uncertainty quantification and numerical a …kn505pc4734/... · this phd thesis is the result of a...

UNCERTAINTY QUANTIFICATION AND NUMERICAL

METHODS FOR CONSERVATION LAWS

A DISSERTATION

SUBMITTED TO THE INSTITUTE FOR COMPUTATIONAL

AND MATHEMATICAL ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Mass Per Pettersson

February 2013

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

This dissertation is online at: http://purl.stanford.edu/kn505pc4734

© 2013 by Mass Per Evald Pettersson. 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/kn505pc4734

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.

Gianluca Iaccarino, Primary Adviser


Jan Nordstrom, Co-Adviser


Margot Gerritsen


Antony Jameson

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

Preface

This PhD thesis is the result of a joint project between the Uncertainty Quantification

Laboratory under Professor Gianluca Iaccarino at Stanford University and the group

for Computational Methods of Unsteady Aerodynamics at Uppsala University under

Jan Nordstrom, now at Linkoping University, Sweden. Accordingly, this monograph

is also available in the form of a published compilation thesis. The monograph is

based on the articles included in the compilation thesis.

iv

Abstract

In many physical problems our knowledge is limited by our ability to measure, by

our bias in the observations and, in general, by an incomplete understanding of the

physical processes. When we attempt to simulate the physical process numerically,

we must account for those limitations, and in addition we must identify the possible

limitations of the numerical techniques and phenomenological models that we employ.

Numerical simulations are subject to uncertainty in boundary or initial conditions,

model parameter values and even in the geometry of the physical domain of the

problem; this results in uncertainty in the output data that must be clearly identified

and quantified.

In this thesis, conservation laws with uncertain initial and boundary conditions

are approximated using a generalized polynomial chaos expansion approach where

the solution is represented as a generalized Fourier series of stochastic, orthogonal

basis functions. The stochastic Galerkin method is used to project the governing

partial differential equation onto the stochastic basis functions to obtain an extended

essentially deterministic system.

The generalized polynomial chaos framework is most suitable for smooth problems

and we first consider a linear conservation law. The stochastic Galerkin and colloca-

tion methods are used to solve an advection-diffusion equation with uncertain and spa-

tially varying viscosity. We investigate well-posedness, monotonicity and stability for

the extended system resulting from the Galerkin projection of the advection-diffusion

equation onto the stochastic basis functions. High-order summation-by-parts opera-

tors and weak imposition of boundary conditions are used to prove stability of the

semi-discrete system. We investigate the total spatial operator of the semi-discretized

v

system and its impact on the convergence to steady-state.

Next we apply the stochastic Galerkin method to a non-linear scalar problem,

Burgers’ equation with uncertain boundary conditions. A characteristic analysis of

the truncated polynomial chaos system is presented and gives a qualitative descrip-

tion of the development of the system over time for different initial and boundary

conditions. An analytical solution is derived and the coefficients of the infinite PC

expansion are shown to be smooth, while the corresponding coefficients of the trun-

cated expansion are discontinuous. We also discuss the problematic implications of

the lack of known boundary data and possible ways of imposing stable and accurate

boundary conditions.

We present a new fully intrusive method for the Euler equations subject to uncer-

tainty based on a variable transformation of the continuous equations. Roe variables

are employed to get quadratic dependence in the flux function and a well-defined

Roe average matrix. The Roe formulation saves computational cost compared to the

formulation based on expansion of conservative variables. Moreover, the Roe formula-

tion is more robust and can handle cases of supersonic flow, for which the conservative

variable formulation fails to produce a bounded solution. We use a multi-wavelet ba-

sis that can be chosen to include a large number of resolution levels to handle more

extreme cases (e.g. strong discontinuities) in a robust way.

Finally, we apply the results of the previous investigations to a two-phase problem.

Based on regularity properties that we derive for the stochastic modes (generalized

polynomial chaos coefficients), we present a hybrid method where solution regions of

varying smoothness are coupled weakly through interfaces. In this way, we couple

smooth solutions solved with high-order finite difference methods with non-smooth

solutions solved for with shock-capturing methods.

vi

Acknowledgements

First and foremost I would like to thank my advisors Professor Gianluca Iaccarino

and Professor Jan Nordstrom for their invaluable support during my PhD. I have ben-

efited a lot from their very different and complementing advice and research styles.

In particular, I have appreciated the encouraging and inspiring atmosphere and many

opportunities provided by Professor Iaccarino as well as Professor Nordstrom’s com-

mitment to his students and for always finding time to give feedback. I am also

very thankful for their courage to accept me as a doctoral student despite my lack of

background in applied mathematics.

