ − uDf−

),

one obtains

1

2

(uDf+ − uDf−


=1

2

[ (uDf+ − uDf−

) (fadv

(uDf+)+ fadv

(uDf−))· n−

(uDf+ − uDf−

)fadv (ξ) · n

− λ(uDf+ − uDf−

)2(

maxη∈[uD

f−,uDf+]


(η) · n∣∣∣∣

)]. (6.49)

The first two terms on the RHS cancel because of the mean value theorem and


therefore equation (6.49) becomes

1

2

(uDf+ − uDf−

)(f⋆adv − fadv (ξ))·n = −λ

2

(uDf+ − uDf−

)2(

maxη∈[uD

f−,uDf+]


(η) · n∣∣∣∣

)≤ 0.

(6.50)

Thus, the Lax-Friedrichs flux is an E-flux (in accordance with equation (6.46)) and

for this choice of the advective numerical flux, Ξadv ≤ 0.

Now, having shown that a particular choice of f⋆adv ensures that Ξadv ≤ 0, one may

follow a similar approach and show that particular choices of u⋆ and f⋆dif ensure

that Ξdif ≤ 0. Towards this end, consider rewriting equation (6.35) in terms of a

summation over interfaces (Γf ) as follows

Ξdif =

Nf∑

f=1

∫

Γf

[ (uDf+ − uDf−

)f⋆dif + u⋆

(fdif(uDf+,q

Df+

)− fdif

(uDf−,q

Df−

))


Df−

)− uDf+fdif

(uDf+,q

Df+

) ]· n dΓf

. (6.51)

Evidently, in order to ensure that Ξdif ≤ 0, one requires that the kernel of the integral

in equation (6.51) is non-positive, i.e.

[ (uDf+ − uDf−

)f⋆dif + u⋆

(fdif(uDf+,q

Df+

)− fdif

(uDf−,q

Df−

))


Df−

)− uDf+fdif

(uDf+,q

Df+

) ]· n (6.52)

must be non-positive. There are a number of popular approaches for computing u⋆

and f⋆dif such that equation (6.52) is non-positive. In what follows, it will be shown

that the LDG approach is one such approach. For the LDG approach, u⋆ and f⋆dif · n


can be defined as follows

u⋆ = uD − β · [[uD]]

=1

2

(uDf+ + uDf−

)+(uDf+ − uDf−

)(β · n) , (6.53)

f⋆dif · n =(fdif+ τ [[uD]] + β[[fdif ]]

)· n

=1

2

(fdif(uDf+,q

Df+

)+ fdif

(uDf−,q

Df−

))· n

− τ(uDf+ − uDf−

)−(fdif(uDf+,q

Df+

)− fdif

(uDf−,q

Df−

))· n (β · n) . (6.54)

On substituting equations (6.53) and (6.54) into equation (6.52), one obtains

[ (uDf+ − uDf−

)(1

2

(fdif(uDf+,q

Df+

)+ fdif

(uDf−,q

Df−

))· n

− τ(uDf+ − uDf−

)−(fdif(uDf+,q

Df+

)− fdif

(uDf−,q

Df−

))· n (β · n)

)

+

(1

2

(uDf+ + uDf−

)+(uDf+ − uDf−

)(β · n)

)(fdif(uDf+,q

Df+

)− fdif

(uDf−,q

Df−

))· n


Df−

)· n− uDf+fdif

(uDf+,q

Df+

)· n]

= −τ(uDf+ − uDf−

)2 ≤ 0. (6.55)

Thus, the LDG approach ensures that equation (6.52) is ≤ 0, and therefore that

Ξdif ≤ 0.

Finally, one may ensure that Ξalias ≤ 0 by utilizing exact L2 projections to construct

the fluxes fDk, adv and fDk, dif on Ωk and Γk, respectively. This will be shown via a brief

examination of these L2 projections.

Consider the exact L2 projection of a flux fDk on Ωk which is defined such that

∀i∫

Ωk

(fDk − f

(uDk))L3Dk,i dΩk = 0, (6.56)

where each L3Dk,i is a member of a 3D orthonormal polynomial basis of degree p − 1

on Ωk. Furthermore, the exact L2 projection of a flux fDk on Γk is defined such that


for each of the f = 1, . . . , 4 faces of the element boundary

∀l∫

Γf

(fDk − f

(uDk))· nf L

2Dk,f,l dΓf = 0, (6.57)

where each L2Dk,f,l is a member of a 2D orthonormal polynomial basis of degree p on

Γf .

Setting equations (6.56) and (6.57) aside for the moment, consider representing ∇uDkon Ωk and

(uDk − u⋆

)on Γf in terms of the orthonormal basis functions L3D

k,i and L2Dk,f,l

as follows

(∇uDk

)m=

Np∑

i=1

ζk,i,mL3Dk,i (x) , for m = 1, 2, 3, (6.58)

(uDk − u⋆

)f=

Nfp∑

l=1

ϕk,f,l L2Dk,f,l (x) , (6.59)

where ζk,i,m and ϕk,f,l are constant coefficients, and Np is the number of points required

to define a polynomial of degree p− 1 in 3D, i.e.

Np =p (p+ 1) (p+ 2)

6. (6.60)

Upon multiplying equation (6.56) by each coefficient ζk,i,m, substituting fadv in place

of f , and summing the result over m and i, one obtains

∫

Ωk

∇uDk ·(fDk, adv − fadv

(uDk))

dΩk = εΩk= 0. (6.61)

Similarly, on multiplying equation (6.57) by each coefficient ϕk,f,l, substituting fdif in

place of f , and summing the result over l and f , one obtains

∫

Γk

[(uDk − u⋆

) (fDk, dif − fdif

(uDk ,q

Dk

))]· n dΓk = εΓk

= 0. (6.62)

From equations (6.61), (6.62), and (6.36), it immediately follows that Ξalias = 0. This

completes the proof of Theorem 6.3.1.


Remark. There is at least one alternative approach for proving the stability of high-

order schemes for nonlinear advection-diffusion problems on unstructured meshes.

This approach (due to Cockburn and Shu [53]) utilizes the following definitions for

fdif and q

fdif =√b (u)q, q =

√b (u)∇u. (6.63)

In accordance with these definitions, the first order system in equations (6.3) and

(6.4) can be rewritten as follows

∂u

∂t+∇ ·

(fadv (u)−

√b (u)q

)= 0, (6.64)

q−√b (u)∇u = 0. (6.65)

It can be shown that if a DG scheme is employed to solve equations (6.64) and (6.65),

the stability of the scheme is governed by the following equation

1

2

N∑

k=1

(d

dt‖Uk‖2

M

)+

N∑

k=1

(QT

k

(B1/2k

)TMk B1/2

k Qk

)= Ξadv + Ξdif + Ξalias, (6.66)

where the quantity QTk

(B1/2k

)TMk B1/2

k Qk is a norm

‖Qk‖2(B1/2k

)TMk B

1/2k

= QTk

(B1/2k

)TMk B1/2

k Qk, (6.67)

because the matrix(B1/2k

)TMk B1/2

k is symmetric positive-definite, and where (as

before) Ξadv, Ξdif , and Ξalias are contributions from the advective flux, the diffusive

flux, and the aliasing error. As in Theorem 6.3.1, it can be shown that if the advective

numerical flux, common solution, and diffusive numerical flux are chosen appropri-

ately, one is assured that Ξadv ≤ 0 and Ξdif ≤ 0, and that if the discontinuous flux is

formed using exact L2 projections, one is assured that Ξalias ≤ 0.

The analysis above may appear to produce a stronger result than that of Theo-

rem 6.3.1, as there are no guidelines governing the mesh resolution or the degree of

nonlinearity of the flux. (Recall that Theorem 6.3.1 only guarantees stability in the

limit of infinite mesh resolution for fluxes for which b(uDk)is highly nonlinear, or

on coarser meshes for weakly nonlinear fluxes for which b(uDk)≈ bk). However, the


definitions of fdif and q used in this analysis (in equation (6.63)) cannot be general-

ized to handle nonlinear advection-diffusion problems that possess arbitrary diffusive

fluxes of the form fdif (u,∇u). In particular, it is unclear how the definitions in equa-

tion (6.63) can be applied to any diffusive flux that is not of the form fdif = −b (u)∇u.Conversely, it is relatively straightforward to rewrite an arbitrary nonlinear diffusive

flux fdif (u,∇u) utilizing the definitions of fdif and q in equations (6.3) and (6.4), as

one may simply form fdif = f (u,q) where q = ∇u. For this reason, the more general

approach (employed in Theorem 6.3.1) is the most frequently used in practice (as

noted in [76]), and is the primary approach that was utilized in this work.

Remark. One is frequently unable to compute exact L2 projections of high dimen-

sional or infinite dimensional nonlinear fluxes fadv(uDk)and fdif

(uDk ,q

Dk

)in problems

of practical interest. However, if numerical quadrature rules of sufficiently high-order

are employed in equations (6.56) and (6.57), the L2 projections can ensure that Ξalias

is of the order of machine zero. Nevertheless, in this case, the L2 projection proce-

dures will be substantially more expensive, from a computational standpoint, than

the collocation projection utilized previously to define fDk (in equation (4.8)).

It turns out that one may utilize collocation projection procedures to form the fluxes

while simultaneously reducing aliasing errors, if the solution points xi and flux points

xf,l are placed at the locations of quadrature points. This can be shown through a

careful examination of the aliasing errors εΩkand εΓk

. Towards this end, consider

utilizing equations (4.19) and (4.20) to transform εΩkand εΓk

from the physical space

to the reference space as follows

εΩk=εΩS

Jk=

1

Jk

∫

ΩS

∇uD ·(fDadv − fadv

(uD))

dΩS, (6.68)

εΓk=εΓS

Jk=

1

Jk

∫

ΓS

[(uD − u⋆

) (fDdif − fdif

(uD, qD

))]· n dΓS, (6.69)

where εΩSand εΓS

are measures of the aliasing errors in the reference space. (Note

that the transformation to reference space has been performed in order to simplify

the subsequent analysis).

Next, one may define ∇uD in equation (6.68) and(uD − u⋆

)in equation (6.69) as


follows

(∇uD

)m=

Np∑

i=1

ζi,mL3Di (x) , for m = 1, 2, 3, (6.70)

(uD − u⋆

)f=

Nfp∑

l=1

ϕf,l L2Df,l (x) , (6.71)

where ζi,m and ϕf,l are constant coefficients, L3Di is a member of the 3D orthonormal

polynomial basis of degree p − 1 inside the element (i.e. L3Di belongs to the class

of orthonormal basis functions that are of one degree lower than the basis functions

defined in Chapter 4), and L2Df,l is a member of the 2D orthonormal polynomial basis

of degree p on face f of the element (i.e. L2Df,l belongs to the class of orthonormal

basis functions that was defined in Chapter 3). Upon substituting equations (6.70)

and (6.71) into the expressions for εΩSand εΓS

in equations (6.68) and (6.69), one

obtains

εΩS=

3∑

m=1

Np∑

i=1

ζi,m

∫

ΩS

(fDm, adv − fm, adv

(uD))L3Di dΩS, (6.72)

εΓS=

4∑

f=1

Nfp∑

l=1

ϕf,l

∫

Γf

[(fDdif − fdif

(uD, qD

))· nf

]L2Df,l dΓf

=

3∑

m=1

4∑

f=1

Nfp∑

l=1

ϕf,l nf,m

∫

Γf

(fDm,f, dif − fm, dif

(uDf , q

Df

))L2Df,l dΓf , (6.73)

or equivalently,

εΩS=

3∑

m=1

Np∑

i=1

ζi,m (εΩS)i,m , (6.74)

εΓS=

3∑

m=1

4∑

f=1

Nfp∑

l=1

ϕf,l nf,m (εΓS)f,l,m , (6.75)


where

(εΩS)i,m =

∫

ΩS

(fDm, adv − fm, adv

(uD))L3Di dΩS, (6.76)

(εΓS)f,l,m =

∫

Γf

(fDm,f, dif − fm, dif

(uDf , q

Df

))L2Df,l dΓf . (6.77)

It should be noted that equations (6.76) and (6.77) are analogous to equations (6.56)

and (6.57) that define the L2 projections in physical space. In fact, equations (6.76)

and (6.77) simply define the component-wise L2 projections of the advective and

diffusive fluxes in reference space. Thus, if exact L2 projections of the fluxes are

performed then (εΩS)i,m and (εΓS

)f,l,m vanish as expected. However, for collocation

projections of the flux, this is not necessarily the case. In order to illustrate this

point, one may consider forming collocation projections of the fluxes as follows

fDm, adv =

Np∑

j=1

fm, adv

((uD)j)

ℓ 3Dj (x) =

Np∑

j=1

(fDm, adv

)jℓ 3Dj , (6.78)

fDm,f, dif =

Nfp∑

r=1

fm, dif

((uDf)r,(qDf

)r)ℓ 2Df,r (x) =

Nfp∑

r=1

(fDm,f, dif

)rℓ 2Df,r , (6.79)

where ℓ 3Dj is the 3D nodal polynomial of degree p that assumes the value of 1 at

solution j and the value of zero at all neighboring solution points (i.e. ℓ 3Dj belongs to

the class of nodal basis functions that was defined in Chapter 4), ℓ 2Df,r is the 2D nodal

polynomial of degree p that assumes the value of 1 at flux point f, r and the value of

zero at all neighboring flux points (i.e. ℓ 2Df,r belongs to the class of nodal basis functions

that was defined in Chapter 3), and where(fDm, adv

)jand

(fDm,f, dif

)rare the mth

components of the pointwise values of the advective and diffusive fluxes in reference

space ((fDadv

)jand

(fDf, dif

)r) that are related to the pointwise values of the fluxes in

physical space ((fDk, adv

)j= fk, adv

((uDk)j)

and(fDk,f, dif

)r= fk, dif

((uDf)r,(qDf

)r)) via

equation (4.15). Upon substituting equations (6.78) and (6.79) into equations (6.76)


and (6.77), one obtains

(εΩS)i,m =

Np∑

j=1

(fDm,adv

)j ∫

ΩS

ℓ 3Dj L3Di ΩS −

∫

ΩS

fm, adv

(uD)L3Di dΩS, (6.80)

(εΓS)f,l,m =

Nfp∑

r=1

(fDm,f, dif

)r ∫

Γf

ℓ 2Df,r L2Df,l dΓf −

∫

Γf

fm, dif

(uDf , q

Df

)L2Df,l dΓf . (6.81)

The terms ℓ 3Dj L3Di and ℓ 2Df,r L

2Df,l that appear in equations (6.80) and (6.81) are poly-

nomials of degree 2p− 1 in 3D and degree 2p in 2D, respectively. Thus, the integrals

of these terms can be computed exactly via quadrature rules of degree 2p− 1 in 3D

and degree 2p in 2D. More generally, a 3D quadrature rule (or equivalently a cubature

rule) of arbitrary degree can approximate the integral of ℓ 3Dj L3Di on ΩS as follows

∫

ΩS

ℓ 3Dj L3Di ΩS =

N3Dq∑

n=1

wn ℓ3Dj (ςn)L

3Di (ςn) + e3Dq , (6.82)

where N3Dq is the number of quadrature points, ςn’s are the point locations, wn’s are

the weights, and e3Dq is the quadrature error that vanishes for quadrature rules of

degree ≥ 2p−1. Similarly, a 2D quadrature rule of arbitrary degree can approximate

the integral of ℓ 2Df,r L2Df,l on Γf as follows

∫

Γf

ℓ 2Df,r L2Df,l dΓf =

N2Dq∑

s=1

ωs ℓ2Df,r (ϑs)L

2Df,l (ϑs) + e2Dq , (6.83)

where N2Dq is the number of quadrature points, ϑs’s are the point locations, ωs’s are

the weights, and e2Dq is the quadrature error that vanishes for quadrature rules of

degree ≥ 2p.

Upon substituting equations (6.82) and (6.83) into equations (6.80) and (6.81), one


obtains

(εΩS)i,m =

Np∑

j=1

(fDm, adv

)j N3Dq∑

n=1

wn ℓ3Dj (ςn)L

3Di (ςn)−

∫

ΩS

fm, adv

(uD)L3Di dΩS + E3D

q ,

(6.84)

(εΓS)f,l,m =

Nfp∑

r=1

(fDm,f, dif

)r N2Dq∑

s=1

ωs ℓ2Df,r (ϑs)L

2Df,l (ϑs)−

∫

Γf

fm,dif

(uDf , q

Df

)L2Df,l dΓf + E2D

q ,

(6.85)

where

E3Dq =

Np∑

j=1

(fDm, adv

)je3Dq , E2D

q =

Nfp∑

r=1

(fDm,f, dif

)re2Dq . (6.86)

Now, if the quadrature rules are required to have a particular number of points,

N3Dq = Np and N2D

q = Nfp, and the solution points xi (or equivalently xj) and flux

points xf,l (or equivalently xf,r) are placed at the locations of the quadrature points

ςn and ϑs, one obtains

(εΩS)i,m =

Np∑

j=1

fm,adv

(uD) ∣∣∣∣

ςj

Np∑

n=1

wn δjn L3Di (ςn)−

∫

ΩS

fm, adv

(uD)L3Di dΩS + E3D

q ,

(6.87)

(εΓS)f,l,m =

Nfp∑

r=1

fm,dif

(uDf , q

Df

) ∣∣∣∣ϑr

Nfp∑

s=1

ωs δr,s L2Df,l (ϑs)−

∫

Γf

fm, dif

(uDf , q

Df

)L2Df,l dΓf + E2D

q ,

(6.88)

or equivalently,

(εΩS)i,m =

Np∑

j=1

wj fm, adv

(uD) ∣∣∣∣

ςj

L3Di (ςj)−

∫

ΩS

fm, adv

(uD)L3Di dΩS + E3D

q , (6.89)

(εΓS)f,l,m =

Nfp∑

r=1

ωr fm, dif

(uDf , q

Df

) ∣∣∣∣ϑr

L2Df,l (ϑr)−

∫

Γf

fm, dif

(uDf , q

Df

)L2Df,l dΓf + E2D

q .

(6.90)

The sums in equations (6.89) and (6.90) act as numerical quadrature approximations

of the integral terms in each of the equations, and evidently, for quadrature rules


of sufficient strength, the integral terms and their numerical approximations will

effectively cancel one another. In addition, (as mentioned previously) for quadrature

rules of order ≥ 2p − 1 in 3D and ≥ 2p in 2D, the terms E3Dq and E2D

q will vanish.

Thus, for quadrature rules of sufficient strength, (εΩS)i,m and (εΓS

)f,l,m approximately

vanish, εΩSand εΓS

approximately vanish, and in turn εΩkand εΓk

approximately

vanish. However, it is important to note that a quadrature rule of a very high degree

maybe required for this to occur, and that there will frequently not be enough solution

points or flux points to allow for this. Therefore, in practice, one should consider the

act of placing the solution points xi and flux points xf,l at the locations of quadrature

points as a procedure for reducing (but most likely not eliminating) aliasing errors.

This procedure results in an effective compromise between the extremes of eliminating

all aliasing errors via expensive L2 projections and creating large aliasing errors via

inexpensive collocation projections that employ solution and flux point locations that

differ from quadrature point locations.

Identifying high-order (or equivalently high-degree) quadrature rules with the correct

numbers of points N3Dq = Np and N2D

q = Nfp is not a trivial task. An effective

process for identifying these rules and thereby identifying desirable locations for the

solution points and flux points is discussed in the next chapter.

Chapter 7

Quadrature Points for Energy Sta-

ble Flux Reconstruction Schemes on

Triangles and Tetrahedra

157

CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 158

7.1 Preamble

This chapter constructs new sets of quadrature rules on triangles and tetrahedra with

Np = (p+1)(p+2)/2 and Np = (p+1)(p+2)(p+3)/6 quadrature points, respectively.

When the quadrature point locations for these rules are utilized as solution and flux

point locations for the VCJH schemes, aliasing errors are minimized or eliminated (as

discussed in the previous chapter).

Note that this chapter is adapted from the article “Symmetric Quadrature Rules for

Simplexes Based on Sphere Closed Packed Lattice Arrangements” by D. M. Williams

and L. Shunn, which is pending submission to the Journal of Computational and

Applied Mathematics. The main results and over 95% of the text for this article were

contributed by D. M. Williams. An overview of this article is given below.

Sphere closed packed (SCP) lattice arrangements of points are well-suited for formu-

lating symmetric quadrature rules on simplexes, as they are symmetric under affine

transformations of the simplex unto itself in 2D and 3D. As a result, SCP lattice ar-

rangements have been utilized to formulate symmetric quadrature rules with Np = 1,

4, 10, 20, 35, and 56 points on the 3-simplex [77] (where it should be noted that the

points from these rules can exactly represent polynomials of order p = 0 to p = 5

on the 3-simplex). In what follows, the work on the 3-simplex is extended, and SCP

lattices are employed to identify symmetric quadrature rules with Np = 1, 3, 6, 10,

15, 21, 28, 36, 45, 55, and 66 points on the 2-simplex (where the points from these

rules can exactly represent polynomials of order p = 0 to p = 10 on the 2-simplex)

and Np = 84 points on the 3-simplex (where the points from this rule can exactly

represent a polynomial of up to order p = 6 on the 3-simplex). These rules are found

to be capable of exactly integrating polynomials of degree ≤ 17 in 2D and ≤ 9 in 3D.

7.2 Introduction

There have been significant efforts to identify high-order quadrature rules on d-simplex

and d-hypercube geometries as evidenced by the surveys in [78], [79], [80], [81], and

[82]. On the d-hypercube, it is well-known that optimal quadrature rules withNp = nd

points can be constructed from tensor products of the 1D Gauss-Legendre quadrature

rules with n points [83]. These quadrature rules are widely used because, in addition


to their optimality, they are symmetric (as they are invariant under reflections and

rotations that map the d-hypercube unto itself), possess positive weights, and have

points located within the interior of the d-hypercube. Due to these desirable prop-

erties, there have been attempts to extend the Gauss-Legendre rules to d-simplexes

(cf. [83] and [84]). The simplest of such approaches has involved constructing Gauss-

Legendre rules on the d-hypercube and then degenerating vertices until the d-simplex

is obtained. However, in general the resulting rules are no longer symmetric on the

d-simplex, as they contain anisotropic clusters of points near the degenerate vertices.

In addition, the optimality of these rules has yet to be shown analytically. In fact, to

the author’s knowledge, no one has identified a family of symmetric quadrature rules

on the d-simplex for which optimality can be rigorously proven. For this reason, the

formulation of quadrature rules on the d-simplex remains an open area of research, as

demonstrated by the recent work presented in [85], [86], [87], [88], [89], [90], and [77].

Of particular interest, is the effort by Shunn and Ham in [77] to construct quadrature

rules on the 3-simplex based on cubic close packing (CCP) arrangements of points.

In their work, the CCP arrangements of points (i.e. the CCP lattices) are defined by

the centers of spheres in the CCP configuration, as shown in Figure (7.1) for the cases

of Np = 1, 4, 10, 20, 35, 56, and 84.


(a) Np = 1 (b) Np = 4 (c) Np = 10

(d) Np = 20 (e) Np = 35 (f) Np = 56

(g) Np = 84

Figure 7.1: Cubic closed packed (CCP) configurations on tetrahedra with Np = 1, 4, 10,20, 35, 56, and 84 points.


The CCP lattices are viewed as the 3-simplex analog to the uniform cartesian lattices

on which the Gauss-Legendre rules on the 3-hypercube are based, in the sense that

they possess the same properties of symmetry on the 3-simplex as uniform cartesian

lattices possess on the 3-hypercube. Based on this fact, it was supposed that an

optimal family of symmetric quadrature rules on the 3-simplex could be obtained

by employing the CCP lattices as initial conditions to optimization procedures for

identifying the rules. Following this approach, a family of symmetric, locally optimal

quadrature rules on the 3-simplex with Np = 1, 4, 10, 20, 35, and 56 points was

obtained [77].

This work attempts to extend the approach in [77] to identify new quadrature rules

on the d-simplex for the cases of d = 2 and d = 3. This extension requires the

construction of CCP lattices in 2-space (sometimes referred to as ‘hexagonal packing’

lattices), which are shown in Figure (7.2) for the cases of Np = 1, 3, 6, 10, 15, 21, 28,

36, 45, 55, and 66.


(a) Np = 1 (b) Np = 3 (c) Np = 6

(d) Np = 10 (e) Np = 15 (f) Np = 21

(g) Np = 28 (h) Np = 36 (i) Np = 45

(j) Np = 55 (k) Np = 66

Figure 7.2: Cubic closed packed (CCP) configurations on triangles with Np = 1, 3, 6, 10,15, 21, 28, 36, 45, 55, and 66 points.

For convenience, the analogs of the CCP lattices in d-space will henceforth be referred


to as d-sphere close packed lattices (d-SCP lattices). It is useful to note that, in

general, the number of points in the d-SCP lattices (the values of Np for the lattices)

can be expressed as a function of Nl, the number of layers of d-spheres in the lattices,

as follows

Np =(Nl + 1)!

d! (Nl + 1− d)! , (7.1)

where each layer is of dimension d− 1.

The remainder of this chapter is structured as follows. Section 3 presents requirements

for the quadrature rules and introduces the theoretical machinery that enables these

requirements to be enforced. Section 4 discusses the practical procedure for obtaining

the quadrature rules through solving nonlinearly constrained optimization problems.

Finally, section 5 and Appendices B and C summarize the quadrature point locations

and weights for the resulting rules.

7.3 Overview of the Theory of Quadrature Rules

on Simplexes

This section will describe the theoretical basis of the standard approach for finding

quadrature rules on d-simplexes. The description presented here is a generalization

of the description of Shunn and Ham in [77].

7.3.1 Preliminary Definition of the Problem

Consider the domain Ω which resides in a d-dimensional space with coordinate vector

x ∈ Rd. Suppose that a scalar-valued function f(x) is well-defined on Ω (such that

f(x) ∈ R ∀x ∈ Ω) and that the integral of f(x) on Ω is also well-defined such that

F ≡∫

Ω

f(x) dΩ, F ∈ R. (7.2)

In this case, F can be approximated using a quadrature rule (F) as follows

F ≡ V (Ω)

[Np∑

i=1

wi f (xi)

]≈ F, (7.3)


where V (Ω) denotes the volume of Ω, and xi and wi denote the locations and weights

of the quadrature points, respectively. Note that the error ǫ introduced by the quadra-

ture approximation in equation (7.3) can be defined as follows

ǫ ≡ F − FV (Ω)

. (7.4)

Now, suppose that Ω is defined to be the d-simplex with vertices v1, . . . ,vd+1. One

may redefine the quadrature rule in equation (7.3) in terms of the d-simplicial def-

initions of the volume V (Ω) and the quadrature point locations xi. In particular,

the volume of the d-simplex can be defined using the Cayley-Menger determinant as

follows

V (Ω) =1

d!det (B) , (7.5)

where B is a matrix which has columns that are constructed from the differences

between vertices of the d-simplex, i.e.

B =(v2 − v1) · · · (vd+1 − v1)

. (7.6)

In addition, the quadrature point locations xi can be expressed as the following linear

combinations of the d-simplex vertices

xi =d+1∑

j=1

ai,jvj. (7.7)

Upon substituting the expressions for V (Ω) and xi from equations (7.5) and (7.7)

into equation (7.3), one obtains the following general expression for the quadrature

rule F on the d-simplex

F =1

d!det (B)

[Np∑

i=1

wi f

(d+1∑

j=1

ai,jvj

)]. (7.8)

7.3.2 Quadrature Rule Requirements

Before proceeding further, it is useful to review the requirements that quadrature

rules are usually required to satisfy. In accordance with the discussions in [80], each


quadrature rule is required to:

• Integrate a polynomial of the highest possible order and minimize the magnitude

of the truncation error term.

• Ensure that all quadrature weights are non-negative in order to minimize the

possibility of cancellation errors.

• Ensure that there are no quadrature points located outside of the d-simplex.

• Ensure that all quadrature points are symmetrically arranged within the d-

simplex. The quadrature rule must be invariant under affine transformations of

the d-simplex unto itself.

In addition, on the 2-simplex it is desirable (but not required) that all quadrature

rules preserve the layered structure of the corresponding 2-SCP configuration. More

precisely, it is convenient if the quadrature rules have rows of points for which the

number of points is the same as the number of points in a complementary row of

the 2-SCP configuration. Here it is important to note that a row of points is defined

such that the y coordinates of all the points in a given row are greater than all y

coordinates of the points in the row below and less than all y coordinates of points

in the row above (with natural exceptions for the top-most and bottom-most rows).

When the points can be partitioned into rows in this way, it facilitates the creation

of mappings between the indices of quadrature points on pairs of 2-simplexes. These

mappings are useful in certain applications, for example, in numerical integration

procedures on triangular (2-simplex) faces of elements in 3D meshes of tetrahedra

(3-simplexes) that are frequently employed in discontinuous Finite Element methods.

Refer to Appendix B for details regarding this topic.

7.3.3 Enforcement of Quadrature Rule Requirements

Evidently, the objective is to find the particular rule (or set of rules) that satisfies

the requirements of the previous section. Of these requirements, it is perhaps most

important to find a quadrature rule which satisfies the first one, namely finding a

quadrature rule which integrates a polynomial of the highest possible degree and

minimizes the approximation error ǫ (equation (7.4)) for a given number of quadrature


points Np. More precisely, ǫ ∼ O (δn) must be minimized, where n is the highest

possible order, and where δ is the ‘characteristic’ length of edges between neighboring

vertices vk and vk+1 of the d-simplex (where k ≤ d). In order to begin finding a

quadrature rule that meets this criterion, consider approximating f (x) with a nth

order Taylor series expansion about the centroid x0 of Ω (denoted by fn (x)) where

fn (x) ≡ f (x) +O (δn) , (7.9)

so that equation (7.8) becomes

F =1

d!det (B)

[Np∑

i=1

wi fn

(d+1∑

j=1

ai,jvj

)]+ O

(δn+d

). (7.10)

Setting equation (7.10) aside for the moment, consider integrating both sides of equa-

tion (7.9) over the domain Ω in order to obtain the following expression for F

F =

∫

Ω

fn (x) dΩ+O(δn+d

)≡ Fn +O

(δn+d

). (7.11)

Upon substituting F and F from equations (7.10) and (7.11), and V (Ω) from equa-

tion (7.5) into equation (7.4), one obtains the following expression for the error term ǫ

ǫ =Fn d!

det (B)−[

Np∑

i=1

wi fn

(d+1∑

j=1

ai,jvj

)]+O (δn) ,

(7.12)

which is equivalent to

ǫ = ǫn +O (δn) , (7.13)

where

ǫn ≡Fn d!

det (B)−[

Np∑

i=1

wi fn

(d+1∑

j=1

ai,jvj

)]. (7.14)

It is well-known that ǫn is proportional to δn [77]. Therefore, upon combining this

result and the result of equation (7.13), one concludes that

ǫ ∼ O (δn) , (7.15)


as expected.

From this analysis, it can be seen that the quadrature rule error scales with the order

of the term ǫn. Therefore, satisfying the first quadrature rule requirement simplifies

to optimizing the quadrature point locations and weights such that ǫn is minimized

for the maximum possible value of n.

The remaining quadrature rule requirements can be satisfied as follows. The sec-

ond requirement states that the quadrature weights must be non-negative, which is

equivalent to requiring that

wi ≥ 0. (7.16)

In addition, the third requirement states that the quadrature points must not lie

outside of the d-simplex, which is equivalent to placing the following conditions on

the coefficients ai,j

ai,j ≥ 0,

d+1∑

j=1

ai,j = 1. (7.17)

The fourth requirement states that the quadrature points must be arranged in a sym-

metric fashion within the d-simplex. In order to satisfy this requirement, one must

enforce additional constraints on the coefficients ai,j. Prior to formulating these con-

straints, it is convenient to first formulate a symmetric d-simplex on which to enforce

them. Of course, symmetry constraints may be enforced on any d-simplex, as a linear

mapping to transform quadrature points from the initial simplex to any other sim-

plex of the same dimension can always be defined. However, it is more convenient to

construct these constraints on the standard, equilateral d-simplex (denoted by ΩS)

whose centroid (x0) is located at the origin. Evidently, ΩS is convenient to utilize

because it has d!-fold symmetry in the sense that all of the d! unique, affine, sym-

metric transformations that are possible for the d-simplex merely result in ΩS being

transformed unto itself. However, it also possesses some additional useful properties.

In particular, the ‘characteristic’ edge length δ is straightforward to define, as the

length of each edge is identical. Furthermore, because x0 is located at the origin,

one is ensured that Taylor series expansions about x0 are simple to formulate, as all

terms which contain components of the vector (x− x0)m can be reduced to contain

only components of xm.


Now, having established the useful properties of the equilateral d-simplex ΩS, it is

important to examine the analytical form for the equilateral 2-simplex and 3-simplex

that will serve as the focus for the remainder of this chapter. The equilateral 2-simplex

with edge length δ is formed from the following three vertices

v1 = δ

(−12,−√3

6

)

v2 = δ

(1

2,−√3

6

)

v3 = δ

(0,

√3

3

). (7.18)

Upon substituting the vertices from equation (7.18) into equation (7.5), one finds that

the 2-simplex has a volume of V (ΩS) = δ2√3/4. In addition, it is useful to note that

equation (7.2) can be reformulated on the 2-simplex as follows

F =

∫ y1

y0

∫ x1

x0

f (x, y)dx dy, (7.19)

where the limits of integration x0, x1, y0, and y1 are defined as follows

x0 = −√3

3

(√3

3δ − y

)x1 =

√3

3

(√3

3δ − y

)(7.20)

y0 = −√3

6δ y1 =

√3

3δ. (7.21)


The equilateral 3-simplex with edge length δ is formed from the following four vertices

v1 = δ

(−12,−√3

6,−√6

12

)

v2 = δ

(1

2,−√3

6,−√6

12

)

v3 = δ

(0,

√3

3,−√6

12

)

v4 = δ

(0, 0,

√6

4

). (7.22)

Upon substituting the vertices from equation (7.22) into equation (7.5), one finds that

the 3-simplex has a volume of V (ΩS) = δ3√2/12. In addition, it is useful to note

that equation (7.2) can be reformulated on the 3-simplex as follows

F =

∫ z1

z0

∫ y1

y0

∫ x1

x0

f (x, y) dx dy dz, (7.23)

where the limits of integration x0, x1, y0, y1, z0, and z1 are defined as follows

x0 = −√3

3

[√2

2

(√6

4δ − z

)− y]

x1 =

√3

3

[√2

2

(√6

4δ − z

)− y]

(7.24)

y0 = −√2

4

(√6

4δ − z

)y1 =

√2

2

(√6

4δ − z

)(7.25)

z0 = −√6

12δ z1 =

√6

4δ. (7.26)

Having established precise definitions for the d = 2 and d = 3 standard simplexes,

symmetry constraints on the coefficients ai,j for each of these simplexes may now

be formulated. For convenience, one may impose the constraints by introducing pa-

rameters αk which symmetrically parameterize the point locations xi, and indirectly

via equation (7.7) parameterize (and constrain) the ai,j’s. In this way, the problem

simplifies to finding the symmetry parameters αk, from which xi (αk) and ai,j (xi)

immediately follow. This is convenient because there are frequently far fewer pa-

rameters αk than there are coefficients ai,j. For example, consider the symmetric


parameterization of the 2-simplex with Np = 15 points illustrated in Figure (7.3).

Figure 7.3: Np = 15 points symmetrically arranged on the 2-simplex (triangle).

In this case, there are a total of 15 (d+ 1) = 45 unknown coefficients ai,j, whereas

there are only five unknown parameters αk (where k = 1, . . . , 5). The reduction in the

number of variables arises naturally from the symmetry requirements. In particular,

due to symmetry, six of the fifteen points are required to lie along lines between

the centroid x0 and the vertices v1, v2, and v3 of the 2-simplex. Because there are

three vertices, there are a total of two points along lines between each vertex and the

centroid, and there are in turn two symmetry parameters α1 and α2 which define the

point locations xi as follows

xi = α1vi where i = 1, 2, 3

xi = α2vi−3 where i = 4, 5, 6. (7.27)

Furthermore, there are three points which are required to lie along lines between the

centroid and the midpoints of the simplex edges. The locations of these points can

be defined using a single symmetry parameter α3 as follows

xi = α3 (vj + vm) where i = 7, 8, 9, 1 ≤ j,m ≤ 3, j 6= m. (7.28)

Finally, there are six points which must lie along lines which emanate from the cen-

troid and intersect the simplex edges at locations between the edge midpoints and the


vertices. The locations of these points can be defined using two symmetry parameters

α4 and α5 as follows

xi = α4vj + α5vm where i = 10, 11, 12

xi = α5vj + α4vm where i = 13, 14, 15. (7.29)

Equations (7.27) - (7.29) provide a complete description of the symmetric point lo-

cations in terms of the five symmetry variables α1, . . . , α5.

Next, in order to recover the coefficients ai,j from the parameters αk, one may refor-

mulate equation (7.7) in terms of the following linear system

(v1

1

)(v2

1

)(v3

1

)ai,1ai,2ai,3

=

xiyi1

, (7.30)

where the fact that ai,1 + ai,2 + ai,3 = 1 (equation (7.17)) has been used.

A similar procedure for enforcing the symmetry constraints can be employed on the

3-simplex. For example, consider the symmetric arrangement of Np = 84 points on

the 3-simplex illustrated in Figure (7.4).

Figure 7.4: Np = 84 points symmetrically arranged on the 3-simplex (tetrahedron).


There are a total of 84 (d+ 1) = 336 parameters ai,j, however, it turns out that

symmetry constraints need only be enforced on fourteen parameters αk (where k =

1, . . . , 14). In particular, due to symmetry, eight of the eighty-four points must be

located along lines from the centroid x0 to each of the vertices v1, v2, v3, and v4.

The locations of these points can be expressed in terms of the symmetry parameters

α1 and α2 as follows

xi = α1vi where i = 1, 2, 3, 4

xi = α2vi−4 where i = 5, 6, 7, 8. (7.31)

In addition, twelve points must located along lines between the centroid and the

midpoints of the simplex edges. The locations of these points can be expressed in

terms of the symmetry parameters α3 and α4 as follows

xi = α3 (vj + vm) where i = 9, . . . , 14 1 ≤ j,m ≤ 4, j 6= m

xi = α4 (vj + vm) where i = 15, . . . , 20. (7.32)

Furthermore, twenty-four points must be located along lines which emanate from the

centroid and intersect the simplex edges at locations between the edge midpoints and

the vertices. The locations of these points can be expressed in terms of the symmetry

parameters α5, α6, α7, and α8 as follows

xi = α5vj + α6vm where i = 21, . . . , 26

xi = α6vj + α5vm where i = 27, . . . , 32

xi = α7vj + α8vm where i = 33, . . . , 38

xi = α8vj + α7vm where i = 39, . . . , 44. (7.33)

Finally, the locations of the remaining forty points are composed from all possible

unique, symmetric, linear combinations of the three vertices of each face. The loca-

tions of these points can be expressed in terms of the symmetry parameters α9, . . . , α14


as follows

xi = α9 (vj + vm + vl) where i = 45, 46, 47, 48

1 ≤ j,m, l ≤ 4, j 6= m 6= l

xi = α10vj + α11 (vm + vl) where i = 49, 50, 51, 52

xi = α10vl + α11 (vj + vm) where i = 53, 54, 55, 56

xi = α10vm + α11 (vl + vj) where i = 57, 58, 59, 60

xi = α12vj + α13vm + α14vl where i = 61, 62, 63, 64

xi = α12vl + α13vj + α14vm where i = 65, 66, 67, 68

xi = α12vm + α13vl + α14vj where i = 69, 70, 71, 72

xi = α12vj + α14vm + α13vl where i = 73, 74, 75, 76

xi = α12vl + α14vj + α13vm where i = 77, 78, 79, 80

xi = α12vm + α14vl + α13vj where i = 81, 82, 83, 84. (7.34)

Equations (7.31) - (7.34) provide a complete description of the symmetric point lo-

cations in terms of the fourteen symmetry variables α1, . . . , α14.

One may recover the coefficients ai,j from the parameters αk by solving the following

linear system of equations

(v1

1

)(v2

1

)(v3

1

)(v4

1

)

ai,1ai,2ai,3ai,4

=

xiyizi1

, (7.35)

where the fact that ai,1 + ai,2 + ai,3 + ai,4 = 1 (equation (7.17)) has been used.

Finally, the fifth requirement (which is a soft requirement) states that the quadrature

points on the 2-simplex should be arranged in a way that is consistent with the 2-

SCP layering. Since this is not a strict requirement, it need not be enforced directly.

However, it can be encouraged by employing 2-SCP configurations (or perturbed

2-SCP configurations) as initial conditions for the optimization process utilized in

identifying the quadrature rules. This process will be discussed in more detail in the

next section.


7.4 Computation of Optimal Quadrature Rules

Quadrature rules on the 2 and 3-simplexes can be obtained by solving a series of

constrained optimization problems. In each of these problems, the symmetry variables

(αk’s) and the quadrature weights (wi’s) are treated as unknowns. These unknowns

are subject to the inequality and equality constraints put forth in section 7.3.3, which,

for convenience, can be reformulated as follows.

7.4.1 Reformulating the Inequality and Equality Constraints

Inequality constraints on the unknowns are obtained from equations (7.16) and (7.17).

The constraints in equation (7.16) apply directly to the wi’s, however the constraints

in equation (7.17) apply to the ai,j’s and cannot be easily applied to the αk’s. Nev-

ertheless, in practice, it is frequently sufficient to weakly enforce the constraints in

equation (7.17) by enforcing the following constraints on the αk’s

1 ≥ αk ≥ 0. (7.36)

These constraints do not ensure that the quadrature points remain inside of the d-

simplex, but they encourage this by ensuring that all points lie inside of the d-sphere

of radius d|v1| centered at the centroid (x0) of the d-simplex.

Equality constraints on the unknowns are obtained by evaluating ǫn in equation (7.14).

In particular, in order to obtain a quadrature rule with truncation error of order

δn, it is required that all error terms of order less than δn vanish, and therefore,

equality constraints are obtained by insisting that all error terms ǫm vanish for m =

0, . . . , (n− 1). The requirement that ǫm = 0 can be written in matrix form as follows

ǫm = δm (∂fm)T Cm um = 0, (7.37)

where (∂fm)T is a vector of partial derivative terms of order m, Cm is a matrix

of constant coefficients (which need not be full rank), and um is a vector containing

monomial terms of the unknown αk’s and wi’s. In order to ensure that equation (7.37)

holds, one requires that

Cm um = 0, (7.38)


or equivalently

Cm um = 0, (7.39)

where Cm is the matrix that arises from reducingCm to row echelon form and removing

all rows of zeros. Evidently rank (Cm) = rank (Cm) if and only if Cm is full rank.

Now, the matrices Cm for m = 0, . . . , (n− 1) can be combined together into the

matrix C, and similarly the vectors um can be combined together in order to form the

vector u. Thus, the final nonlinear system of equality constraints for the quadrature

rule of order n becomes Cu = 0.

Evidently, the number of rows in C is equivalent to the number of equality constraints

that must be satisfied. In order to ensure that it is possible for the system of equations

Cu = 0 to have a solution, it is necessary for the number of unknowns (αk’s and wi’s)

to equal or exceed the number of equations. Therefore, for each of the quadrature rules

obtained in this work, the value of n was chosen such that the number of unknowns

was greater than or equal to the number of unique equality constraints associated

with the error terms of order less than n.

Having established a methodology for forming the constraints and identifying the

order of a quadrature rule n, one may proceed to form an objective function and

solve the resulting optimization problem.

7.4.2 Forming the Objective Function and Solving the Opti-

mization Problem

In following the approach of [77], the objective function Jn can be formed as the sum

of the squares of all unique equality constraints of degree n, i.e.