I would like to thank my Reading Committee members Professor Margot Gerritsen

and Professor Antony Jameson for reading my thesis and Professor Juan J. Alonso

for being the Chair of the Oral Committee. Many thanks to Professor Gerritsen for

the wide range of support - from providing great research opportunities to help with

moving furniture. With the unforgettable experience of being the only student of one

of Professor Jameson’s courses, I have truly enjoyed the privilege of private education

of one of the biggest names in the field.

I would like to express my gratitude to Professor Alireza Doostan for the collab-

oration on the paper about the stochastic advection-diffusion equation.

I am grateful to Professor Remi Abgrall, Dr. Pietro Congedo and Gianluca Geraci

for hosting and welcoming me to INRIA Bordeaux Sud-Ouest and for giving me

valuable feedback on my work.

Thanks to the staff at the Division of Scientific Computing (TDB) at Uppsala

University, and in particular my research group, Dr. Qaiser Abbas, Dr. Sofia Eriksson

and Dr. Jens Berg. Special thanks to Dr. Anna Nissen for proof-reading this thesis

vii

and for support and interesting discussions throughout the PhD project.

I am grateful for fruitful discussions with my present and former colleagues at

the Center for Turbulence Research and the Uncertainty Quantification group at

Stanford University, in particular Gary Tang, Paul Covington, Daniele Schiavazzi,

Paul Constantine and Nicolas Kseib.

I would like to thank Dr. Xiangyu Hu and Dr. Kwok Kai So for ideas and fruitful

discussions regarding numerical methods for two-phase flow.

I am grateful for the help from the administrators at TDB, Center for Turbu-

lence Research and ICME, Carina Lindgren, Marlene Lomuljo-Bautista and Indira

Choudhury, respectively.

The funding of this PhD has generously been provided by King Abdullah Univer-

sity of Science and Technology (KAUST) in Saudi Arabia. Financial support for the

participation in the summer program 2011 at TU Munchen, Germany, was provided

by the German Research Foundation (Deutsche Forschungsgemeinschaft - DFG) in

the framework of the Sonderforschungsbereich Transregio 40 and the IGSSE (Interna-

tional Graduate School of Science and Engineering). I am thankful for the resources

that have been provided for me during my visits and extended stays at TDB, Uppsala.

viii

Contents

Preface iv

Abstract v

Acknowledgements vii

1 Introduction 1

1.1 Uncertainty in physical systems . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Random field representation 6

2.1 Karhunen-Loeve expansion . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Generalized chaos expansions . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Generalized polynomial chaos expansion . . . . . . . . . . . . 8

2.2.2 Haar wavelet expansion . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Multiwavelet expansion . . . . . . . . . . . . . . . . . . . . . . 12

3 Polynomial chaos methods 16

3.1 Intrusive methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Stochastic Galerkin methods . . . . . . . . . . . . . . . . . . . 16

3.1.2 Semi-intrusive methods . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Non-intrusive methods . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Stochastic collocation methods . . . . . . . . . . . . . . . . . 18

ix

3.2.2 Spectral projection . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.3 Stochastic multi-elements . . . . . . . . . . . . . . . . . . . . 20

4 Spatial discretization 21

4.1 Summation-by-parts operators . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Artificial dissipation operators . . . . . . . . . . . . . . . . . . 23

4.2 Shock capturing methods . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.1 MUSCL scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.2 HLL Riemann solver . . . . . . . . . . . . . . . . . . . . . . . 25

5 Linear stochastic conservation laws 26

5.1 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1.1 Uncertainty and solution procedure . . . . . . . . . . . . . . . 28

5.1.2 Stochastic Galerkin projection . . . . . . . . . . . . . . . . . . 29

5.1.3 Diagonalization of the stochastic Galerkin system . . . . . . . 33

5.2 The eigenvalues of the diffusion matrix B . . . . . . . . . . . . . . . . 34

5.2.1 General bounds on the eigenvalues of B . . . . . . . . . . . . . 34

5.2.2 Legendre polynomial representation . . . . . . . . . . . . . . . 35

5.2.3 Hermite polynomial representation . . . . . . . . . . . . . . . 36

5.3 Boundary conditions for well-posedness . . . . . . . . . . . . . . . . . 37

5.4 Monotonicity of the solution . . . . . . . . . . . . . . . . . . . . . . . 38

5.4.1 Second order operators . . . . . . . . . . . . . . . . . . . . . . 38

5.4.2 Fourth order operators . . . . . . . . . . . . . . . . . . . . . . 40

5.5 Stability of the semi-discretized problem . . . . . . . . . . . . . . . . 42

5.5.1 The initial value problem: von Neumann analysis . . . . . . . 43

5.5.2 The initial boundary value problem . . . . . . . . . . . . . . . 45

5.5.3 Eigenvalues of the total system matrix . . . . . . . . . . . . . 51

5.5.4 Convergence to steady-state . . . . . . . . . . . . . . . . . . . 54

5.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.6.1 The inviscid limit . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.6.2 Steady-state calculations . . . . . . . . . . . . . . . . . . . . . 62