Jn = uTnCTn Cn un. (7.40)


In accordance with this definition, the constrained optimization problem for each

quadrature rule of degree n becomes

minimize Jn = uTnCTn Cn un

subject to Cu = 0

wi ≥ 0, 1 ≥ αk ≥ 0. (7.41)

In order to obtain the quadrature rules in this work, equation (7.41) was solved iter-

atively using a sequential quadratic programming (SQP) algorithm. This algorithm

was chosen due to its ability to treat equality and inequality constraints of linear and

nonlinear type. In accordance with the standard SQP approach (described in [91],

[92], [93], and [94]), all constraints were incorporated into a Lagrangian function, and

approximate Hessians of this function (computed via the Broyden-Fletcher-Goldfarb-

Shanno (BFGS) method [95], [96], [97], and [98]) were used to create a sequence of

quadratic programming subproblems. The subproblems were solved using an Active-

Set strategy (described in [99] and [100]) in order to obtain search directions, and

a line search algorithm (utilizing a merit function described in [101] and [102]) was

employed in order to identify step lengths. In turn, these step lengths and search

directions were used to successively update the solution.

The converged, optimal solutions were sensitive to initial conditions, indicating that

the optimization problems possessed multiple local minima. In an effort to obtain

the global minima, 106 initial conditions were employed during the optimization of

each quadrature rule with Np = 3, 6, 10, 15, 21, and 28 points on the 2-simplex,

105 initial conditions were employed for the quadrature rules with Np = 36, 45, 55,

and 66 points on the 2-simplex, and 2.5 × 104 initial conditions were employed for

the quadrature rule with Np = 84 points on the 3-simplex. These initial conditions

were obtained by subjecting the d-SCP configurations to random perturbations. The

optimal quadrature rules obtained through this process are discussed in the next

section.

7.5 Results

Table (C.1) in Appendix C contains a set of optimal quadrature rules on the 2-simplex

with Np = 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and 66 points. Note that the quadrature


rule parameters in the table are given to fifteen decimal places, in accordance with

the limits on the precision of the computations. Figures (7.5)–(7.7) illustrate the

arrangements of points for each of the rules. From the figures, it is apparent that the

points are arranged symmetrically on the 2-simplex, the points lie within the interior

of the 2-simplex, and all point arrangements can be partitioned into rows, and are

thus consistent with the corresponding 2-SCP configurations.

(a) Np = 1 (b) Np = 3

(c) Np = 6 (d) Np = 10

Figure 7.5: Quadrature point locations for the quadrature rules with 2-SCP structure andNp = 1, 3, 6, and 10 points for the triangle (the 2-simplex). The size of each point is scaledby the absolute value of the logarithm of the associated quadrature weight.


(a) Np = 15 (b) Np = 21

(c) Np = 28 (d) Np = 36

Figure 7.6: Quadrature point locations for the quadrature rules with 2-SCP structure andNp = 15, 21, 28, and 36 points for the triangle (the 2-simplex). The size of each point isscaled by the absolute value of the logarithm of the associated quadrature weight.


(a) Np = 45 (b) Np = 55

(c) Np = 66

Figure 7.7: Quadrature point locations for the quadrature rules with 2-SCP structure andNp = 45, 55, and 66 points for the triangle (the 2-simplex). The size of each point is scaledby the absolute value of the logarithm of the associated quadrature weight.

For the sake of completeness, the requirement on consistency with the 2-SCP config-

urations was relaxed, and an alternate set of quadrature rules with Np = 45, 55, and

66 points was obtained on the 2-simplex. These quadrature rules (summarized by Ta-

ble (C.2) and Figure (7.8)) are more optimal than their counterparts in Table (C.1),

as they produce roughly an additional order of magnitude decrease in the objective

function.


(a) Np = 45 (b) Np = 55

(c) Np = 66

Figure 7.8: Quadrature point locations for the quadrature rules with non-(2-SCP) structureand Np = 45, 55, and 66 points for the triangle (the 2-simplex). The size of each point isscaled by the absolute value of the logarithm of the associated quadrature weight.

Nevertheless, both sets of quadrature rules produce the same order of truncation error

(cf. Table (7.1)), as the magnitude of the objective function only governs the size of

the constant that multiplies the truncation error term, and not the order of the term

itself. Recall that the order of the truncation error term is determined by the number

of degrees of freedom and the number of nonlinear constraints, which are identical

for both sets of quadrature rules.


Rules that are consistent with 2-SCP configurationsNl Np Order1 1 12 3 33 6 54 10 65 15 86 21 97 28 118 36 139 45 1510 55 1611 66 18

Rules that are inconsistent with 2-SCP configurationsNl Np Order9 45 1510 55 1611 66 18

Table 7.1: Order of accuracy estimates for quadrature rules on the 2-simplex with Np = 1,3, 6, 10, 15, 21, 28, 36, 45, 55, and 66 points. Estimates are provided for quadrature rulesthat are consistent and inconsistent with the 2-SCP configurations.

Based on the truncation errors in Table (7.1), the quadrature rules in Tables (C.1)

and (C.2) are capable of exactly integrating all polynomials of degree 0 ≤ p ≤ 17 on

the 2-simplex.

Finally, Table (C.3) contains the optimal quadrature rule with Np = 84 points on the

3-simplex. Figure (7.9) illustrates this rule.


Figure 7.9: Quadrature point locations for the quadrature rule with Np = 84 points for thetetrahedron (the 3-simplex). The size of each point is scaled by the associated quadratureweight.

The truncation error for this rule was found to be O(δ10), enabling it to exactly

integrate polynomials of degree ≤ 9 on the 3-simplex.

Part III

Nonlinear Numerical Experiments

183

Chapter 8

Governing Equations

184

CHAPTER 8. GOVERNING EQUATIONS 185

8.1 Preamble

In order to evaluate the nonlinear behavior of the VCJH schemes, they were employed

to solve the nonlinear Navier-Stokes (NS) equations on 2D and 3D grids of triangular

and tetrahedral elements. In order to help ensure the stability of the schemes, the

flux points and solution points for the VCJH schemes on each grid were placed at the

locations of the SCP quadrature points from the previous chapter, unless otherwise

indicated.

Before proceeding further, it should be noted that the NS equations in 2D and 3D

are of particular interest because they have the capability to accurately model the

flows of Newtonian fluids that arise in many real-world applications. In addition,

the NS equations are (in some sense) a natural extension of the linear and nonlin-

ear advection-diffusion equations that were used in previous chapters to develop the

theoretical foundation of the VCJH schemes.

8.2 Definitions of the Navier-Stokes Equations

The NS equations in 2D and 3D can be written as follows

∂U

∂t+∇ · F(U,∇U) = 0, (8.1)

where U represents the conserved variables (which are scalars) and F represents

the flux vector that is composed from inviscid and viscous parts: F = Finv (U) −Fvisc (U,∇U). In 2D, the conserved variables are defined as follows

U =

ρρuρvE

, (8.2)

where ρ = ρ (x, y, t) is the density, u = u (x, y, t) and v = v (x, y, t) are the velocity

components, E = p/ (γ − 1) + (1/2)ρ (u2 + v2) is the total energy, p = p (x, y, t) is

the pressure, and γ is the ratio of specific heats. In addition, the inviscid and viscous

fluxes in 2D can be defined in terms of their components along the x and y coordinate

directions, i.e. Finv = (finv, ginv) and Fvisc = (fvisc, gvisc). Here, the inviscid flux


components are defined such that

finv =

ρuρu2 + pρuv

u(E + p)

, ginv =

ρvρuv

ρv2 + pv(E + p)

, (8.3)

and the viscous flux components are defined such that

fvisc = µ

02ux + λ(ux + vy)

vx + uyu[2ux + λ(ux + vy)] + v(vx + uy) +

Cp

PrTx

,

gvisc = µ

0vx + uy

2vy + λ(ux + vy)

v[2vy + λ(ux + vy)] + u(vx + uy) +Cp

PrTy

, (8.4)

where µ is the dynamic viscosity, λ is the bulk viscosity coefficient, T = p/ (ρR) is

the temperature, R is the gas constant, Cp is the specific heat capacity at constant

pressure, and Pr is the Prandtl number. It should be noted that the terms with

subscripts x and y in equation (8.4) signify first derivatives in x and y (for example

Ty =∂T∂y).

In 3D, the conserved variables U are defined as follows

U =

ρρuρvρwE

, (8.5)

where ρ = ρ (x, y, z, t), u = u (x, y, z, t), v = v (x, y, z, t), w = w (x, y, z, t) (where w

is the velocity component in the z direction), E = p/ (γ − 1)+ (1/2)ρ (u2 + v2 + w2),

and p = p (x, y, z, t). In addition, the inviscid and viscous fluxes in 3D can be

defined in terms of their components along the coordinate directions, i.e. Finv =


(finv, ginv, hinv) and Fvisc = (fvisc, gvisc, hvisc), where

finv =

ρuρu2 + pρuvρuw

u(E + p)

, ginv =

ρvρuv

ρv2 + pρvw

v(E + p)

, hinv =

ρwρuwρvw

ρw2 + pw(E + p)

, (8.6)

and

fvisc = µ

02ux + λ(ux + vy + wz)

vx + uywx + uz

u[2ux + λ(ux + vy + wz)] + v(vx + uy) + w(wx + uz) +Cp

PrTx

,

gvisc = µ

0vx + uy

2vy + λ(ux + vy + wz)wy + vz

v[2vy + λ(ux + vy + wz)] + u(vx + uy) + w(wy + vz) +Cp

PrTy

,

hvisc = µ

0wx + uzwy + vz

2wz + λ(ux + vy + wz)

w[2wz + λ(ux + vy + wz)] + u(wx + uz) + v(wy + vz) +Cp

PrTz

. (8.7)

Before proceeding further, it is useful to rewrite the 2D and 3D versions of the NS

equations in non-dimensional form. Non-dimensionalization reduces the round-off

errors associated with numerical computations by ensuring that certain critical quan-

tities (e.g. the density, the velocities, the spatial coordinates, and the temperature)

are normalized so that they are O (1). In order to facilitate the computation of non-

dimensional quantities, it is necessary to define dimensional reference values of the

density (ρ∞), the flow speed (u∞), the characteristic length scale (L), and the tem-

perature (T∞). Using these quantities, the following additional dimensional reference

quantities can be computed

t∞ =L

u∞, p∞ = ρ∞u

2∞, µ∞ = ρ∞u∞L. (8.8)


Now that these quantities are specified, it is straightforward to construct non-dimensional

forms of all the quantities in the NS equations as follows

ρ⋆ =ρ

ρ∞, t⋆ =

t

t∞, x⋆ =

x

L, y⋆ =

y

L, z⋆ =

z

L, (8.9)

T ⋆ =T

T∞, u⋆ =

u

u∞, v⋆ =

v

u∞, w⋆ =

w

u∞, (8.10)

p⋆ =p

p∞= ρ⋆R⋆T ⋆, R⋆ =

RT∞u2∞

, µ⋆ =µ

µ∞

, (8.11)

E⋆ =p⋆

γ − 1+

1

2ρ⋆(u⋆

2

+ v⋆2

+ w⋆2), C⋆

p =CpT∞u2∞

, (8.12)

where each non-dimensional quantity is denoted with a (⋆) superscript. Upon substi-

tuting equations (8.9) – (8.12) into the expressions for the conservative variables and

the flux vectors, one obtains a set of non-dimensional conservative variables denoted

by U⋆, and a set of non-dimensional fluxes denoted by F⋆. Utilizing these quantities,

the non-dimensional NS equations can be written as follows

∂U⋆

∂t⋆+∇⋆ · F⋆(U⋆,∇⋆U⋆) = 0, (8.13)

where ∇⋆ =(

∂∂x⋆ ,

∂∂y⋆, ∂∂z⋆

)is the non-dimensional form of the gradient operator.

It should be noted that the behavior of solutions to equation (8.13) is (to a large

extent) dictated by the Prandtl number Pr (discussed previously), and the Reynolds

(Re) and Mach (M) numbers, which are non-dimensional quantities that are defined

as follows

Re =ρ∞u∞L

µ∞, M =

u∞√γRT∞

. (8.14)

Finally, it should be noted that equation (8.13) contains second derivatives of the non-

dimensional conservative variables and can therefore be classified as a ‘2nd-order’

system of PDEs. However, this system can easily be reformulated as a first-order

system by following the approach of sections 3.2, 4.2, and 6.2, i.e. by eliminating∇⋆U⋆

from equation (8.13) and replacing it with the non-dimensional auxiliary variable


denoted by Q⋆ as follows

∂U⋆

∂t⋆+∇⋆ · F⋆(U⋆, Q⋆) = 0, (8.15)

Q⋆ −∇⋆U⋆ = 0. (8.16)

This operation transforms equation (8.13) into a form that is more amenable to

treatment by the VCJH schemes.

8.3 Definitions of Viscosity

In this section, the viscosity µ which appears in the NS equations will be more pre-

cisely defined.

It is well-known that the local value of viscosity in a compressible flow of gas depends

on the type of gas and on the local value of the temperature of the flow. In fact,

the assumption that µ = const is only defensible when the Mach number is such

that the flow is in the incompressible range (e.g. when there is less than 5 percent

compressibility as is the case for M ≤ 0.3). In order to model the variations in

viscosity as a function of temperature, it is common practice to use Sutherland’s law

which takes the following form

µ = µref

(Tref + S

T + S

)(T

Tref

)3/2

, (8.17)

where µref is a reference value of the viscosity at the reference temperature Tref , and

S is ‘Sutherland’s temperature’ which is an empirically determined temperature that

depends on the type of gas.

Now, having formulated a dimensional definition for µ, it is convenient to rewrite this

definition in non-dimensional form in accordance with equation (8.11), as follows

µ⋆ =µ

µ∞=

1

Reref

(T ⋆ref + S⋆

T ⋆ + S⋆

)(T ⋆

T ⋆ref

)3/2

(8.18)

where

Reref =ρ∞u∞L

µref, T ⋆

ref =TrefT∞

, S⋆ =S

T∞. (8.19)


This definition of the viscosity was used frequently in the numerical experiments

described in the next chapter.

Chapter 9

Results of Numerical Experiments

191

CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 192

9.1 Couette Flow

In what follows, the VCJH schemes are used to solve the NS equations for the ‘Couette

flow’ problem. Couette flow occurs between two infinite walls which are unbounded

in the x − z plane, and separated by a distance of H in the y direction. One wall

is stationary with a temperature of Tw, and the other wall moves in the x direction

with a speed of Uw and (the same) temperature of Tw. For µ = const, the flow has an

analytical solution where p = const and the total energy E takes the following form

E = p

[1

γ − 1+

U2w

2R

(yH

)2

Tw + Pr U2w

2Cp

(yH

) [1−

(yH

)]]. (9.1)

It is common practice to generate approximate solutions for Couette flow on finite

domains with periodic boundary conditions imposed. In following this approach,

experiments were performed in 2D on the rectangular domain [−1, 1] × [0, 1], with

periodic conditions imposed on the left and right boundaries, and isothermal wall

conditions imposed on the upper and lower boundaries. In addition, experiments

were performed in 3D on the cuboid domain [−1, 1] × [0, 1] × [−1, 1], with periodic

conditions imposed on the left, right, front, and back boundaries, and isothermal

wall conditions imposed on the upper and lower boundaries. Figure (9.1) shows the

computational domains and their associated boundary conditions in 2D and 3D.


Periodic

(left)

Periodic

(right)

Isothermal wall

(top)

Isothermal wall

(bottom)

(a) 2D rectangular domain

Periodic

(left)

Periodic

(back)

Isothermal wall

(top)

Isothermal wall

(bottom)

Periodic

(right)

Periodic

(front)

(b) 3D cuboid domain

Figure 9.1: Boundary conditions for Couette flow on 2D and 3D computational domains.


For the upper and lower isothermal walls, the temperature was given the value of

T = Tw = 300K and the pressure was held constant. In addition, no-slip conditions

were imposed on the lower walls, i.e. the velocity components on the lower wall were

given the values u = 0 and v = 0 in 2D, and u = 0, v = 0, and w = 0 in 3D. Similarly,

no-slip conditions were imposed on the upper walls (each of which were required to

move at a speed of Uw in the x direction), i.e. the velocity components on the upper

wall were given the values u = Uw and v = 0 in 2D, and u = Uw, v = 0, and w = 0

in 3D.

The boundary conditions (described above) were imposed on discretizations of the

rectangular and cuboid domains in 2D and 3D. The rectangular domain in 2D was

discretized by forming 2N × N regular quadrilateral meshes and then splitting these

meshes into grids with N = 4N2 triangle elements. In this manner, structured

grids with N = 2, 4, 8, and 16 were formed. In addition, a complementary set of

unstructured triangular grids with N = 16, 64, 256, and 1024 elements was formed.

Figure (9.2) shows examples of the structured and unstructured triangular grids for

the cases of N = 4 and N = 64, respectively.


x

y

-1 -0.5 0 0.5 10

0.5

1

(a) Structured grid

x

y

-1 -0.5 0 0.5 10

0.5

1

(b) Unstructured grid

Figure 9.2: Structured and unstructured triangular grids for the cases of N = 4 and N = 64.

Similarly, the cuboid domain in 3D was discretized by forming 2N × N × 2N regular

hexahedral meshes and then splitting these meshes into grids with N = 24N3 tetra-

hedron elements. In this manner, structured grids with N = 1, 2, 3, 4, 6, 8, 12, and

16 were formed. In addition, a complementary set of unstructured tetrahedral grids

with N = 24, 192, 648, 1536, 5184, 12288, 41472, and 98304 elements was formed.

Figures (9.3) and (9.4) show the structured tetrahedral grid with N = 4, as well as

the unstructured tetrahedral grid with N = 1536 elements.


X

Y

Z

(a) Structured grid

(b) Cutaway of structured grid

Figure 9.3: Structured tetrahedral grid with N = 4.


X

Y

Z

(a) Unstructured grid

(b) Cutaway of unstructured grid

Figure 9.4: Unstructured tetrahedral grid with N = 1536 elements.

At time t = 0, the flow on each domain was initialized with gas parameters Pr = 0.72

and γ = 1.4, and with velocities of u = Uw and v = 0 in 2D and u = Uw, v = 0,

and w = 0 in 3D, where Uw was chosen such that the Mach number M = 0.2, and

the Reynolds number (based on H) was 200. In both 2D and 3D, the solution was

marched forward in time using the RK54 approach and, at each time-step, the inviscid

and viscous numerical fluxes were computed using the Rusanov approach (with λ = 1)

and the LDG approach (with β = ±0.5n− and τ = 0.1 in 2D and β = ±0.5n− and


τ = 1.0 in 3D). Each simulation was terminated after the residual reached machine

zero. In 2D, results were obtained on structured grids with N = 2, 4, 8, and 16 and on

unstructured grids with N = 16, 64, 256, and 1024 for polynomial orders p = 2 and 3.

In 3D, results were obtained on structured grids with N = 2, 3, 4, 6, 8, 12, and 16

for p = 2 and 3, N = 2, 3, 4, and 6 for p = 4, and N = 1, 2, and 3 for p = 5, and on

the unstructured grids with N = 192, 648, 1536, 5184, 12288, 41472, and 98304 for

p = 2 and 3, N = 192, 648, 1536, and 5184 for p = 4, and N = 24, 192, and 648 for

p = 5.

9.1.1 Orders of Accuracy and Explicit Time-Step Limits (2D)

Solutions in 2D were obtained for four different VCJH schemes defined by the fol-

lowing pairings of c and κ: (c = cdg, κ = κdg), (c = csd, κ = κsd), (c = chu, κ = κhu),

and (c = c+, κ = κ+). Tables (9.1) – (9.6) contain the absolute errors, orders of

accuracy, and maximum time-step limits for each of these schemes, and Figures (9.5)

and (9.6) show the density and energy contours obtained via the scheme with c = c+

and κ = κ+ on the unstructured mesh with N = 64 elements for the case of p = 3.

c, κ Grid L2 err. L2s err. O(L2) O(L2s)

cdg, κdg N = 2 3.02e-05 5.58e-04

N = 4 3.19e-06 1.09e-04 3.24 2.36

N = 8 3.61e-07 2.37e-05 3.14 2.20

N = 16 4.37e-08 6.11e-06 3.05 1.95

csd, κsd N = 2 3.02e-05 4.94e-04

N = 4 3.19e-06 9.93e-05 3.24 2.32

N = 8 3.61e-07 2.19e-05 3.14 2.18

N = 16 4.37e-08 5.61e-06 3.05 1.96

chu, κhu N = 2 3.02e-05 4.61e-04

N = 4 3.19e-06 9.44e-05 3.24 2.29

N = 8 3.61e-07 2.10e-05 3.14 2.17

N = 16 4.37e-08 5.34e-06 3.05 1.98

c+, κ+ N = 2 3.02e-05 4.20e-04

N = 4 3.19e-06 8.90e-05 3.24 2.24

N = 8 3.61e-07 2.01e-05 3.14 2.15

N = 16 4.39e-08 5.04e-06 3.04 1.99

Table 9.1: Accuracy of VCJH schemes for the Couette flow problem on structured triangulargrids, for the case of p = 2. The inviscid and viscous numerical fluxes were computed usinga Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.



cdg, κdg N = 2 9.54e-07 2.80e-05

N = 4 5.99e-08 3.93e-06 3.99 2.83

N = 8 3.78e-09 5.01e-07 3.99 2.97

N = 16 2.40e-10 6.11e-08 3.97 3.04

csd, κsd N = 2 9.53e-07 2.76e-05

N = 4 5.98e-08 3.88e-06 3.99 2.83

N = 8 3.77e-09 4.96e-07 3.99 2.97

N = 16 2.39e-10 6.05e-08 3.98 3.04

chu, κhu N = 2 9.53e-07 2.77e-05

N = 4 5.97e-08 3.89e-06 4.00 2.83

N = 8 3.76e-09 4.96e-07 3.99 2.97

N = 16 2.39e-10 6.05e-08 3.98 3.04

c+, κ+ N = 2 9.52e-07 2.81e-05

N = 4 5.97e-08 3.91e-06 4.00 2.84

N = 8 3.76e-09 4.99e-07 3.99 2.97

N = 16 2.38e-10 6.08e-08 3.98 3.04

Table 9.2: Accuracy of VCJH schemes for the Couette flow problem on structured triangulargrids, for the case of p = 3. The inviscid and viscous numerical fluxes were computed usinga Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.

c, κ Grid L2 err. L2s err. O(L2) O(L2s)cdg, κdg N = 16 6.32e-05 1.28e-03

N = 64 3.88e-06 1.38e-04 4.03 3.22N = 256 3.87e-07 2.54e-05 3.33 2.44N = 1024 4.51e-08 6.27e-06 3.10 2.02

csd, κsd N = 16 6.31e-05 1.15e-03N = 64 3.88e-06 1.27e-04 4.03 3.18N = 256 3.87e-07 2.36e-05 3.33 2.43N = 1024 4.50e-08 5.77e-06 3.10 2.03

chu, κhu N = 16 6.31e-05 1.09e-03N = 64 3.87e-06 1.22e-04 4.03 3.15N = 256 3.86e-07 2.27e-05 3.32 2.43N = 1024 4.50e-08 5.49e-06 3.10 2.05

c+, κ+ N = 16 6.30e-05 1.00e-03N = 64 3.86e-06 1.16e-04 4.03 3.11N = 256 3.86e-07 2.17e-05 3.32 2.41N = 1024 4.52e-08 5.19e-06 3.09 2.07

Table 9.3: Accuracy of VCJH schemes for the Couette flow problem on unstructured trian-gular grids, for the case of p = 2. The inviscid and viscous numerical fluxes were computedusing a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.



N = 64 7.77e-08 4.89e-06 4.50 3.33N = 256 4.22e-09 5.47e-07 4.20 3.16N = 1024 2.48e-10 6.27e-08 4.09 3.13

csd, κsd N = 16 1.75e-06 4.69e-05N = 64 7.76e-08 4.79e-06 4.50 3.29N = 256 4.20e-09 5.40e-07 4.21 3.15N = 1024 2.47e-10 6.21e-08 4.09 3.12

chu, κhu N = 16 1.75e-06 4.66e-05N = 64 7.75e-08 4.79e-06 4.50 3.28N = 256 4.20e-09 5.39e-07 4.21 3.15N = 1024 2.47e-10 6.21e-08 4.09 3.12

c+, κ+ N = 16 1.75e-06 4.65e-05N = 64 7.74e-08 4.81e-06 4.50 3.27N = 256 4.19e-09 5.41e-07 4.21 3.15N = 1024 2.46e-10 6.24e-08 4.09 3.12

Table 9.4: Accuracy of VCJH schemes for the Couette flow problem on unstructured trian-gular grids, for the case of p = 3. The inviscid and viscous numerical fluxes were computedusing a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.

p = 2 p = 3

cdg, κdg 3.19e-03 1.84e-03csd, κsd 4.03e-03 2.29e-03chu, κhu 4.58e-03 2.44e-03c+, κ+ 5.56e-03 2.67e-03

Table 9.5: Explicit time-step limits (∆tmax) of VCJH schemes on the structured triangulargrid with N = 8 for the Couette flow problem, for the cases of p = 2 and 3. The inviscidand viscous numerical fluxes were computed using a Rusanov flux with λ = 1 and a LDGflux with τ = 0.1 and β = ±0.5n−.

p = 2 p = 3

cdg, κdg 2.88e-03 1.57e-03csd, κsd 3.62e-03 1.93e-03chu, κhu 4.07e-03 2.06e-03c+, κ+ 4.96e-03 2.27e-03

Table 9.6: Explicit time-step limits (∆tmax) of VCJH schemes on the unstructured trian-gular grid with N = 256 for the Couette flow problem, for the cases of p = 2 and 3. Theinviscid and viscous numerical fluxes were computed using a Rusanov flux with λ = 1 anda LDG flux with τ = 0.1 and β = ±0.5n−.


Figure 9.5: Contours of density obtained using the VCJH scheme with c = c+ and κ = κ+on the unstructured grid with N = 64 for the case of p = 3. The inviscid and viscousnumerical fluxes were computed using a Rusanov flux with λ = 1 and a LDG flux withτ = 0.1 and β = ±0.5n−.

Figure 9.6: Contours of energy obtained using the VCJH scheme with c = c+ and κ = κ+on the unstructured grid with N = 64 for the case of p = 3. The inviscid and viscousnumerical fluxes were computed using a Rusanov flux with λ = 1 and a LDG flux withτ = 0.1 and β = ±0.5n−.

Absolute errors (and by extension orders of accuracy) were determined using L2 norms

and seminorms of the errors in the total energy E and its gradient ∇E. The time-step

limits (∆tmax) were determined using an iterative method on the structured grid with

N = 8 and on the unstructured grid with N = 256.

Tables (9.1) – (9.4) demonstrate that the expected order of accuracy is obtained for

all four schemes, for p = 2 and 3, on structured and unstructured grids. In addition,

Tables (9.5) and (9.6) demonstrate that increasing c and κ increases the maximum


time-step limit on structured and unstructured grids. In particular, for p = 2 on the

structured grid with N = 8, the scheme with c = c+ and κ = κ+ has ∆tmax = 5.56e-

03, while the collocation-based nodal DG scheme (recovered with c = 0 and κ = 0)

has ∆tmax = 3.19e-03. Similarly, for p = 2 on the unstructured grid with N = 256,

the scheme with c = c+ and κ = κ+ has ∆tmax = 4.96e-03, while the collocation-based

nodal DG scheme has ∆tmax = 2.88e-03.


Solutions in 3D were obtained for two different VCJH schemes defined by the following

pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Tables (9.7) – (9.16)

contain the absolute errors, orders of accuracy, and maximum time-step limits for

each of these schemes, and Figures (9.7) and (9.8) show the density and energy

contours obtained via the scheme with c = c+ and κ = κ+ on the unstructured mesh

with N = 1536 elements for the case of p = 3.


cdg, κdg N = 2 3.84e-05 8.05e-04

N = 3 1.04e-05 3.06e-04 3.21 2.38

N = 4 4.18e-06 1.57e-04 3.18 2.33

N = 6 1.17e-06 6.24e-05 3.15 2.27

N = 8 4.75e-07 3.31e-05 3.12 2.20

N = 12 1.35e-07 1.41e-05 3.09 2.10

N = 16 5.62e-08 7.97e-06 3.05 1.98

c+, κ+ N = 2 3.83e-05 6.11e-04

N = 3 1.04e-05 2.45e-04 3.22 2.25

N = 4 4.16e-06 1.29e-04 3.18 2.22

N = 6 1.16e-06 5.31e-05 3.16 2.19

N = 8 4.71e-07 2.87e-05 3.13 2.14

N = 12 1.34e-07 1.23e-05 3.10 2.08

N = 16 5.55e-08 6.94e-06 3.06 1.99

Table 9.7: Accuracy of VCJH schemes for the Couette flow problem on structured tetrahe-dral grids, for the case of p = 2. The inviscid and viscous numerical fluxes were computedusing a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.



cdg, κdg N = 2 1.22e-06 3.81e-05

N = 3 2.42e-07 1.19e-05 3.99 2.88

N = 4 7.68e-08 5.09e-06 3.99 2.94

N = 6 1.53e-08 1.52e-06 3.98 2.98

N = 8 4.87e-09 6.41e-07 3.98 3.01

N = 12 9.72e-10 1.88e-07 3.97 3.02

N = 16 3.10e-10 7.86e-08 3.98 3.03

c+, κ+ N = 2 1.21e-06 3.83e-05

N = 3 2.38e-07 1.19e-05 4.00 2.89

N = 4 7.53e-08 5.10e-06 4.00 2.94

N = 6 1.49e-08 1.52e-06 4.00 2.98

N = 8 4.70e-09 6.42e-07 4.00 3.00

N = 12 9.26e-10 1.89e-07 4.00 3.02

N = 16 2.93e-10 7.87e-08 4.00 3.04



cdg, κdg N = 2 2.68e-09 1.02e-07

N = 3 3.53e-10 1.91e-08 5.00 4.14

N = 4 8.39e-11 5.92e-09 4.99 4.07

N = 6 1.10e-11 1.18e-09 5.02 3.99

c+, κ+ N = 2 2.67e-09 8.96e-08

N = 3 3.51e-10 1.71e-08 5.01 4.08

N = 4 8.34e-11 5.37e-09 4.99 4.03

N = 6 1.11e-11 1.08e-09 4.97 3.95



cdg, κdg N = 1 1.01e-08 2.79e-07

N = 2 1.58e-10 9.28e-09 6.01 4.91

N = 3 1.35e-11 1.19e-09 6.06 5.07

c+, κ+ N = 1 1.01e-08 2.49e-07

N = 2 1.56e-10 8.38e-09 6.01 4.89

N = 3 1.36e-11 1.11e-09 6.03 4.99




N = 648 1.13e-05 3.40e-04 3.32 2.54N = 1536 4.44e-06 1.69e-04 3.26 2.42N = 5184 1.21e-06 6.53e-05 3.20 2.35N = 12288 4.90e-07 3.42e-05 3.16 2.25N = 41472 1.38e-07 1.43e-05 3.13 2.14N = 98304 5.69e-08 8.06e-06 3.07 2.00

c+, κ+ N = 192 4.35e-05 7.32e-04N = 648 1.13e-05 2.73e-04 3.33 2.43N = 1536 4.42e-06 1.39e-04 3.26 2.35N = 5184 1.21e-06 5.55e-05 3.20 2.27N = 12288 4.85e-07 2.96e-05 3.16 2.19N = 41472 1.36e-07 1.25e-05 3.14 2.12N = 98304 5.61e-08 7.01e-06 3.07 2.01

Table 9.11: Accuracy of VCJH schemes for the Couette flow problem on unstructuredtetrahedral grids, for the case of p = 2. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.


N = 648 2.83e-07 1.39e-05 4.12 3.01N = 1536 8.67e-08 5.78e-06 4.12 3.05N = 5184 1.64e-08 1.63e-06 4.10 3.12N = 12288 5.09e-09 6.68e-07 4.07 3.10N = 41472 9.92e-10 1.92e-07 4.03 3.07N = 98304 3.13e-10 7.96e-08 4.01 3.06

c+, κ+ N = 192 1.50e-06 4.59e-05N = 648 2.80e-07 1.36e-05 4.14 3.00N = 1536 8.52e-08 5.69e-06 4.13 3.03N = 5184 1.60e-08 1.62e-06 4.13 3.10N = 12288 4.91e-09 6.66e-07 4.10 3.08N = 41472 9.46e-10 1.92e-07 4.06 3.07N = 98304 2.97e-10 7.96e-08 4.03 3.06




N = 648 4.62e-10 2.57e-08 5.17 4.32N = 1536 1.02e-10 7.43e-09 5.23 4.32N = 5184 1.25e-11 1.34e-09 5.19 4.23

c+, κ+ N = 192 3.75e-09 1.26e-07N = 648 4.60e-10 2.27e-08 5.18 4.23N = 1536 1.02e-10 6.58e-09 5.24 4.30N = 5184 1.26e-11 1.22e-09 5.15 4.16



N = 192 2.43e-10 1.46e-08 5.38 4.25N = 648 1.85e-11 1.65e-09 6.35 5.37

c+, κ+ N = 24 1.01e-08 2.49e-07N = 192 2.41e-10 1.32e-08 5.38 4.24N = 648 1.85e-11 1.52e-09 6.33 5.32


p = 2 p = 3 p = 4 p = 5

cdg, κdg 1.56e-03 9.77e-04 2.14e-03 1.53e-03c+, κ+ 2.73e-03 1.42e-03 2.88e-03 1.91e-03

Table 9.15: Explicit time-step limits (∆tmax) of VCJH schemes for the Couette flow problemon the structured tetrahedral grids with N = 8 for the cases of p = 2 and 3, and N = 3 forthe cases of p = 4 and 5. The inviscid and viscous numerical fluxes were computed using aRusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.

p = 2 p = 3 p = 4 p = 5

cdg, κdg 1.48e-03 8.92e-04 1.42e-03 9.49e-04c+, κ+ 2.53e-03 1.27e-03 1.89e-03 1.20e-03

Table 9.16: Explicit time-step limits (∆tmax) of VCJH schemes for the Couette flow problemon the unstructured grids with N = 12288 for the cases of p = 2 and 3, and N = 648 forthe cases of p = 4 and 5. The inviscid and viscous numerical fluxes were computed using aRusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.


Figure 9.7: Contours of density obtained using the VCJH scheme with c = c+ and κ = κ+on the unstructured grid with N = 1536 for the case of p = 3. The inviscid and viscousnumerical fluxes were computed using a Rusanov flux with λ = 1 and a LDG flux withτ = 1.0 and β = ±0.5n−.

Figure 9.8: Contours of energy obtained using the VCJH scheme with c = c+ and κ = κ+on the unstructured grid with N = 1536 for the case of p = 3. The inviscid and viscousnumerical fluxes were computed using a Rusanov flux with λ = 1 and a LDG flux withτ = 1.0 and β = ±0.5n−.


In following the approach of the 2D experiments, absolute errors (and by extension

orders of accuracy) were determined using L2 norms and seminorms of the errors in

the total energy E and its gradient∇E. The time-step limits (∆tmax) were determined

using an iterative method on the structured grid with N = 8 for p = 2 and p = 3,

the structured grid with N = 3 for p = 4 and p = 5, the unstructured grid with

N = 12288 for p = 2 and p = 3, and the unstructured grid with N = 648 for p = 4

and p = 5.

Tables (9.7) – (9.14) demonstrate that the expected order of accuracy is obtained for

both schemes, on structured and unstructured grids for p = 2 to p = 5. In addition,

Tables (9.15) and (9.16) demonstrate that the scheme with c = c+ and κ = κ+ has

a larger time-step limit than the scheme with c = cdg and κ = κdg. In particular,

for p = 2 on the structured grid with N = 8, the scheme with c = c+ and κ = κ+

has ∆tmax = 2.73e-03, while the collocation-based nodal DG scheme has ∆tmax =

1.56e-03. Similarly, for p = 2 on the unstructured grid with N = 12288, the scheme

with c = c+ and κ = κ+ has ∆tmax = 2.53e-03, while the collocation-based nodal DG

scheme has ∆tmax = 1.48e-03.

Overall, the time-step improvements originally observed for linear problems, for larger

values of c and κ, have been shown to extend to nonlinear problems in 2D and 3D

on structured and unstructured grids. Furthermore, these improvements appear to

preserve accuracy, as the expected order of accuracy has been obtained in the vast

majority of the nonlinear experiments in this section.

9.2 Flow Generated by a Time-Dependent Source

Term

In what follows, the VCJH schemes are employed to solve the NS equations with a

time-dependent source term S. In general, the term S is incorporated into the NS

equations as follows

∂U

∂t+∇ · F(U,Q) = S, (9.2)

Q−∇U = 0. (9.3)


It turns out that for certain choices of S, the NS equations have well-known exact

solutions. In particular, Gassner et al. [103] have shown that if the source term is

defined in 2D as follows

S =

s1s2s3s4

,

s1 = (2k − ω) cos (k (x+ y)− ωt) ,s2 = cos (k (x+ y)− ωt) [(3 + 2a (γ − 1)− γ) k − ω + 2 (γ − 1) k sin (k (x+ y)− ωt)] ,s3 = cos (k (x+ y)− ωt) [(3 + 2a (γ − 1)− γ) k − ω + 2 (γ − 1) k sin (k (x+ y)− ωt)] ,

s4 =

(2γk2µ

Pr

)sin (k (x+ y)− ωt) + 2 cos (k (x+ y)− ωt)

[(1− γ + 2aγ) k − aω + (2γk − ω) sin (k (x+ y)− ωt)] , (9.4)

the following exact solution can be obtained

U =

sin (k (x+ y)− ωt) + asin (k (x+ y)− ωt) + asin (k (x+ y)− ωt) + a

(sin (k (x+ y)− ωt) + a)2

. (9.5)

Similarly, it can be shown that if the source term is defined in 3D as follows

S =

s1s2s3s4s5

,


s1 = (3k − ω) cos (k (x+ y + z)− ωt) ,

s2 =1

2cos (k (x+ y + z)− ωt)

[(9 + 4a (γ − 1)− 3γ) k − 2ω + 4 (γ − 1) k sin (k (x+ y + z)− ωt)] ,

s3 =1

2cos (k (x+ y + z)− ωt)


s4 =1

2cos (k (x+ y + z)− ωt)


s5 =

(3γk2µ

Pr

)sin (k (x+ y + z)− ωt) + 1

2cos (k (x+ y + z)− ωt)

[3 (3− 3γ + 4aγ) k − 4aω + 4 (3γk − ω) sin (k (x+ y + z)− ωt)] , (9.6)

the following exact solution can be obtained

U =

sin (k (x+ y)− ωt) + asin (k (x+ y)− ωt) + asin (k (x+ y)− ωt) + asin (k (x+ y)− ωt) + a

(sin (k (x+ y)− ωt) + a)2

. (9.7)

Now, the non-dimensionalized NS equations with the source terms in 2D and 3D can

be written as follows

∂U⋆

∂t⋆+∇⋆ · F⋆(U⋆,∇⋆U⋆) = S⋆, (9.8)

Q⋆ −∇⋆U⋆ = 0, (9.9)

where non-dimensionalization has been performed in accordance with equations (8.9)

– (8.12).

Approximate solutions to equations (9.8) and (9.9) were sought on a square domain

Ω = [−1, 1] × [−1, 1] and a cube domain Ω = [−1, 1] × [−1, 1] × [−1, 1]. Periodic

boundary conditions were imposed on the boundaries of the square and cube domains,

as shown in Figure (9.9).


Periodic

(top)

Periodic

(bottom)

Periodic

(left)

Periodic

(right)

(a) 2D square domain

Periodic

(top)

Periodic

(bottom)

Periodic

(left)

Periodic

(right)

Periodic

(front)

Periodic

(back)

(b) 3D cube domain

Figure 9.9: Boundary conditions on 2D and 3D computational domains.


The boundary conditions (described above) were imposed on discretizations of the

square and cube domains. The square domain was discretized by forming N × N

regular quadrilateral meshes and then splitting these meshes into grids with N = 2N2

triangle elements. In this manner, structured triangular grids with N = 24, 32, 48,

and 64 were formed. Figure (9.10) shows the triangular grid with N = 24.

Figure 9.10: Structured triangular grid for the case of N = 24.

Similarly, the cube domain was discretized by forming N × N × N regular hexahe-

dral meshes and then splitting these meshes into grids with N = 6N3 tetrahedron

elements. In this manner, structured tetrahedral grids with N = 24, 32, 48, and 64

were formed. Figure (9.11) shows the tetrahedral grid with N = 24.


Figure 9.11: Structured tetrahedral grid for the case of N = 24.

At time t = 0, the flow on each domain was initialized with source term parameters

Pr = 0.72, γ = 1.4, k = π, ω = π, a = 3.0, and µ = 0.001. In both 2D and 3D, the

solution was marched forward in time using the RK54 approach and, at each time-

step, the inviscid and viscous numerical fluxes were computed using the Rusanov

approach (with λ = 1) and the LDG approach (with β = ±0.5n− and τ = 0.1 in 2D

and β = ±0.5n− and τ = 1.0 in 3D). Each simulation was terminated at time t = 1.

Results in 2D and 3D were obtained for polynomial orders p = 2 and p = 3.


Solutions in 2D were obtained for two different VCJH schemes defined by the following

pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Tables (9.17) – (9.19)

contain the absolute errors, orders of accuracy, and maximum time-step limits for

each of these schemes, and Figure (9.12) shows the energy contours obtained via the

scheme with c = c+ and κ = κ+ on the mesh with N = 32 for the case of p = 3.



N = 32 3.60e-04 7.95e-02 2.98 2.00N = 48 1.01e-04 3.30e-02 3.16 2.18N = 64 3.98e-05 1.72e-02 3.27 2.30

c+, κ+ N = 24 5.65e-03 7.23e-01N = 32 2.34e-03 4.12e-01 3.07 1.99N = 48 6.41e-04 1.79e-01 3.19 2.09N = 64 2.48e-04 9.61e-02 3.30 2.21

Table 9.17: Accuracy of VCJH schemes for flow generated by a time-dependent source termon triangular grids, for the case of p = 2. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.


N = 32 4.02e-06 1.86e-03 3.99 3.06N = 48 7.99e-07 5.35e-04 3.97 3.06N = 64 2.54e-07 2.26e-04 3.96 2.99

c+, κ+ N = 24 9.48e-05 1.71e-02N = 32 2.86e-05 6.87e-03 4.17 3.16N = 48 5.27e-06 1.91e-03 4.17 3.15N = 64 1.58e-06 7.73e-04 4.18 3.15

Table 9.18: Accuracy of VCJH schemes for flow generated by a time-dependent source termon triangular grids, for the case of p = 3. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.

p = 2 p = 3

cdg, κdg 2.67e-03 1.70e-03c+, κ+ 4.39e-03 2.35e-03

Table 9.19: Explicit time-step limits (∆tmax) of VCJH schemes for flow generated by atime-dependent source term on the triangular grid with N = 48, for the cases of p = 2and 3. The inviscid and viscous numerical fluxes were computed using a Rusanov flux withλ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.


Figure 9.12: Contours of energy obtained using the VCJH scheme with c = c+ and κ = κ+on the triangular grid with N = 32 for the case of p = 3. The inviscid and viscous numericalfluxes were computed using a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 andβ = ±0.5n−.

Absolute errors (and by extension orders of accuracy) were determined using L2 norms

and seminorms of the errors in the total energy E and its gradient ∇E. The time-step

limits (∆tmax) were determined using an iterative method on the grid with N = 48

for p = 2 and p = 3.

Tables (9.17) and (9.18) demonstrate that the expected order of accuracy is obtained

for both schemes, for p = 2 and p = 3. In addition, Table (9.19) demonstrates that

the scheme with c = c+ and κ = κ+ has a larger time-step limit than the scheme with

c = cdg and κ = κdg. In particular, for p = 2 on the grid with N = 48, the scheme




Solutions were also obtained in 3D for the VCJH schemes with (c = cdg, κ = κdg)

and (c = c+, κ = κ+). Tables (9.20) – (9.22) contain the absolute errors, orders of

accuracy, and maximum time-step limits for each of these schemes, and Figure (9.13)

shows the energy contours obtained via the scheme with c = c+ and κ = κ+ on the

mesh with N = 32 for the case of p = 3.



N = 32 1.20e-03 2.78e-01 2.95 2.03N = 48 3.58e-04 1.20e-01 2.99 2.07N = 64 1.48e-04 6.54e-02 3.07 2.11

c+, κ+ N = 24 1.05e-01 1.06e+01N = 32 4.85e-02 6.45e+00 2.68 1.72N = 48 1.53e-02 3.02e+00 2.85 1.87N = 64 6.40e-03 1.69e+00 3.03 2.02

Table 9.20: Accuracy of VCJH schemes for flow generated by a time-dependent source termon tetrahedral grids, for the case of p = 2. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.


N = 32 2.70e-05 9.11e-03 4.11 3.13N = 48 5.17e-06 2.57e-03 4.07 3.12N = 64 1.62e-06 1.06e-03 4.04 3.10

c+, κ+ N = 24 3.86e-03 4.90e-01N = 32 1.29e-03 2.17e-01 3.80 2.84N = 48 2.62e-04 6.51e-02 3.94 2.96N = 64 8.14e-05 2.69e-02 4.06 3.08

Table 9.21: Accuracy of VCJH schemes for flow generated by a time-dependent source termon tetrahedral grids, for the case of p = 3. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.

p = 2 p = 3

cdg, κdg 1.07e-03 7.35e-04c+, κ+ 1.81e-03 1.03e-03

Table 9.22: Explicit time-step limits (∆tmax) of VCJH schemes for flow generated by atime-dependent source term on the tetrahedral grid with N = 48, for the cases of p = 2and 3. The inviscid and viscous numerical fluxes were computed using a Rusanov flux withλ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.


Figure 9.13: Contours of energy obtained using the VCJH scheme with c = c+ and κ = κ+on the tetrahedral grid with N = 32 for the case of p = 3. The inviscid and viscousnumerical fluxes were computed using a Rusanov flux with λ = 1 and a LDG flux withτ = 1.0 and β = ±0.5n−.

In following the approach of the 2D experiments, absolute errors (and by extension

orders of accuracy) were determined using L2 norms and seminorms of the errors in

the total energy E and its gradient∇E. The time-step limits (∆tmax) were determined

using an iterative method on the grid with N = 48 for p = 2 and p = 3.

Tables (9.20) and (9.21) demonstrate that the expected order of accuracy is obtained

for both schemes, for p = 2 and p = 3. In addition, Table (9.22) demonstrates that

the scheme with c = c+ and κ = κ+ has a larger time-step limit than the scheme with

c = cdg and κ = κdg. In particular, for p = 2 on the grid with N = 48, the scheme



Overall, the time-step improvements that were observed for steady, nonlinear prob-

lems, for larger values of c and κ (cf. section 9.1), have been shown to extend to

unsteady nonlinear problems in 2D and 3D. Furthermore, these improvements appear

to preserve accuracy, as the expected order of accuracy has been obtained in the vast


majority of the nonlinear experiments in this section.

9.2.3 Aliasing Errors in 2D and 3D

Thus far, in an effort to minimize aliasing errors, all the results in this section have

been obtained with the solution and flux points placed at the locations of the quadra-

ture points from Chapter 7. Figure (9.14) illustrates the locations of the solution and

flux points on the reference triangle and tetrahedron for the case of p = 3.

(a) Point arrangement in 2D (b) Point arrangement in 3D

Figure 9.14: Placement of the solution points (circles) and flux points (squares) at thelocations of the quadrature points described in Chapter 7 for the case of p = 3.

Having established the preliminary effectiveness of the quadrature points via experi-

ments, it was important to more carefully test their ability to reduce aliasing errors.

Towards this end, the errors produced by simulations with the quadrature points

were compared to the errors produced by simulations with an alternative set of points:

namely the α-optimized points discovered by Hesthaven andWarburton [104, 105, 13].

The α-optimized points are a well-known set of points that are designed to optimize

the conditioning of the Vandermonde matrices used in discontinuous FEM procedures.

Figure (9.15) illustrates the α-optimized locations of the solution and flux points on

the reference triangle and tetrahedron for the case of p = 3.


(a) Point arrangement in 2D (b) Point arrangement in 3D

Figure 9.15: Placement of the solution points (circles) and flux points (squares) at thelocations of the α-optimized points for the case of p = 3.

The α-optimized points are not designed to minimize aliasing errors, and thus, it

was instructive to compare the performance of these points to the performance of the

quadrature points. In order to evaluate the performance of the two sets of points,

each set was employed to solve equations (9.8) and (9.9) with source term parameters

Pr = 0.72, γ = 1.4, k = π, ω = π, a = 3.0, and µ = 0.001. Approximate solutions

to this problem were sought on coarsely meshed square domains with N = 12 and

N = 4 for polynomials orders p = 2 and p = 3, and coarsely meshed cube domains

with N = 12, 8, 6, and 4 for polynomial orders p = 2, 3, 4, and 5. In order to highlight

the effects of aliasing errors, the approximate solutions were computed for long times

(until time t = 10) using the scheme with c = c+ and κ = κ+. L2 errors in the energy

E for each set of points (on triangles and tetrahedra) are shown in Tables (9.23) and

(9.24), and contours of the energy obtained with the α-optimized points for the case

of N = 4 and p = 3 in 2D, and the case of N = 8 and p = 3 in 3D are shown in

Figures (9.16) and (9.17), respectively.


p N L2 err. (quad-points) L2 err. (α-points) % difference

2 12 4.53e-02 4.56e-02 0.573 4 7.18e-02 1.14e-01 58.22

Table 9.23: Comparison of errors produced by experiments with the quadrature points andthe α-optimized points for the VCJH scheme with c = c+ and κ = κ+, for the problem withflow driven by a time-dependent source term on triangular grids, for the cases of p = 2 andp = 3. The inviscid and viscous numerical fluxes were computed using a Rusanov flux withλ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.

p N L2 err. (quad-points) L2 err. (α-points) % difference

2 12 5.42e-01 5.45e-01 0.523 8 1.82e-01 2.28e-01 25.474 6 1.47e-01 1.51e-01 2.215 4 3.07e-01 4.63e-01 50.86

Table 9.24: Comparison of errors produced by experiments with the quadrature points andthe α-optimized points for the VCJH scheme with c = c+ and κ = κ+, for the problem withflow driven by a time-dependent source term on tetrahedral grids, for the cases of p = 2 top = 5. The inviscid and viscous numerical fluxes were computed using a Rusanov flux withλ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.

Figure 9.16: Contours of energy obtained via the VCJH scheme with c = c+, κ = κ+,and the solution and flux points placed at the locations of the α-optimized points, on thetriangular grid with N = 4 for the case of p = 3. The inviscid and viscous numericalfluxes were computed using a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 andβ = ±0.5n−.


Figure 9.17: Contours of energy obtained via the VCJH scheme with c = c+, κ = κ+,and the solution and flux points placed at the locations of the α-optimized points, on thetetrahedral grid with N = 8 for the case of p = 3. The inviscid and viscous numericalfluxes were computed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 andβ = ±0.5n−.

Excellent examples of the aliasing errors that frequently arose in each of the simu-

lations can be seen in the form of the oscillations that distort the contours in Fig-

ures (9.16) and (9.17).

The data in Tables (9.23) and (9.24) demonstrates that the simulations with the

quadrature points produce less error than the simulations with the α-optimized points.

The reduction in error can be seen most notably for odd polynomial orders p = 3

and p = 5, where the error is reduced by roughly 25 – 60 percent. This suggests

the existence of a diffusive phenomena that tends to dampen aliasing errors for even

polynomial orders p = 2 and p = 4. More importantly, the tabulated data suggests

that (regardless of whether the order is even or odd), the quadrature points are

effective in reducing the total errors by producing smaller aliasing errors than those

produced by the α-optimized points.


9.3 Flow Around Circular Cylinder

In this section, the VCJH schemes are employed to simulate the flow around a circular

cylinder. This particular flow was chosen because it is frequently used as a benchmark

for evaluating unsteady, compressible, viscous flow solvers. For low Reynolds numbers

and moderate Mach numbers, i.e., Re = 103 – 105 and M ≤ Mcritical, the freestream

flow encounters the circular cylinder, forms a boundary layer, separates, and then

forms a laminar wake of vortices commonly referred to as a ‘von Karman vortex

sheet’. The vortices in the wake are shed at a particular frequency f . The temporal

behavior of the flow is quantified by the non-dimensional ‘Strouhal’ number, which

is defined in terms of f , the cylinder diameter D, and the flow speed u as follows:

St = fD/u. In some cases, St is a known function of Re. For example, for Re = 200,

St for a circular cylinder is approximately 0.2 [106, 107, 108]. Based on this fact,

it was convenient to evaluate the VCJH schemes by simulating flow with Re = 200

around the circular cylinder.

For each VCJH scheme, the flow with Re = 200 was simulated on a 2D circular domain

with the circular cylinder located at the center. The boundaries of the domain were

placed at a distance of 100 diameters from the center of the cylinder. Isothermal wall

boundary conditions were imposed on the surface of the cylinder and characteristic

boundary conditions were imposed on the farfield boundaries. Figure (9.18) illustrates

the circular domain along with the boundary conditions that were imposed on it.


Characteristic

Isothermal wall

Figure 9.18: Computational domain for the circular cylinder with associated boundaryconditions.

The domain was discretized by forming an unstructured triangular mesh with N =

63472 elements. The mesh was refined in the neighborhood of the cylinder surface in

order to resolve the boundary layer, and it was refined in the wake region in order

to resolve the von Karman vortex sheet. Figures (9.19) and (9.20) illustrate the

unstructured triangular mesh with refined boundary layer and wake regions.


(a) Farfield mesh (b) Cylinder and wake mesh

Figure 9.19: Faraway views of the unstructured triangular grid with N = 63472 elementsaround the circular cylinder geometry.

(a) Cylinder mesh (b) Boundary layer mesh

Figure 9.20: Close-up views of the unstructured triangular grid with N = 63472 elementsaround the circular cylinder geometry.

The flow on each domain was initialized with gas parameters Pr = 0.72, γ = 1.4,

and Re = 200 (as mentioned previously). The flow was simulated for two different


Mach numbers: M = 0.3 and M = 0.5. After the flow velocity had been set in

accordance with the Mach number, the solution was marched forward in time using

the RK54 approach and, at each time-step, the inviscid and viscous numerical fluxes

were computed using the Rusanov approach with λ = 1 and the LDG approach

with β = ±0.5n− and τ = 0.1. Results from each simulation were evaluated after

the lift and drag reached a pseudo-periodic state. The results were obtained on the

aforementioned unstructured triangular grid with N = 63472 elements, for the case

of p = 2.

9.3.1 Time-Averaged Results

The simulations were performed with two different VCJH schemes defined by the

following pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Table (9.25)

contains the time-averaged drag coefficients and Strouhal numbers for each of these

schemes, and Figures (9.21) – (9.24) show the density, pressure, Mach number, and

vorticity contours obtained via the scheme with c = c+ and κ = κ+, for the cases of

M = 0.3 and M = 0.5.

M = 0.3 M = 0.5

c, κ CD St CD St

cdg, κdg 1.3928 0.190 1.5669 0.190c+, κ+ 1.3923 0.190 1.5666 0.190

Table 9.25: Values of the time-averaged drag coefficient and Strouhal number for the circularcylinder in flows with M = 0.3 and M = 0.5. The flows were simulated using the VCJHschemes with p = 2, c = cdg, κ = κdg and c = c+, κ = κ+ in conjunction with the Rusanovflux with λ = 1 and the LDG flux with β = ±0.5n− and τ = 0.1 on the unstructuredtriangular grid with N = 63472 elements.


(a) Density contours (b) Pressure contours

Figure 9.21: Density and pressure contours for the flow with M = 0.3 around the circularcylinder. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, and p = 2in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− andτ = 0.1 on the unstructured triangular grid with N = 63472 elements.

(a) Mach contours (b) Vorticity contours

Figure 9.22: Mach number and vorticity contours for the flow with M = 0.3 around thecircular cylinder. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, andp = 2 in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n−

and τ = 0.1 on the unstructured triangular grid with N = 63472 elements.


(a) Density contours (b) Pressure contours

Figure 9.23: Density and pressure contours for the flow with M = 0.5 around the circularcylinder. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, and p = 2in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− andτ = 0.1 on the unstructured triangular grid with N = 63472 elements.

(a) Mach contours (b) Vorticity contours

Figure 9.24: Mach number and vorticity contours for the flow with M = 0.5 around thecircular cylinder. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, andp = 2 in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n−

and τ = 0.1 on the unstructured triangular grid with N = 63472 elements.


The two VCJH schemes produce values of the time-averaged drag coefficient and

Strouhal number that are within a few percent of each other. Furthermore, the

values of the Strouhal number produced by each scheme are in close agreement with

the expected value of St = 0.2. These results suggest that the VCJH schemes are

capable of accurately simulating time-dependent, strongly nonlinear, viscous flows.

In particular, these experiments have shown that the VCJH schemes are capable of

handling highly compressible (high-Mach number) flows, as Figure (9.24) shows that

the scheme with c = c+ and κ = κ+ is capable of producing stable results for a flow

in which the local Mach number exceeds M = 0.7.

9.4 Flow Around SD7003 Airfoil and Wing at 4

Degrees Angle of Attack

In what follows, the VCJH schemes are employed to simulate the flow around the

SD7003 airfoil (in 2D), and the SD7003 infinite wing (in 3D) at 4 degrees angle of

attack. Simulating the flow around these geometries is a convenient way to evaluate

the VCJH schemes, due to the abundance of data that has been collected in physical

experiments (cf. [109] and [110]) and numerical experiments (cf. [111], [112], [113],

[114], and [115]) on these geometries. In 2D, the SD7003 geometry is that of a low

Reynolds number Selig/Donovan (SD) airfoil that has a maximum thickness of 9.2

percent at 30.9 percent chord, and a maximum camber of 1.2 percent at 38.3 percent

chord [116]. Figure (9.25) shows an illustration of the SD7003 airfoil with a rounded

trailing edge, which is a version of the airfoil that is frequently used in numerical

experiments.


0 0.2 0.4 0.6 0.8 1

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

(a) Airfoil geometry

0.975 0.98 0.985 0.99 0.995

−0.01

−0.005

0

0.005

0.01

(b) Trailing edge

Figure 9.25: SD7003 airfoil section (left) and a magnified view of its rounded trailing edge(right).

A SD7003 wing can be formed by extruding the SD7003 airfoil geometry. In fact, it is

common-practice to approximate the infinite SD7003 wing by extruding the SD7003

airfoil a finite distance (0.2c where c is the chord) in the z direction and prescribing

periodic boundary conditions on the x − y planes that border the resulting wing-

section. Figure (9.26) presents an illustration of the finite SD7003 wing-section with

a span of 0.2c.


(a) Wing geometry (b) Trailing edge

Figure 9.26: SD7003 wing-section (left) and a magnified view of its rounded trailing edge(right).

Numerical experiments on the SD7003 geometry in 2D were performed on a circular

domain with a radius of 50c, centered at the leading edge of the airfoil. Characteristic

boundary conditions were prescribed on the outermost edge of the circular domain

and adiabatic wall boundary conditions were prescribed on the surface of the airfoil.

Similarly, numerical experiments on the SD7003 geometry in 3D were performed on

a cylindrical domain formed by extruding a circular domain (with a radius of 20c) by

0.2c in the z direction. Periodic boundary conditions were prescribed on the front and

back of the cylindrical domain, characteristic boundary conditions were prescribed on

the sides of the domain, and adiabatic wall boundary conditions were prescribed on

the surface of the wing-section. Figure (9.27) illustrates the boundary conditions that

were prescribed on the computational domains in 2D and 3D.


Adiabatic wall

Characteristic

(a) 2D circular domain

Periodic (front)

Periodic (back)

Adiabatic wall

Characteristic (side)

(b) 3D cylindrical domain

Figure 9.27: Boundary conditions for 2D and 3D computational domains for simulating theSD7003 airfoil and wing-section. Note: The domains are not drawn to scale.

The 2D computational domain was discretized into an unstructured grid with N =


25810 triangular elements, and the 3D computational domain was discretized into

an unstructured grid with N = 711332 tetrahedral elements. Figures (9.28) – (9.31)

illustrate the unstructured triangular and tetrahedral grids.

(a) Farfield mesh (b) Airfoil and wake mesh

Figure 9.28: Faraway views of the unstructured triangular grid with N = 25810 elementsaround the SD7003 airfoil geometry.


(a) Airfoil mesh (b) Boundary layer mesh

Figure 9.29: Close-up views of the unstructured triangular grid with N = 25810 elementsaround the SD7003 airfoil geometry.

(a) Farfield mesh (b) Wing and wake mesh

Figure 9.30: Faraway views of the unstructured tetrahedral grid with N = 711332 elementsaround the SD7003 wing geometry.


(a) Wing mesh (b) Boundary layer mesh

Figure 9.31: Close-up views of the unstructured tetrahedral grid with N = 711332 elementsaround the SD7003 wing geometry.

It should be noted, that, in order to facilitate the accurate representation of the

SD7003 geometries, the edges and faces of the triangular and tetrahedral elements on

each grid were defined in terms of 2nd-order, quadratic polynomials in 2D and 3D,

respectively.

At time t = 0, a uniform flow with the properties of air (Pr = 0.72, γ = 1.4) was

initialized on each unstructured grid. The incoming flow was given a Mach number of

M = 0.2, and an angle of entry of 4 (in order to simulate flight at an angle of attack

of α = 4). In 2D, the flow was initialized with Reynolds numbers (based on c) of

Re = 10000, Re = 22000, and Re = 60000, and in 3D the flow was initialized with

Re = 10000. In both 2D and 3D, the solution was marched forward in time using


were computed using the Rusanov approach (with λ = 1) and the LDG approach

(with β = ±0.5n− and τ = 0.1 in 2D and β = ±0.5n− and τ = 1.0 in 3D). Results

from each simulation were evaluated after the lift and drag reached a pseudo-periodic

state. In 2D, results were obtained on the unstructured grid with N = 25810 for

p = 2, and in 3D, results were obtained on the unstructured grid with N = 711332

for p = 3.


9.4.1 Lift and Drag Results (2D)

Solutions were obtained in 2D using the VCJH schemes defined by the following

pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Table (9.26) compares

the time-averaged values of the lift and drag coefficients for each of the schemes with

the values from [115]. In addition, Figures (9.32) – (9.34) show portions of the time

histories of the lift and drag coefficients obtained via the scheme with c = c+ and

κ = κ+, and Figures (9.35) – (9.37) show the density and vorticity contours obtained

via the scheme with c = c+ and κ = κ+.

Re = 10K Re = 22K Re = 60K

Source CL CD CL CD CL CD

Uranga et al. [115] 0.3755 0.04978 0.6707 0.04510 0.5730 0.02097cdg, κdg 0.3719 0.04940 0.6722 0.04295 0.5831 0.01975c+, κ+ 0.3713 0.04935 0.6655 0.04275 0.5774 0.02005

Table 9.26: Time-averaged values of the lift and drag coefficients for the SD7003 airfoil inflows with Re = 10000, Re = 22000, and Re = 60000. The flows were simulated usingthe VCJH schemes with p = 2, c = cdg, κ = κdg and c = c+, κ = κ+ in conjunction withthe Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− and τ = 0.1 on theunstructured triangular grid with N = 25810 elements.

75 76 77 78 79 80

0.34

0.36

0.38

0.4

0.42

t

CL

(a) Lift coefficient

75 76 77 78 79 80

0.046

0.048

0.05

0.052

0.054

t

CD

(b) Drag coefficient

Figure 9.32: Temporal variation of the lift and drag coefficients for the SD7003 airfoil inflow with Re = 10000. The flow was simulated using the VCJH scheme with c = c+,κ = κ+, and p = 2 in conjunction with the Rusanov flux with λ = 1 and the LDG flux withβ = ±0.5n− and τ = 0.1 on the unstructured triangular grid with N = 25810 elements.


70 75 800.3

0.4

0.5

0.6

0.7

0.8

0.9

t

CL


70 75 800.01

0.02

0.03

0.04

0.05

0.06

0.07

t

CD



70 75 800.45

0.5

0.55

0.6

0.65

0.7

t

CL


70 75 800.01

0.015

0.02

0.025

0.03

0.035

t

CD




(a) Density contours (b) Vorticity contours

Figure 9.35: Density and vorticity contours for the flow with Re = 10000 around the SD7003airfoil. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, and p = 2in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− andτ = 0.1 on the unstructured triangular grid with N = 25810 elements.






The data in Table (9.26) demonstrates that the results produced by the VCJH schemes

are in close agreement with the results independently obtained by [115]. Overall,

based on the tabulated results and the results in Figures (9.32) – (9.37), it appears

that the VCJH schemes are successful in simulating changes in the frequencies and

amplitudes of the lift and drag forces as a function of Reynolds number, and in

accurately representing the rapidly evolving vortex structures that are formed in the

shear layer that emanates from the trailing edge of the airfoil.

9.4.2 Lift and Drag Results (3D)

Solutions were obtained in 3D using the VCJH schemes defined by the following

pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Table (9.27) compares

the time-averaged values of the lift and drag coefficients for these schemes with the

values from [115]. In addition, Figure (9.38) shows portions of the time histories of

the lift and drag coefficients obtained via the scheme with c = c+ and κ = κ+, and

Figure (9.39) shows isosurfaces of the density and vorticity obtained via the scheme

with c = c+ and κ = κ+.


Re = 10K

Source CL CD

Uranga et al. [115] 0.3743 0.04967cdg, κdg 0.3466 0.04908c+, κ+ 0.3454 0.04903

Table 9.27: Time-averaged values of the lift and drag coefficients for the SD7003 wing-section in a flow with Re = 10000. The flow was simulated using the VCJH schemes withp = 3, c = cdg, κ = κdg and c = c+, κ = κ+ in conjunction with the Rusanov flux withλ = 1 and the LDG flux with β = ±0.5n− and τ = 1.0 on the unstructured tetrahedralgrid with N = 711332 elements.

40 41 42 43 44 450.31

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

t

CL


40 41 42 43 44 450.046

0.047

0.048

0.049

0.05

0.051

0.052

0.053

t

CD


Figure 9.38: Temporal variation of the lift and drag coefficients for the SD7003 wing-sectionin flow with Re = 10000. The flow was simulated using the VCJH scheme with c = c+,κ = κ+, and p = 3 in conjunction with the Rusanov flux with λ = 1 and the LDG flux withβ = ±0.5n− and τ = 1.0 on the unstructured tetrahedral grid with N = 711332 elements.


(a) Density isosurfaces (b) Vorticity isosurfaces

Figure 9.39: Density and vorticity isosurfaces colored by Mach number for the flow withRe = 10000 around the SD7003 wing-section. The flow was simulated using the VCJHscheme with c = c+, κ = κ+, and p = 3 in conjunction with the Rusanov flux with λ = 1and the LDG flux with β = ±0.5n− and τ = 1.0 on the unstructured tetrahedral grid withN = 711332 elements.

The data in Table (9.27) demonstrates that the results produced by the VCJH schemes

are in close agreement with the results obtained by [115]. Thus, the favorable perfor-

mance that was originally observed in 2D on the SD7003 airfoil appears to extend to

3D on the SD7003 wing-section.

9.5 Flow Around SD7003 Wing at 30 Degrees An-

gle of Attack

In this section, the VCJH schemes are employed to simulate flow around an infinite

SD7003 wing at 30 degrees angle of attack. For this case, the incoming freestream

encounters the wing and forms a large wake of unsteady, separated, vortical flow.

In [114], Persson successfully simulated the flow using a DG scheme in conjunction

with a hybrid implicit-explicit time-stepping scheme. In this thesis, the focus is on

successfully employing the VCJH schemes in conjunction with explicit time-stepping

schemes, and thus (naturally) an explicit time-stepping scheme was utilized during

each simulation of the flow with the VCJH schemes. The objective of the simulations


was not to compare the computational cost of the VCJH schemes in conjunction

with an explicit time-stepping scheme to the computational cost of Persson’s DG

scheme in conjunction with a hybrid implicit-explicit time-stepping scheme. Rather,

the objective was to demonstrate that the VCJH schemes in conjunction with an

explicit time-stepping scheme could produce stable results for a highly-separated,

vortex-dominated flow.

In the numerical experiments, the flow was simulated on a cuboid domain Ω =

[−2.0c, 7.0c] × [−2.0c, 2.0c] × [0, 0.18c]. Periodic boundary conditions were imposed

on the front and back faces of the domain, characteristic boundary conditions were

imposed on the left, right, top and bottom faces of the domain, and adiabatic wall

boundary conditions were imposed on the wing-section. Figure (9.40) shows the

domain Ω along with the associated boundary conditions.

Adiabatic wall

Characteristic(right)

Characteristic (bottom)

Characteristic(left)

Periodic (back)

Periodic (front)

Characteristic (top)

Figure 9.40: Boundary conditions for the computational domain for simulating the SD7003wing-section. Note: The domain is not drawn to scale.

The domain was discretized into an unstructured grid of N = 311958 tetrahedral

elements. The grid, which was similar to the one that was utilized in [114], was

generously provided by Persson. Figure (9.41) illustrates the grid.


(a) Farfield mesh

(b) Airfoil and wake mesh (c) Boundary layer mesh

Figure 9.41: Views of the unstructured tetrahedral grid with N = 311958 elements aroundthe SD7003 wing-section.

It should be noted, that, in order to facilitate the accurate representation of the

SD7003 geometry, the faces of the tetrahedral elements on the grid were defined in

terms of 2nd-order, quadratic polynomials.

At time t = 0, a uniform flow with the properties of air (Pr = 0.72, γ = 1.4)

was initialized on the unstructured tetrahedral grid. The incoming flow was given


a Mach number of M = 0.2 and a Reynolds number of Re = 100000. The solution

was marched forward in time using the RK54 approach and, at each time-step, the

inviscid and viscous numerical fluxes were computed using the Rusanov approach

with λ = 1 and the LDG approach with β = ±0.5n− and τ = 1.0. Results were

obtained on the aforementioned unstructured grid with N = 311958 for polynomial

order p = 2 at time t = 3.

9.5.1 Isosurfaces and Time-Step Limits

Solutions were obtained using the VCJH schemes defined by the following pairings

of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Figure (9.42) shows the density

and vorticity isosurfaces obtained via the scheme with c = c+ and κ = κ+.

(a) Density isosurfaces (b) Vorticity isosurfaces

Figure 9.42: Density and vorticity isosurfaces colored by Mach number for the flow withRe = 100000 around the SD7003 wing-section. The flow was simulated using the VCJHscheme with c = c+, κ = κ+, and p = 2 in conjunction with the Rusanov flux with λ = 1and the LDG flux with β = ±0.5n− and τ = 1.0 on the unstructured tetrahedral grid withN = 311958 elements.

Figure (9.42) demonstrates that the VCJH scheme with c = c+ and κ = κ+ is capable

of producing a stable simulation of the initial vortices that are generated by the wing

at time t = 3. In addition, further experiments demonstrated that the VCJH scheme

with c = c+ and κ = κ+ has a larger time-step limit than the scheme with c = cdg


and κ = κdg. In particular, an iterative method was used to show that the VCJH

scheme with c = c+ and κ = κ+ has ∆tmax = 5.09e-06, while the collocation-based

nodal DG scheme has ∆tmax = 2.84e-06.

Chapter 10

Conclusion

244

CHAPTER 10. CONCLUSION 245

10.1 Summary

A new class of energy stable FR schemes has been proposed for triangular and tetra-

hedral elements. These schemes (referred to as VCJH schemes) utilize the VCJH

correction functions and fields in order to correct the solution and the flux in the

advection-diffusion equation. It has been shown that these schemes are provably

stable for linear advection-diffusion problems in 2D and 3D, for all orders of accu-

racy on unstructured grids. In addition, it has been shown that the schemes can be

parameterized by two constant scalars c and κ, and that for appropriate choices of

these scalars, a class of filtered collocation-based nodal DG schemes can be recovered.

Furthermore, numerical experiments have demonstrated that the VCJH schemes are

stable and accurate for linear model problems, and that a certain VCJH scheme pos-

sesses explicit time-step limits that are (in some cases) more than 2x greater than

those of the collocation-based nodal DG scheme.

The behavior of the VCJH schemes was also investigated in the context of nonlin-

ear advection-diffusion problems. Here, it was shown that the VCJH schemes are

guaranteed to be stable if the nonlinearity of the flux is sufficiently weak, and if the

approximate flux is constructed using an exact L2-projection that eliminates all alias-

ing errors. In addition, if the approximate flux is constructed using a (less expensive)

collocation-projection instead of an L2-projection, it was shown that placing the so-

lution and flux points at the locations of quadrature points minimizes aliasing errors.

As a result, it was recommended that the solution and flux points be placed at the

locations of a class of newly discovered quadrature points on triangles and tetrahedra.

Finally, a series of numerical experiments on nonlinear problems were executed in

order to assess the practical performance of the VCJH schemes in conjunction with the

new quadrature points. The behavior of the schemes was evaluated on structured and

unstructured grids around straight-edged and curved-edged geometries. Overall, the

results of the experiments indicated that the favorable properties of the VCJH schemes

that were initially observed for linear problems (i.e. the stability and increased time-

step limits that were initially observed for linear problems) extended to nonlinear

problems. Furthermore, it was shown that the schemes are capable of obtaining high-

order accuracy and producing results that are in good agreement with independently

performed numerical experiments.


10.2 Future Avenues for Research

A natural extension of this research entails adapting the VCJH schemes to simulate a

wider range of fluid-flow phenomena. In particular, the current schemes are tailored

towards obtaining high-order approximate solutions to unsteady linear and nonlin-

ear problems that have relatively smooth exact solutions (i.e unsteady linear and

nonlinear problems whose exact solutions are differentiable at least once). Further

efforts must be employed to extend the schemes to treat problems involving flows

with sharp discontinuities, such as flows with shock waves. Furthermore, additional

efforts are required in order to extend the VCJH schemes to mixed grids of triangular

and quadrilateral elements in 2D and hexahedral and tetrahedral elements in 3D. In

particular, tensor product extensions of the 1D VCJH schemes to quadrilateral and

hexahedral grids have yet to be proven stable. In addition, the VCJH schemes have

yet to be proven stable for transitional elements that are frequently required for the

construction of mixed grids, such as prisms and pyramids.

Although the VCJH schemes have been tailored towards treating unsteady problems,

there may be some benefit in a detailed evaluation of the schemes’ behavior for steady

problems. In the midst of unsteady problems, there are frequently regions in which

the flow is predominantly steady. Thus far, the stability of the schemes for quasi-

steady, or steady flows has yet to be investigated in rigorous mathematical detail. In

particular, the VCJH schemes have yet to be proven stable for the canonical steady-

state advection-diffusion problem, i.e. the linear advection-diffusion problem with a

time-independent forcing term. Preliminary experiments (cf. Appendix D), indicate

that the schemes are stable for this problem, but further theoretical and experimental

investigations are required.

The parameterizing coefficients for the VCJH schemes can also be examined in more

detail. Thus far, the coefficients have been chosen with the objective of maximizing

the explicit time-step limits of the schemes relative to those of the collocation-based

nodal DG scheme. However, it may be possible to choose the coefficients such that

the resulting schemes accurately resolve a larger range of wave numbers relative to

those that are resolved by the collocation-based nodal DG scheme. In order to iden-

tify such values of the parameterizing coefficients, it will be necessary to perform

a detailed Fourier analysis of the schemes. This type of analysis has the potential


to significantly improve the utility of the schemes, as it may incentivize their usage

alongside implicit time-stepping schemes for which time-step limits are no longer an

issue (from a stability standpoint).

There are also compelling opportunities to compare the VCJH schemes to other classes

of schemes. In particular, it may prove useful to explore the relationship between

the VCJH schemes and the class of ‘enriched’ DG schemes equipped with Lagrange

multipliers [117, 118, 119] or the closely related class of Hybridizable DG schemes [120,

121, 122, 123, 124].

In conclusion, energy stable high-order methods, particularly the VCJH schemes, re-

main an open area of research. In accordance with the favorable results obtained in

this thesis, it is hoped that researchers will continue to develop the VCJH schemes

for application to complex problems in the field of fluid dynamics. More generally,

it is hoped that the continued development of innovative high-order methods will en-

courage the adoption of such methods by a broader community of CFD practitioners,

and will result in a revolutionary improvement in accuracy and efficiency.

Appendix

248

APPENDIX 249

Appendix A. Energy Stable (VCJH) Correction

Functions in 1D

A.1 Solution Correction Functions

The energy stable (VCJH) solution correction functions in 1D (denoted by gL and

gR) are required to satisfy the following identities

∫ 1

−1

gLdℓidx

dx− κ(dpℓidxp

)(dp+1gLdxp+1

)= 0, (A.1)

∫ 1

−1

gRdℓidx

dx− κ(dpℓidxp

)(dp+1gRdxp+1

)= 0, (A.2)

where κ is a constant scalar that must lie within the range

−2(2p+ 1) (app!)

2 < κ <∞, (A.3)

and where

ap =(2p)!

2p (p!)2. (A.4)

In order to satisfy equations (A.1) – (A.4), and equations (2.21) – (2.23) from Chap-

ter 2, gL and gR must be defined as follows

gL =(−1)p2

[Υp −

(ηpΥp−1 +Υp+1

1 + ηp

)], (A.5)

gR =1

2

[Υp +

(ηpΥp−1 +Υp+1

1 + ηp

)], (A.6)

where

ηp =κ (2p+ 1) (app!)

2

2, (A.7)

and Υp is a Legendre polynomial of degree p.

It is important to note that, based on equations (A.5) – (A.7), κ parameterizes gL and

gR. In fact, for different values of κ, different FR schemes are obtained (cf. section

A.3).

APPENDIX 250

A.2 Flux Correction Functions

The energy stable (VCJH) flux correction functions in 1D (denoted by hL and hR)

are required to satisfy the following identities

∫ 1

−1

hLdℓidx

dx− c(dpℓidxp

)(dp+1hLdxp+1

)= 0, (A.8)

∫ 1

−1

hRdℓidx

dx− c(dpℓidxp

)(dp+1hRdxp+1

)= 0, (A.9)

where c is a constant scalar that must lie within the range

−2(2p+ 1) (app!)

2 < c <∞. (A.10)

In order to satisfy equations (A.8) – (A.10), and equations (2.28) – (2.30) from Chap-

ter 2, hL and hR must be defined as follows

hL =(−1)p2

[Υp −

(ζpΥp−1 +Υp+1

1 + ζp

)], (A.11)

hR =1

2

[Υp +

(ζpΥp−1 +Υp+1

1 + ζp

)], (A.12)

where

ζp =c (2p+ 1) (app!)

2

2. (A.13)

It is important to note that, based on equations (A.11) – (A.13), c parameterizes hL

and hR.

A.3 Choosing the Parameters c and κ

It turns out that judicious choices of c and κ can result in the recovery of well-known

high-order schemes. In particular, if κ = κdg = cdg and c = cdg = 0, one recovers a

collocation-based nodal DG scheme. Alternatively, if κ = κsd = csd and

c = csd =2p

(2p+ 1) (p+ 1) (app!)2 , (A.14)

APPENDIX 251

then one recovers the SD scheme that Jameson showed to be stable for linear advection

problems in 1D [48]. In addition, if κ = κhu = chu and

c = chu =2 (p+ 1)

(2p+ 1) p (app!)2 , (A.15)

then one recovers the so-called g2 scheme, which was originally identified by Huynh

in [23]. Finally, if κ = κ+ = c+ and c = c+, where c+ is tabulated in [125], one obtains

a scheme with an increased explicit time-step limit compared to schemes with κ = κdg,

κ = κsd, κ = κhu, c = cdg, c = csd, or c = chu (as shown in [51]).

APPENDIX 252

Appendix B. Mapping the Indices of Quadrature

Points on the 2-Simplex

When discontinuous Finite Element methods are employed on 3D meshes of simplex

elements, the solution is (in general) double valued at the 2-simplex interfaces between

the 3-simplex elements, as the solution on the faces of each element is not constrained

to be identical to the solution on the faces of neighboring elements. Because of the

double valued nature of the solution, it is convenient to integrate over faces in the

mesh by defining sets of quadrature points associated with each face of each element in

the mesh, where it should be noted that the quadrature points on adjacent faces must

be collocated. However, while these points are collocated, they are not guaranteed

to be ordered (index-wise) in the same way, as their indexing may depend on the

orientation of the element with which they are associated. Thus, it maybe necessary

to construct mappings between indexes of the collocated points in order to determine

the appropriate pairs of solution values at these points, and to enable the numerical

integration of single-valued functions of the pairs of solution values at these points.

In practice, creating a mapping between the indices of the collocated points on a pair

of adjacent 2-simplex faces is a simpler task if the points on each 2-simplex are ar-

ranged in a layered structure consistent with the corresponding 2-SCP configuration.

In this case, the points can be numbered in increasing order from left to right and

bottom to top. An example of the numbering convention is shown in Figure (B.1) for

the case of Np = 15.

1 2 3 4 5

6 7 8 9

10 11 12

13 14

15

Figure B.1: Baseline convention for numbering quadrature points on a triangle for the caseof Np = 15.

APPENDIX 253

In general, the index i = 0, . . . , (Np − 1) for each point can be computed based on

the number of layers Nl via the following pseudo code

for j = 0→ (Nl − 1) do

for k = 0→ (Nl − j − 1) do

i← jNl −⌊j(j−1)

2

⌋+ k

end for

end for

Now, the points (with index i) must be related to the collocated points (with in-

dex in) on the adjacent face. In order to relate these points, one must consider all

three possible orientations of points (with indices i0, i1, and i2) on the adjacent face.

Figure (B.2), shows an example of the three possible orientations of points on the

adjacent face for the case of Np = 15.

1 2 3 4 5

6 7 8 9

10 11 12

13 14

15

(a) Baseline

15 13 10 6 1

14 11 7 2

12 8 3

9 4

5

(b) Rotated 120 CCW

5 9 12 14 15

4 8 11 13

3 7 10

2 6

1

(c) Rotated 240 CCW

Figure B.2: Potential orientations of quadrature point numberings on a triangle for the caseof Np = 15.

In general, the points with indices i0, i1, or i2 can be mapped to the appropriate

(collocated) points with index i via the mappings Mi0 , Mi1 , and Mi2 that are defined

(in pseudocode) as follows

for j = 0→ (Nl − 1) do

for k = 0→ (Nl − j − 1) do

Mi0 ← kNl −⌊k(k−1)

2

⌋+ j

end for

end for

APPENDIX 254

for j = 0→ (Nl − 1) do

for k = 0→ (Nl − j − 1) do

Mi1 ← Nl(Nl+1)2

−⌊(k+j)(k+j+1)

2

⌋− j − 1

end for

end for

for j = 0→ (Nl − 1) do

for k = 0→ (Nl − j − 1) do

Mi2 ← jNl −⌊j(j−1)

2

⌋+Nl − j − k − 1

end for

end for

These mappings are convenient, as they hold for all quadrature rules for which the

points are arranged in layers that are consistent with the corresponding 2-SCP con-

figurations. For quadrature rules that do not possess this structure, the mappings do

not hold, and bespoke mappings must be created. The creation of bespoke mappings

is inconvenient and maybe intractable (from a practical standpoint) for quadrature

rules with a large number of points. For this reason, quadrature rules with points

that are arranged in layers (similar to the 2-SCP layers) are preferred.

APPENDIX 255

Appendix C. Quadrature Rules on the 2-Simplex

and 3-Simplex

Nl i ai,1 ai,2 ai,3 wi

1 1 0.333333333333333 0.333333333333333 0.333333333333333 1.000000000000000

2 1 0.666666666666667 0.166666666666667 0.166666666666667 0.3333333333333332 0.166666666666667 0.666666666666667 0.166666666666667 0.3333333333333333 0.166666666666667 0.166666666666667 0.666666666666667 0.333333333333333

3 1 0.816847572980440 0.091576213509780 0.091576213509780 0.1099517436553332 0.091576213509780 0.816847572980440 0.091576213509780 0.1099517436553333 0.091576213509780 0.091576213509780 0.816847572980440 0.1099517436553334 0.445948490915964 0.445948490915964 0.108103018168071 0.2233815896780005 0.445948490915964 0.108103018168071 0.445948490915964 0.2233815896780006 0.108103018168071 0.445948490915964 0.445948490915964 0.223381589678000

4 1 0.888871894660413 0.055564052669793 0.055564052669793 0.0419555129966492 0.055564052669793 0.888871894660413 0.055564052669793 0.0419555129966493 0.055564052669793 0.055564052669793 0.888871894660413 0.0419555129966494 0.295533711735893 0.634210747745723 0.070255540518384 0.1120984120708875 0.295533711735893 0.070255540518384 0.634210747745723 0.1120984120708876 0.070255540518384 0.295533711735893 0.634210747745723 0.1120984120708877 0.634210747745723 0.295533711735893 0.070255540518384 0.1120984120708878 0.634210747745723 0.070255540518384 0.295533711735893 0.1120984120708879 0.070255540518384 0.634210747745723 0.295533711735893 0.11209841207088710 0.333333333333333 0.333333333333333 0.333333333333333 0.201542988584730

5 1 0.928258244608533 0.035870877695734 0.035870877695734 0.0179154550123032 0.035870877695734 0.928258244608533 0.035870877695734 0.0179154550123033 0.035870877695734 0.035870877695734 0.928258244608533 0.0179154550123034 0.516541208464066 0.241729395767967 0.241729395767967 0.1277121958812655 0.241729395767967 0.516541208464066 0.241729395767967 0.1277121958812656 0.241729395767967 0.241729395767967 0.516541208464066 0.1277121958812657 0.474308787777079 0.474308787777079 0.051382424445843 0.0762060623855358 0.474308787777079 0.051382424445843 0.474308787777079 0.0762060623855359 0.051382424445843 0.474308787777079 0.474308787777079 0.07620606238553510 0.201503881881800 0.751183631106484 0.047312487011716 0.05574981002711511 0.201503881881800 0.047312487011716 0.751183631106484 0.05574981002711512 0.047312487011716 0.201503881881800 0.751183631106484 0.05574981002711513 0.751183631106484 0.201503881881800 0.047312487011716 0.05574981002711514 0.751183631106484 0.047312487011716 0.201503881881800 0.05574981002711515 0.047312487011716 0.751183631106484 0.201503881881800 0.055749810027115

6 1 0.943774095634672 0.028112952182664 0.028112952182664 0.0103593746965382 0.028112952182664 0.943774095634672 0.028112952182664 0.0103593746965383 0.028112952182664 0.028112952182664 0.943774095634672 0.0103593746965384 0.645721803061365 0.177139098469317 0.177139098469317 0.0753948843267385 0.177139098469317 0.645721803061365 0.177139098469317 0.0753948843267386 0.177139098469317 0.177139098469317 0.645721803061365 0.0753948843267387 0.405508595867433 0.405508595867433 0.188982808265134 0.0975478023732428 0.405508595867433 0.188982808265134 0.405508595867433 0.0975478023732429 0.188982808265134 0.405508595867433 0.405508595867433 0.09754780237324210 0.148565812270887 0.817900980028499 0.033533207700614 0.02896926937247311 0.148565812270887 0.033533207700614 0.817900980028499 0.02896926937247312 0.033533207700614 0.148565812270887 0.817900980028499 0.02896926937247313 0.817900980028499 0.148565812270887 0.033533207700614 0.02896926937247314 0.817900980028499 0.033533207700614 0.148565812270887 0.02896926937247315 0.033533207700614 0.817900980028499 0.148565812270887 0.02896926937247316 0.357196298615681 0.604978911775132 0.037824789609186 0.04604636659593517 0.357196298615681 0.037824789609186 0.604978911775132 0.04604636659593518 0.037824789609186 0.357196298615681 0.604978911775132 0.04604636659593519 0.604978911775132 0.357196298615681 0.037824789609186 0.046046366595935

APPENDIX 256

20 0.604978911775132 0.037824789609186 0.357196298615681 0.04604636659593521 0.037824789609186 0.604978911775132 0.357196298615681 0.046046366595935

7 1 0.960045625755613 0.019977187122193 0.019977187122193 0.0052721702804952 0.019977187122193 0.960045625755613 0.019977187122193 0.0052721702804953 0.019977187122193 0.019977187122193 0.960045625755613 0.0052721702804954 0.736556464940005 0.131721767529998 0.131721767529998 0.0445529366795045 0.131721767529998 0.736556464940005 0.131721767529998 0.0445529366795046 0.131721767529998 0.131721767529998 0.736556464940005 0.0445529366795047 0.333333333333333 0.333333333333333 0.333333333333333 0.0836082122156378 0.485135346793461 0.485135346793461 0.029729306413079 0.0338157128041989 0.485135346793461 0.029729306413079 0.485135346793461 0.03381571280419810 0.029729306413079 0.485135346793461 0.485135346793461 0.03381571280419811 0.107951981846011 0.867911210117951 0.024136808036039 0.01571046134018312 0.107951981846011 0.024136808036039 0.867911210117951 0.01571046134018313 0.024136808036039 0.107951981846011 0.867911210117951 0.01571046134018314 0.867911210117951 0.107951981846011 0.024136808036039 0.01571046134018315 0.867911210117951 0.024136808036039 0.107951981846011 0.01571046134018316 0.024136808036039 0.867911210117951 0.107951981846011 0.01571046134018317 0.270840772921567 0.700872570380723 0.028286656697710 0.02820513628061618 0.270840772921567 0.028286656697710 0.700872570380723 0.02820513628061619 0.028286656697710 0.270840772921567 0.700872570380723 0.02820513628061620 0.700872570380723 0.270840772921567 0.028286656697710 0.02820513628061621 0.700872570380723 0.028286656697710 0.270840772921567 0.02820513628061622 0.028286656697710 0.700872570380723 0.270840772921567 0.02820513628061623 0.316549598844617 0.536654684206138 0.146795716949245 0.06699595712783024 0.316549598844617 0.146795716949245 0.536654684206138 0.06699595712783025 0.146795716949245 0.316549598844617 0.536654684206138 0.06699595712783026 0.536654684206138 0.316549598844617 0.146795716949245 0.06699595712783027 0.536654684206138 0.146795716949245 0.316549598844617 0.06699595712783028 0.146795716949245 0.536654684206138 0.316549598844617 0.066995957127830

8 1 0.957657154441070 0.021171422779465 0.021171422779465 0.0056391237869102 0.021171422779465 0.957657154441070 0.021171422779465 0.0056391237869103 0.021171422779465 0.021171422779465 0.957657154441070 0.0056391237869104 0.798831205208225 0.100584397395888 0.100584397395888 0.0271489681922785 0.100584397395888 0.798831205208225 0.100584397395888 0.0271489681922786 0.100584397395888 0.100584397395888 0.798831205208225 0.0271489681922787 0.457923384576135 0.271038307711932 0.271038307711932 0.0631009125333598 0.271038307711932 0.457923384576135 0.271038307711932 0.0631009125333599 0.271038307711932 0.271038307711932 0.457923384576135 0.06310091253335910 0.440191258403832 0.440191258403832 0.119617483192335 0.05175279567989911 0.440191258403832 0.119617483192335 0.440191258403832 0.05175279567989912 0.119617483192335 0.440191258403832 0.440191258403832 0.05175279567989913 0.101763679498021 0.879979641427232 0.018256679074748 0.00986675357464614 0.101763679498021 0.018256679074748 0.879979641427232 0.00986675357464615 0.018256679074748 0.101763679498021 0.879979641427232 0.00986675357464616 0.879979641427232 0.101763679498021 0.018256679074748 0.00986675357464617 0.879979641427232 0.018256679074748 0.101763679498021 0.00986675357464618 0.018256679074748 0.879979641427232 0.101763679498021 0.00986675357464619 0.394033271669987 0.582562022863673 0.023404705466341 0.02200820480014720 0.394033271669987 0.023404705466341 0.582562022863673 0.02200820480014721 0.023404705466341 0.394033271669987 0.582562022863673 0.02200820480014722 0.582562022863673 0.394033271669987 0.023404705466341 0.02200820480014723 0.582562022863673 0.023404705466341 0.394033271669987 0.02200820480014724 0.023404705466341 0.582562022863673 0.394033271669987 0.02200820480014725 0.226245530909229 0.751530614542782 0.022223854547989 0.01664457007673626 0.226245530909229 0.022223854547989 0.751530614542782 0.01664457007673627 0.022223854547989 0.226245530909229 0.751530614542782 0.01664457007673628 0.751530614542782 0.226245530909229 0.022223854547989 0.01664457007673629 0.751530614542782 0.022223854547989 0.226245530909229 0.01664457007673630 0.022223854547989 0.751530614542782 0.226245530909229 0.01664457007673631 0.635737183263105 0.249079227621332 0.115183589115563 0.04432623811891432 0.635737183263105 0.115183589115563 0.249079227621332 0.04432623811891433 0.115183589115563 0.635737183263105 0.249079227621332 0.044326238118914

APPENDIX 257

34 0.249079227621332 0.635737183263105 0.115183589115563 0.04432623811891435 0.249079227621332 0.115183589115563 0.635737183263105 0.04432623811891436 0.115183589115563 0.249079227621332 0.635737183263105 0.044326238118914

9 1 0.843284788002473 0.078357605998763 0.078357605998763 0.0170244686950012 0.078357605998763 0.843284788002473 0.078357605998763 0.0170244686950013 0.078357605998763 0.078357605998763 0.843284788002473 0.0170244686950014 0.559371813714564 0.220314093142718 0.220314093142718 0.0454846169853155 0.220314093142718 0.559371813714564 0.220314093142718 0.0454846169853156 0.220314093142718 0.220314093142718 0.559371813714564 0.0454846169853157 0.490320948006900 0.490320948006900 0.019358103986199 0.0167775837271798 0.490320948006900 0.019358103986199 0.490320948006900 0.0167775837271799 0.019358103986199 0.490320948006900 0.490320948006900 0.01677758372717910 0.197918789679510 0.710726414278087 0.091354796042403 0.02915450502887711 0.197918789679510 0.091354796042403 0.710726414278087 0.02915450502887712 0.091354796042403 0.197918789679510 0.710726414278087 0.02915450502887713 0.710726414278087 0.197918789679510 0.091354796042403 0.02915450502887714 0.710726414278087 0.091354796042403 0.197918789679510 0.02915450502887715 0.091354796042403 0.710726414278087 0.197918789679510 0.02915450502887716 0.542373714730306 0.360400011893839 0.097226273375855 0.03793870318362417 0.542373714730306 0.097226273375855 0.360400011893839 0.03793870318362418 0.097226273375855 0.542373714730306 0.360400011893839 0.03793870318362419 0.360400011893839 0.542373714730306 0.097226273375855 0.03793870318362420 0.360400011893839 0.097226273375855 0.542373714730306 0.03793870318362421 0.097226273375855 0.360400011893839 0.542373714730306 0.03793870318362422 0.964116986998421 0.017941506500789 0.017941506500789 0.00413973995184523 0.017941506500789 0.964116986998421 0.017941506500789 0.00413973995184524 0.017941506500789 0.017941506500789 0.964116986998421 0.00413973995184525 0.896950733986638 0.089133098716849 0.013916167296513 0.00664832510992126 0.896950733986638 0.013916167296513 0.089133098716849 0.00664832510992127 0.013916167296513 0.896950733986638 0.089133098716849 0.00664832510992128 0.089133098716849 0.896950733986638 0.013916167296513 0.00664832510992129 0.089133098716849 0.013916167296513 0.896950733986638 0.00664832510992130 0.013916167296513 0.089133098716849 0.896950733986638 0.00664832510992131 0.790422017699194 0.191573237936673 0.018004744364133 0.01060238114445832 0.790422017699194 0.018004744364133 0.191573237936673 0.01060238114445833 0.018004744364133 0.790422017699194 0.191573237936673 0.01060238114445834 0.191573237936673 0.790422017699194 0.018004744364133 0.01060238114445835 0.191573237936673 0.018004744364133 0.790422017699194 0.01060238114445836 0.018004744364133 0.191573237936673 0.790422017699194 0.01060238114445837 0.326066002157021 0.655463812371175 0.018470185471804 0.01447360057203238 0.326066002157021 0.018470185471804 0.655463812371175 0.01447360057203239 0.018470185471804 0.326066002157021 0.655463812371175 0.01447360057203240 0.655463812371175 0.326066002157021 0.018470185471804 0.01447360057203241 0.655463812371175 0.018470185471804 0.326066002157021 0.01447360057203242 0.018470185471804 0.655463812371175 0.326066002157021 0.01447360057203243 0.386135747159797 0.386135747159797 0.227728505680405 0.05227189389616844 0.386135747159797 0.227728505680405 0.386135747159797 0.05227189389616845 0.227728505680405 0.386135747159797 0.386135747159797 0.052271893896168

10 1 0.978284862480007 0.010857568759997 0.010857568759997 0.0016443746829212 0.010857568759997 0.978284862480007 0.010857568759997 0.0016443746829213 0.010857568759997 0.010857568759997 0.978284862480007 0.0016443746829214 0.647599341031373 0.176200329484314 0.176200329484314 0.0329741136750505 0.176200329484314 0.647599341031373 0.176200329484314 0.0329741136750506 0.176200329484314 0.176200329484314 0.647599341031373 0.0329741136750507 0.333333333333333 0.333333333333333 0.333333333333333 0.0462793692013868 0.459125218363044 0.459125218363044 0.081749563273912 0.0307214412075599 0.459125218363044 0.081749563273912 0.459125218363044 0.03072144120755910 0.081749563273912 0.459125218363044 0.459125218363044 0.03072144120755911 0.140275175172566 0.847808234372640 0.011916590454793 0.00607361572460112 0.140275175172566 0.011916590454793 0.847808234372640 0.00607361572460113 0.011916590454793 0.140275175172566 0.847808234372640 0.00607361572460114 0.847808234372640 0.140275175172566 0.011916590454793 0.00607361572460115 0.847808234372640 0.011916590454793 0.140275175172566 0.006073615724601

APPENDIX 258

16 0.011916590454793 0.847808234372640 0.140275175172566 0.00607361572460117 0.575062807338271 0.409151378413380 0.015785814248348 0.01317785592736118 0.575062807338271 0.015785814248348 0.409151378413380 0.01317785592736119 0.015785814248348 0.575062807338271 0.409151378413380 0.01317785592736120 0.409151378413380 0.575062807338271 0.015785814248348 0.01317785592736121 0.409151378413380 0.015785814248348 0.575062807338271 0.01317785592736122 0.015785814248348 0.409151378413380 0.575062807338271 0.01317785592736123 0.322649911034091 0.487189977189360 0.190160111776549 0.04131172678705224 0.322649911034091 0.190160111776549 0.487189977189360 0.04131172678705225 0.190160111776549 0.322649911034091 0.487189977189360 0.04131172678705226 0.487189977189360 0.322649911034091 0.190160111776549 0.04131172678705227 0.487189977189360 0.190160111776549 0.322649911034091 0.04131172678705228 0.190160111776549 0.487189977189360 0.322649911034091 0.04131172678705229 0.875353991115967 0.062323004442017 0.062323004442017 0.00936131969521630 0.062323004442017 0.875353991115967 0.062323004442017 0.00936131969521631 0.062323004442017 0.062323004442017 0.875353991115967 0.00936131969521632 0.059613529821276 0.927550680224530 0.012835789954194 0.00423233019611833 0.059613529821276 0.012835789954194 0.927550680224530 0.00423233019611834 0.012835789954194 0.059613529821276 0.927550680224530 0.00423233019611835 0.927550680224530 0.059613529821276 0.012835789954194 0.00423233019611836 0.927550680224530 0.012835789954194 0.059613529821276 0.00423233019611837 0.012835789954194 0.927550680224530 0.059613529821276 0.00423233019611838 0.154806239308904 0.776325074178857 0.068868686512238 0.01902805982482739 0.154806239308904 0.068868686512238 0.776325074178857 0.01902805982482740 0.068868686512238 0.154806239308904 0.776325074178857 0.01902805982482741 0.776325074178857 0.154806239308904 0.068868686512238 0.01902805982482742 0.776325074178857 0.068868686512238 0.154806239308904 0.01902805982482743 0.068868686512238 0.776325074178857 0.154806239308904 0.01902805982482744 0.257119504441304 0.728131380172750 0.014749115385946 0.01034226238154545 0.257119504441304 0.014749115385946 0.728131380172750 0.01034226238154546 0.014749115385946 0.257119504441304 0.728131380172750 0.01034226238154547 0.728131380172750 0.257119504441304 0.014749115385946 0.01034226238154548 0.728131380172750 0.014749115385946 0.257119504441304 0.01034226238154549 0.014749115385946 0.728131380172750 0.257119504441304 0.01034226238154550 0.293604225335278 0.627586163158369 0.078809611506352 0.02743696299455951 0.293604225335278 0.078809611506352 0.627586163158369 0.02743696299455952 0.078809611506352 0.293604225335278 0.627586163158369 0.02743696299455953 0.627586163158369 0.293604225335278 0.078809611506352 0.02743696299455954 0.627586163158369 0.078809611506352 0.293604225335278 0.02743696299455955 0.078809611506352 0.627586163158369 0.293604225335278 0.027436962994559

11 1 0.919998537494297 0.040000731252851 0.040000731252851 0.0076322380362942 0.040000731252851 0.919998537494297 0.040000731252851 0.0076322380362943 0.040000731252851 0.040000731252851 0.919998537494297 0.0076322380362944 0.426232284257801 0.286883857871099 0.286883857871099 0.0364039862105005 0.286883857871099 0.426232284257801 0.286883857871099 0.0364039862105006 0.286883857871099 0.286883857871099 0.426232284257801 0.0364039862105007 0.694969928569889 0.152515035715055 0.152515035715055 0.0233729265849068 0.152515035715055 0.694969928569889 0.152515035715055 0.0233729265849069 0.152515035715055 0.152515035715055 0.694969928569889 0.02337292658490610 0.417558476506865 0.417558476506865 0.164883046986269 0.03255682218407911 0.417558476506865 0.164883046986269 0.417558476506865 0.03255682218407912 0.164883046986269 0.417558476506865 0.417558476506865 0.03255682218407913 0.246568021743037 0.685908909127481 0.067523069129483 0.01995617223328114 0.246568021743037 0.067523069129483 0.685908909127481 0.01995617223328115 0.067523069129483 0.246568021743037 0.685908909127481 0.01995617223328116 0.685908909127481 0.246568021743037 0.067523069129483 0.01995617223328117 0.685908909127481 0.067523069129483 0.246568021743037 0.01995617223328118 0.067523069129483 0.685908909127481 0.246568021743037 0.01995617223328119 0.127043668117795 0.812419250675340 0.060537081206864 0.01438515200415620 0.127043668117795 0.060537081206864 0.812419250675340 0.01438515200415621 0.060537081206864 0.127043668117795 0.812419250675340 0.01438515200415622 0.812419250675340 0.127043668117795 0.060537081206864 0.01438515200415623 0.812419250675340 0.060537081206864 0.127043668117795 0.01438515200415624 0.060537081206864 0.812419250675340 0.127043668117795 0.014385152004156

APPENDIX 259

25 0.541105928987454 0.389267091481018 0.069626979531528 0.02308400305558326 0.541105928987454 0.069626979531528 0.389267091481018 0.02308400305558327 0.069626979531528 0.541105928987454 0.389267091481018 0.02308400305558328 0.389267091481018 0.541105928987454 0.069626979531528 0.02308400305558329 0.389267091481018 0.069626979531528 0.541105928987454 0.02308400305558330 0.069626979531528 0.389267091481018 0.541105928987454 0.02308400305558331 0.275533440114186 0.562228604219100 0.162237955666714 0.02993808983668632 0.275533440114186 0.162237955666714 0.562228604219100 0.02993808983668633 0.162237955666714 0.275533440114186 0.562228604219100 0.02993808983668634 0.562228604219100 0.275533440114186 0.162237955666714 0.02993808983668635 0.562228604219100 0.162237955666714 0.275533440114186 0.02993808983668636 0.162237955666714 0.562228604219100 0.275533440114186 0.02993808983668637 0.988833333392586 0.005583333303707 0.005583333303707 0.00001157615726638 0.005583333303707 0.988833333392586 0.005583333303707 0.00001157615726639 0.005583333303707 0.005583333303707 0.988833333392586 0.00001157615726640 0.026811189665324 0.972365555790341 0.000823254544335 0.00122223095295141 0.026811189665324 0.000823254544335 0.972365555790341 0.00122223095295142 0.000823254544335 0.026811189665324 0.972365555790341 0.00122223095295143 0.972365555790341 0.026811189665324 0.000823254544335 0.00122223095295144 0.972365555790341 0.000823254544335 0.026811189665324 0.00122223095295145 0.000823254544335 0.972365555790341 0.026811189665324 0.00122223095295146 0.104989079145226 0.884326028755204 0.010684892099570 0.00526045994486047 0.104989079145226 0.010684892099570 0.884326028755204 0.00526045994486048 0.010684892099570 0.104989079145226 0.884326028755204 0.00526045994486049 0.884326028755204 0.104989079145226 0.010684892099570 0.00526045994486050 0.884326028755204 0.010684892099570 0.104989079145226 0.00526045994486051 0.010684892099570 0.884326028755204 0.104989079145226 0.00526045994486052 0.214914778317184 0.772280033821947 0.012805187860868 0.00806423419576953 0.214914778317184 0.012805187860868 0.772280033821947 0.00806423419576954 0.012805187860868 0.214914778317184 0.772280033821947 0.00806423419576955 0.772280033821947 0.214914778317184 0.012805187860868 0.00806423419576956 0.772280033821947 0.012805187860868 0.214914778317184 0.00806423419576957 0.012805187860868 0.772280033821947 0.214914778317184 0.00806423419576958 0.347894845976132 0.638704860716161 0.013400293307707 0.00966153276233659 0.347894845976132 0.013400293307707 0.638704860716161 0.00966153276233660 0.013400293307707 0.347894845976132 0.638704860716161 0.00966153276233661 0.638704860716161 0.347894845976132 0.013400293307707 0.00966153276233662 0.638704860716161 0.013400293307707 0.347894845976132 0.00966153276233663 0.013400293307707 0.638704860716161 0.347894845976132 0.00966153276233664 0.493220303857103 0.493220303857103 0.013559392285795 0.01021203418904765 0.493220303857103 0.013559392285795 0.493220303857103 0.01021203418904766 0.013559392285795 0.493220303857103 0.493220303857103 0.010212034189047

Table C.1: Quadrature rules with Np = 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and 66 points forthe triangle (the 2-simplex). Note that the quadrature point locations for these rules areconsistent with the 2-SCP configurations.

Nl i ai,1 ai,2 ai,3 wi

9 1 0.565662308452164 0.217168845773918 0.217168845773918 0.0472730782838262 0.217168845773918 0.565662308452164 0.217168845773918 0.0472730782838263 0.217168845773918 0.217168845773918 0.565662308452164 0.0472730782838264 0.772586942778009 0.113706528610996 0.113706528610996 0.0173758146386815 0.113706528610996 0.772586942778009 0.113706528610996 0.0173758146386816 0.113706528610996 0.113706528610996 0.772586942778009 0.0173758146386817 0.489925932913270 0.489925932913270 0.020148134173461 0.0189425819639098 0.489925932913270 0.020148134173461 0.489925932913270 0.0189425819639099 0.020148134173461 0.489925932913270 0.489925932913270 0.01894258196390910 0.148248121611418 0.834688090615778 0.017063787772804 0.01193778309346111 0.148248121611418 0.017063787772804 0.834688090615778 0.01193778309346112 0.017063787772804 0.148248121611418 0.834688090615778 0.011937783093461

APPENDIX 260

13 0.834688090615778 0.148248121611418 0.017063787772804 0.01193778309346114 0.834688090615778 0.017063787772804 0.148248121611418 0.01193778309346115 0.017063787772804 0.834688090615778 0.148248121611418 0.01193778309346116 0.707247992654248 0.210211628838570 0.082540378507182 0.02469792154263517 0.707247992654248 0.082540378507182 0.210211628838570 0.02469792154263518 0.082540378507182 0.707247992654248 0.210211628838570 0.02469792154263519 0.210211628838570 0.707247992654248 0.082540378507182 0.02469792154263520 0.210211628838570 0.082540378507182 0.707247992654248 0.02469792154263521 0.082540378507182 0.210211628838570 0.707247992654248 0.02469792154263522 0.895845867331362 0.052077066334319 0.052077066334319 0.01322597607050623 0.052077066334319 0.895845867331362 0.052077066334319 0.01322597607050624 0.052077066334319 0.052077066334319 0.895845867331362 0.01322597607050625 0.962901916642929 0.037059242391890 0.000038840965181 0.00219288363391126 0.962901916642929 0.000038840965181 0.037059242391890 0.00219288363391127 0.000038840965181 0.962901916642929 0.037059242391890 0.00219288363391128 0.037059242391890 0.962901916642929 0.000038840965181 0.00219288363391129 0.037059242391890 0.000038840965181 0.962901916642929 0.00219288363391130 0.000038840965181 0.037059242391890 0.962901916642929 0.00219288363391131 0.674006111412898 0.308960120493180 0.017033768093922 0.01543751914995332 0.674006111412898 0.017033768093922 0.308960120493180 0.01543751914995333 0.017033768093922 0.674006111412898 0.308960120493180 0.01543751914995334 0.308960120493180 0.674006111412898 0.017033768093922 0.01543751914995335 0.308960120493180 0.017033768093922 0.674006111412898 0.01543751914995336 0.017033768093922 0.308960120493180 0.674006111412898 0.01543751914995337 0.541948958822830 0.360324176839321 0.097726864337849 0.03749072353273738 0.541948958822830 0.097726864337849 0.360324176839321 0.03749072353273739 0.097726864337849 0.541948958822830 0.360324176839321 0.03749072353273740 0.360324176839321 0.541948958822830 0.097726864337849 0.03749072353273741 0.360324176839321 0.097726864337849 0.541948958822830 0.03749072353273742 0.097726864337849 0.360324176839321 0.541948958822830 0.03749072353273743 0.386072774361323 0.386072774361323 0.227854451277355 0.05300222047101844 0.386072774361323 0.227854451277355 0.386072774361323 0.05300222047101845 0.227854451277355 0.386072774361323 0.386072774361323 0.053002220471018

10 1 0.897387770939597 0.051306114530201 0.051306114530201 0.0092371898171242 0.051306114530201 0.897387770939597 0.051306114530201 0.0092371898171243 0.051306114530201 0.051306114530201 0.897387770939597 0.0092371898171244 0.696074909255265 0.151962545372368 0.151962545372368 0.0302213713993785 0.151962545372368 0.696074909255265 0.151962545372368 0.0302213713993786 0.151962545372368 0.151962545372368 0.696074909255265 0.0302213713993787 0.333333333333333 0.333333333333333 0.333333333333333 0.0359781987080308 0.456735551506448 0.456735551506448 0.086528896987104 0.0341506893006829 0.456735551506448 0.086528896987104 0.456735551506448 0.03415068930068210 0.086528896987104 0.456735551506448 0.456735551506448 0.03415068930068211 0.285087236937195 0.640399601327240 0.074513161735566 0.02754985042469012 0.285087236937195 0.074513161735566 0.640399601327240 0.02754985042469013 0.074513161735566 0.285087236937195 0.640399601327240 0.02754985042469014 0.640399601327240 0.285087236937195 0.074513161735566 0.02754985042469015 0.640399601327240 0.074513161735566 0.285087236937195 0.02754985042469016 0.074513161735566 0.640399601327240 0.285087236937195 0.02754985042469017 0.412110984988823 0.374864151991723 0.213024863019454 0.01516974101315118 0.412110984988823 0.213024863019454 0.374864151991723 0.01516974101315119 0.213024863019454 0.412110984988823 0.374864151991723 0.01516974101315120 0.374864151991723 0.412110984988823 0.213024863019454 0.01516974101315121 0.374864151991723 0.213024863019454 0.412110984988823 0.01516974101315122 0.213024863019454 0.374864151991723 0.412110984988823 0.01516974101315123 0.095949618766213 0.899786478796084 0.004263902437703 0.00410050405380724 0.095949618766213 0.004263902437703 0.899786478796084 0.00410050405380725 0.004263902437703 0.095949618766213 0.899786478796084 0.00410050405380726 0.899786478796084 0.095949618766213 0.004263902437703 0.00410050405380727 0.899786478796084 0.004263902437703 0.095949618766213 0.00410050405380728 0.004263902437703 0.899786478796084 0.095949618766213 0.00410050405380729 0.968561337845200 0.015719331077400 0.015719331077400 0.00327180460570130 0.015719331077400 0.968561337845200 0.015719331077400 0.00327180460570131 0.015719331077400 0.015719331077400 0.968561337845200 0.003271804605701

APPENDIX 261

32 0.142259815786171 0.805988955570894 0.051751228642936 0.01711700763992133 0.142259815786171 0.051751228642936 0.805988955570894 0.01711700763992134 0.051751228642936 0.142259815786171 0.805988955570894 0.01711700763992135 0.805988955570894 0.142259815786171 0.051751228642936 0.01711700763992136 0.805988955570894 0.051751228642936 0.142259815786171 0.01711700763992137 0.051751228642936 0.805988955570894 0.142259815786171 0.01711700763992138 0.238246511759627 0.749684765380752 0.012068722859621 0.00986146202320439 0.238246511759627 0.012068722859621 0.749684765380752 0.00986146202320440 0.012068722859621 0.238246511759627 0.749684765380752 0.00986146202320441 0.749684765380752 0.238246511759627 0.012068722859621 0.00986146202320442 0.749684765380752 0.012068722859621 0.238246511759627 0.00986146202320443 0.012068722859621 0.749684765380752 0.238246511759627 0.00986146202320444 0.289270385412491 0.529942943598071 0.180786670989438 0.03388739229008245 0.289270385412491 0.180786670989438 0.529942943598071 0.03388739229008246 0.180786670989438 0.289270385412491 0.529942943598071 0.03388739229008247 0.529942943598071 0.289270385412491 0.180786670989438 0.03388739229008248 0.529942943598071 0.180786670989438 0.289270385412491 0.03388739229008249 0.180786670989438 0.529942943598071 0.289270385412491 0.03388739229008250 0.403978386074363 0.579643945351165 0.016377668574472 0.01454381520903051 0.403978386074363 0.016377668574472 0.579643945351165 0.01454381520903052 0.016377668574472 0.403978386074363 0.579643945351165 0.01454381520903053 0.579643945351165 0.403978386074363 0.016377668574472 0.01454381520903054 0.579643945351165 0.016377668574472 0.403978386074363 0.01454381520903055 0.016377668574472 0.579643945351165 0.403978386074363 0.014543815209030

11 1 0.882896472355606 0.058551763822197 0.058551763822197 0.0086789829264682 0.058551763822197 0.882896472355606 0.058551763822197 0.0086789829264683 0.058551763822197 0.058551763822197 0.882896472355606 0.0086789829264684 0.510598871357414 0.244700564321293 0.244700564321293 0.0283922241192655 0.244700564321293 0.510598871357414 0.244700564321293 0.0283922241192656 0.244700564321293 0.244700564321293 0.510598871357414 0.0283922241192657 0.975074699200870 0.012462650399565 0.012462650399565 0.0020484757364318 0.012462650399565 0.975074699200870 0.012462650399565 0.0020484757364319 0.012462650399565 0.012462650399565 0.975074699200870 0.00204847573643110 0.377866568269033 0.377866568269033 0.244266863461934 0.03608765481900911 0.377866568269033 0.244266863461934 0.377866568269033 0.03608765481900912 0.244266863461934 0.377866568269033 0.377866568269033 0.03608765481900913 0.065537553074503 0.923203740779200 0.011258706146296 0.00420780780122914 0.065537553074503 0.011258706146296 0.923203740779200 0.00420780780122915 0.011258706146296 0.065537553074503 0.923203740779200 0.00420780780122916 0.923203740779200 0.065537553074503 0.011258706146296 0.00420780780122917 0.923203740779200 0.011258706146296 0.065537553074503 0.00420780780122918 0.011258706146296 0.923203740779200 0.065537553074503 0.00420780780122919 0.831234993753852 0.155719036161244 0.013045970084903 0.00712156854049920 0.831234993753852 0.013045970084903 0.155719036161244 0.00712156854049921 0.013045970084903 0.831234993753852 0.155719036161244 0.00712156854049922 0.155719036161244 0.831234993753852 0.013045970084903 0.00712156854049923 0.155719036161244 0.013045970084903 0.831234993753852 0.00712156854049924 0.013045970084903 0.155719036161244 0.831234993753852 0.00712156854049925 0.303481647730415 0.558086639760159 0.138431712509426 0.02536300638992026 0.303481647730415 0.138431712509426 0.558086639760159 0.02536300638992027 0.138431712509426 0.303481647730415 0.558086639760159 0.02536300638992028 0.558086639760159 0.303481647730415 0.138431712509426 0.02536300638992029 0.558086639760159 0.138431712509426 0.303481647730415 0.02536300638992030 0.138431712509426 0.558086639760159 0.303481647730415 0.02536300638992031 0.277464678634328 0.709512160997863 0.013023160367809 0.00897113404401732 0.277464678634328 0.013023160367809 0.709512160997863 0.00897113404401733 0.013023160367809 0.277464678634328 0.709512160997863 0.00897113404401734 0.709512160997863 0.277464678634328 0.013023160367809 0.00897113404401735 0.709512160997863 0.013023160367809 0.277464678634328 0.00897113404401736 0.013023160367809 0.709512160997863 0.277464678634328 0.00897113404401737 0.675703468866176 0.162148265566912 0.162148265566912 0.02626548714732738 0.162148265566912 0.675703468866176 0.162148265566912 0.02626548714732739 0.162148265566912 0.162148265566912 0.675703468866176 0.02626548714732740 0.569498661228012 0.422918149828929 0.007583188943059 0.006139451525626

APPENDIX 262

41 0.569498661228012 0.007583188943059 0.422918149828929 0.00613945152562642 0.007583188943059 0.569498661228012 0.422918149828929 0.00613945152562643 0.422918149828929 0.569498661228012 0.007583188943059 0.00613945152562644 0.422918149828929 0.007583188943059 0.569498661228012 0.00613945152562645 0.007583188943059 0.422918149828929 0.569498661228012 0.00613945152562646 0.254961656922477 0.677518274954083 0.067520068123440 0.01838228242843547 0.254961656922477 0.067520068123440 0.677518274954083 0.01838228242843548 0.067520068123440 0.254961656922477 0.677518274954083 0.01838228242843549 0.677518274954083 0.254961656922477 0.067520068123440 0.01838228242843550 0.677518274954083 0.067520068123440 0.254961656922477 0.01838228242843551 0.067520068123440 0.677518274954083 0.254961656922477 0.01838228242843552 0.142212054671463 0.791204827541060 0.066583117787477 0.01475951143888553 0.142212054671463 0.066583117787477 0.791204827541060 0.01475951143888554 0.066583117787477 0.142212054671463 0.791204827541060 0.01475951143888555 0.791204827541060 0.142212054671463 0.066583117787477 0.01475951143888556 0.791204827541060 0.066583117787477 0.142212054671463 0.01475951143888557 0.066583117787477 0.791204827541060 0.142212054671463 0.01475951143888558 0.551832851583548 0.400248959302507 0.047918189113945 0.01806748982960759 0.551832851583548 0.047918189113945 0.400248959302507 0.01806748982960760 0.047918189113945 0.551832851583548 0.400248959302507 0.01806748982960761 0.400248959302507 0.551832851583548 0.047918189113945 0.01806748982960762 0.400248959302507 0.047918189113945 0.551832851583548 0.01806748982960763 0.047918189113945 0.400248959302507 0.551832851583548 0.01806748982960764 0.437842298747172 0.437842298747172 0.124315402505657 0.02583600458839665 0.437842298747172 0.124315402505657 0.437842298747172 0.02583600458839666 0.124315402505657 0.437842298747172 0.437842298747172 0.025836004588396

Table C.2: Quadrature rules with Np = 45, 55, and 66 points for the triangle (the 2-simplex). Note that the quadrature point locations for these rules are inconsistent with the2-SCP configurations.

APPENDIX 263

Nl i ai,1 ai,2 ai,3 ai,4 wi

28 1 0.919364576755549 0.026878474414817 0.026878474414817 0.026878474414817 0.0021449351443162 0.026878474414817 0.919364576755549 0.026878474414817 0.026878474414817 0.0021449351443163 0.026878474414817 0.026878474414817 0.919364576755549 0.026878474414817 0.0021449351443164 0.026878474414817 0.026878474414817 0.026878474414817 0.919364576755549 0.0021449351443165 0.438577972589591 0.187140675803470 0.187140675803470 0.187140675803470 0.0208266416907696 0.187140675803470 0.438577972589591 0.187140675803470 0.187140675803470 0.0208266416907697 0.187140675803470 0.187140675803470 0.438577972589591 0.187140675803470 0.0208266416907698 0.187140675803470 0.187140675803470 0.187140675803470 0.438577972589591 0.0208266416907699 0.473575835127937 0.473575835127937 0.026424164872063 0.026424164872063 0.00721013606445510 0.473575835127937 0.026424164872063 0.473575835127937 0.026424164872063 0.00721013606445511 0.473575835127937 0.026424164872063 0.026424164872063 0.473575835127937 0.00721013606445512 0.026424164872063 0.473575835127937 0.473575835127937 0.026424164872063 0.00721013606445513 0.026424164872063 0.473575835127937 0.026424164872063 0.473575835127937 0.00721013606445514 0.026424164872063 0.026424164872063 0.473575835127937 0.473575835127937 0.00721013606445515 0.352045262027356 0.352045262027356 0.147954737972644 0.147954737972644 0.03079891915971216 0.352045262027356 0.147954737972644 0.352045262027356 0.147954737972644 0.03079891915971217 0.352045262027356 0.147954737972644 0.147954737972644 0.352045262027356 0.03079891915971218 0.147954737972644 0.352045262027356 0.352045262027356 0.147954737972644 0.03079891915971219 0.147954737972644 0.352045262027356 0.147954737972644 0.352045262027356 0.03079891915971220 0.147954737972644 0.147954737972644 0.352045262027356 0.352045262027356 0.03079891915971221 0.225783205866940 0.732309909692947 0.020953442220056 0.020953442220056 0.00435784481386422 0.225783205866940 0.020953442220056 0.732309909692947 0.020953442220056 0.00435784481386423 0.225783205866940 0.020953442220056 0.020953442220056 0.732309909692947 0.00435784481386424 0.020953442220056 0.225783205866940 0.732309909692947 0.020953442220056 0.00435784481386425 0.020953442220056 0.225783205866940 0.020953442220056 0.732309909692947 0.00435784481386426 0.020953442220056 0.020953442220056 0.225783205866940 0.732309909692947 0.00435784481386427 0.732309909692947 0.225783205866940 0.020953442220056 0.020953442220056 0.00435784481386428 0.732309909692947 0.020953442220056 0.225783205866940 0.020953442220056 0.00435784481386429 0.732309909692947 0.020953442220056 0.020953442220056 0.225783205866940 0.00435784481386430 0.020953442220056 0.732309909692947 0.225783205866940 0.020953442220056 0.00435784481386431 0.020953442220056 0.732309909692947 0.020953442220056 0.225783205866940 0.00435784481386432 0.020953442220056 0.020953442220056 0.732309909692947 0.225783205866940 0.00435784481386433 0.158462939666092 0.647557594086975 0.096989733123466 0.096989733123466 0.00859353067783334 0.158462939666092 0.096989733123466 0.647557594086975 0.096989733123466 0.00859353067783335 0.158462939666092 0.096989733123466 0.096989733123466 0.647557594086975 0.00859353067783336 0.096989733123466 0.158462939666092 0.647557594086975 0.096989733123466 0.00859353067783337 0.096989733123466 0.158462939666092 0.096989733123466 0.647557594086975 0.00859353067783338 0.096989733123466 0.096989733123466 0.158462939666092 0.647557594086975 0.00859353067783339 0.647557594086975 0.158462939666092 0.096989733123466 0.096989733123466 0.00859353067783340 0.647557594086975 0.096989733123466 0.158462939666092 0.096989733123466 0.00859353067783341 0.647557594086975 0.096989733123466 0.096989733123466 0.158462939666092 0.00859353067783342 0.096989733123466 0.647557594086975 0.158462939666092 0.096989733123466 0.00859353067783343 0.096989733123466 0.647557594086975 0.096989733123466 0.158462939666092 0.00859353067783344 0.096989733123466 0.096989733123466 0.647557594086975 0.158462939666092 0.00859353067783345 0.322111431830857 0.322111431830857 0.322111431830857 0.033665704507429 0.02300068166928646 0.322111431830857 0.322111431830857 0.033665704507429 0.322111431830857 0.02300068166928647 0.322111431830857 0.033665704507429 0.322111431830857 0.322111431830857 0.02300068166928648 0.033665704507429 0.322111431830857 0.322111431830857 0.322111431830857 0.02300068166928649 0.792939256469618 0.097608162890442 0.097608162890442 0.011844417749498 0.00486306390491250 0.097608162890442 0.792939256469618 0.097608162890442 0.011844417749498 0.00486306390491251 0.097608162890442 0.097608162890442 0.792939256469618 0.011844417749498 0.00486306390491252 0.792939256469618 0.097608162890442 0.011844417749498 0.097608162890442 0.00486306390491253 0.097608162890442 0.792939256469618 0.011844417749498 0.097608162890442 0.00486306390491254 0.097608162890442 0.097608162890442 0.011844417749498 0.792939256469618 0.00486306390491255 0.792939256469618 0.011844417749498 0.097608162890442 0.097608162890442 0.00486306390491256 0.097608162890442 0.011844417749498 0.792939256469618 0.097608162890442 0.00486306390491257 0.097608162890442 0.011844417749498 0.097608162890442 0.792939256469618 0.00486306390491258 0.011844417749498 0.792939256469618 0.097608162890442 0.097608162890442 0.00486306390491259 0.011844417749498 0.097608162890442 0.792939256469618 0.097608162890442 0.00486306390491260 0.011844417749498 0.097608162890442 0.097608162890442 0.792939256469618 0.00486306390491261 0.541184412800237 0.133558160703568 0.296501020543124 0.028756405953071 0.01559514007825962 0.296501020543124 0.541184412800237 0.133558160703568 0.028756405953071 0.01559514007825963 0.133558160703568 0.296501020543124 0.541184412800237 0.028756405953071 0.015595140078259

APPENDIX 264

64 0.541184412800237 0.296501020543124 0.133558160703568 0.028756405953071 0.01559514007825965 0.133558160703568 0.541184412800237 0.296501020543124 0.028756405953071 0.01559514007825966 0.296501020543124 0.133558160703568 0.541184412800237 0.028756405953071 0.01559514007825967 0.541184412800237 0.133558160703568 0.028756405953071 0.296501020543124 0.01559514007825968 0.296501020543124 0.541184412800237 0.028756405953071 0.133558160703568 0.01559514007825969 0.133558160703568 0.296501020543124 0.028756405953071 0.541184412800237 0.01559514007825970 0.541184412800237 0.296501020543124 0.028756405953071 0.133558160703568 0.01559514007825971 0.133558160703568 0.541184412800237 0.028756405953071 0.296501020543124 0.01559514007825972 0.296501020543124 0.133558160703568 0.028756405953071 0.541184412800237 0.01559514007825973 0.541184412800237 0.028756405953071 0.133558160703568 0.296501020543124 0.01559514007825974 0.296501020543124 0.028756405953071 0.541184412800237 0.133558160703568 0.01559514007825975 0.133558160703568 0.028756405953071 0.296501020543124 0.541184412800237 0.01559514007825976 0.541184412800237 0.028756405953071 0.296501020543124 0.133558160703568 0.01559514007825977 0.133558160703568 0.028756405953071 0.541184412800237 0.296501020543124 0.01559514007825978 0.296501020543124 0.028756405953071 0.133558160703568 0.541184412800237 0.01559514007825979 0.028756405953071 0.541184412800237 0.133558160703568 0.296501020543124 0.01559514007825980 0.028756405953071 0.296501020543124 0.541184412800237 0.133558160703568 0.01559514007825981 0.028756405953071 0.133558160703568 0.296501020543124 0.541184412800237 0.01559514007825982 0.028756405953071 0.541184412800237 0.296501020543124 0.133558160703568 0.01559514007825983 0.028756405953071 0.133558160703568 0.541184412800237 0.296501020543124 0.01559514007825984 0.028756405953071 0.296501020543124 0.133558160703568 0.541184412800237 0.015595140078259

Table C.3: Quadrature rule with Np = 84 points for the tetrahedron (the 3-simplex).

APPENDIX 265

Appendix D. Experiments on a Steady Linear

Advection-Diffusion Problem

In this section, the VCJH schemes are employed to solve the steady linear advection-

diffusion equation in 3D which takes the following form

∇ · (au− bq) = S, (D.1)

q−∇u = 0. (D.2)

Here, it should be noted that the advection and diffusion of the scalar u is driven

by the source term S. There is a steady solution to equations (D.1) and (D.2) if the

boundary conditions on u are well-posed, and if the source term S is constant in time

and (at most) varies in space.

D.1 Problem Definition

In particular, a solution to equations (D.1) and (D.2) can be obtained within an

L-shaped domain Ω = [0, 1]× [0, 1]×[−1

2, 12

]\[−1

2, 12

]×[12, 1]×[−1

2, 12

](shown in

Figure (D.1)), if no-slip Dirichlet boundary conditions are imposed on all faces of the

domain, i.e. u = 0 is enforced on all of Γ, and flow within the domain is driven by

the source term S = 1.

APPENDIX 266

x

y

z

Figure D.1: L-shaped 3D computational domain.

The resulting steady flow does not have an analytical solution, however, it does yield

numerical solutions which have been studied in considerable detail by Brogniez [119].

It turns out that the resulting steady flow is a useful test of a scheme’s stability.

In particular, for a = (cos θ, sin θ, 0), θ = π/6, and b = 5 × 10−3, the flow has

an extremely thin boundary-layer region at x = 1. This region frequently causes

overshoots or undershoots to appear and these spurious oscillations may cause the

approximate solution to diverge during the course of its iterative march towards the

final steady state. As a result, it was deemed necessary to perform experiments on

this problem with the VCJH schemes in order to evaluate the schemes’ robustness

in the presence of this particular phenomena. For the experiments, the domain Ω

was tessellated with an unstructured tetrahedral mesh with N = 144847 elements as

shown in Figure (D.2).

APPENDIX 267

(a) Transparent view (b) Solid view

Figure D.2: Unstructured tetrahedral mesh with N = 144847 elements.

At time t = 0, the flow on the domain was initialized with velocity magnitude

|a| = 1. The solution was marched forward in time towards a steady state using


were computed using the Rusanov approach with λ = 1 and the LDG approach with

β = ±0.5n− and τ = 1.0. Each simulation was terminated after the residual reached

machine zero. Steady solutions were obtained on the aforementioned unstructured

tetrahedral grid for polynomials orders p = 2 to p = 5.

D.2 Steady-State Results

Solutions were obtained for two different VCJH schemes defined by the following

pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Figures (D.3) and

(D.4) show the velocity profiles for the schemes, and Figure (D.5) shows the solution

contours obtained via the scheme with c = c+ and κ = κ+.

APPENDIX 268

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

c = c

dg, κ = κ

dg

c = c+, κ = κ

+

(a) p = 2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

(b) p = 3

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

(c) p = 4

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

(d) p = 5

Figure D.3: Velocity profiles at y = 0.3 and z = 0 for the steady linear advection-diffusionproblem, for the cases of p = 2 to p = 5. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.

APPENDIX 269

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y

c = c

dg, κ = κ

dg

c = c+, κ = κ

+

(a) p = 2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y

(b) p = 3

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y

(c) p = 4

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y

(d) p = 5

Figure D.4: Velocity profiles at x = 0.9 and z = 0 for the steady linear advection-diffusionproblem, for the cases of p = 2 to p = 5. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.

APPENDIX 270

(a) p = 2 (b) p = 3

(c) p = 4 (d) p = 5

Figure D.5: Solution contours for the steady linear advection-diffusion problem, for thecases of p = 2 to p = 5. The inviscid and viscous numerical fluxes were computed using aRusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.

For smaller values of the polynomial order p, i.e., p = 2 and p = 3, the schemes

produce weak spurious oscillations in the velocity profiles. However, for p = 4 and

p = 5, the velocity profiles are devoid of oscillations, indicating that the boundary

layer at x = 1 is now sufficiently resolved. Most importantly, for all cases, for even

the smaller values of p, both VCJH schemes remain stable. Overall, the favorable

properties of the schemes with regard to stability, that were originally observed for

unsteady linear problems, have been shown (in this case) to extend to steady linear

problems.

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