5.7 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 63

x

6 Burgers’ equation and boundary conditions 66

6.1 Polynomial chaos expansion of Burgers’ equation . . . . . . . . . . . 68

6.1.1 Entropy and energy estimate for the M = 2 case . . . . . . . . 69

6.1.2 Diagonalization of the system matrix A(u) . . . . . . . . . . . 70

6.2 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2.1 Regularity determined by the order of gPC . . . . . . . . . . . 73

6.3 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4 Energy estimates for stability analysis . . . . . . . . . . . . . . . . . . 77

6.4.1 Artificial dissipation for enhanced stability . . . . . . . . . . . 79

6.5 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.6 Eigenvalue approximation . . . . . . . . . . . . . . . . . . . . . . . . 81

6.7 Efficiency of the polynomial chaos method . . . . . . . . . . . . . . . 82

6.7.1 Numerical convergence . . . . . . . . . . . . . . . . . . . . . . 84

6.8 Theoretical results and interpretation . . . . . . . . . . . . . . . . . . 88

6.8.1 Analysis of characteristics: disturbed cosine wave . . . . . . . 88

6.9 Dependence on available data . . . . . . . . . . . . . . . . . . . . . . 97

6.9.1 Complete set of data . . . . . . . . . . . . . . . . . . . . . . . 97

6.9.2 Incomplete set of boundary data . . . . . . . . . . . . . . . . . 99

6.9.3 Discussion of the results with incomplete set of data . . . . . . 106

6.10 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 107

7 Euler equations 109

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.2 Euler equations with input uncertainty . . . . . . . . . . . . . . . . . 110

7.2.1 Formulation in Roe variables . . . . . . . . . . . . . . . . . . . 111

7.2.2 Stochastic Galerkin formulation of the Euler equations . . . . 111

7.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.3.1 Expansion of conservative variables . . . . . . . . . . . . . . . 113

7.3.2 Expansion of Roe’s variables . . . . . . . . . . . . . . . . . . . 114

7.3.3 Stochastic Galerkin Roe average matrix for Roe variables . . . 115


xi

7.4.1 Spatial convergence . . . . . . . . . . . . . . . . . . . . . . . . 120

7.4.2 Initial conditions and discontinuous solutions . . . . . . . . . . 122

7.4.3 Initial conditions and resolution requirements . . . . . . . . . 124

7.4.4 Convergence of multiwavelet expansions . . . . . . . . . . . . 125

7.4.5 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.4.6 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . 129

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8 A hybrid scheme for two-phase flow 133

8.1 Two-phase flow problem . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.2 Smoothness properties of the solution . . . . . . . . . . . . . . . . . . 137

8.2.1 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . 137

8.2.2 The stochastic modes . . . . . . . . . . . . . . . . . . . . . . . 139

8.2.3 The stochastic Galerkin solution modes . . . . . . . . . . . . . 140

8.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.3.1 Summation-by-parts operators . . . . . . . . . . . . . . . . . . 142

8.3.2 HLL Riemann solver . . . . . . . . . . . . . . . . . . . . . . . 142

8.3.3 Hybrid scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 143


8.4.1 Convergence of smooth solutions . . . . . . . . . . . . . . . . 153

8.4.2 Non-smooth Riemann problem . . . . . . . . . . . . . . . . . . 156

8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

A Generation of multiwavelets 160

B Eigenvectors of MW matrices 162

B.1 Proof of constant eigenvectors of A . . . . . . . . . . . . . . . . . . . 162

B.2 Eigenvalue decompositions of A . . . . . . . . . . . . . . . . . . . . . 164

B.2.1 Piecewise constant multiwavelets (Haar wavelets) . . . . . . . 164

B.2.2 Piecewise linear multiwavelets . . . . . . . . . . . . . . . . . . 166

Bibliography 168

xii

List of Tables

6.1 Convergence to (6.12) and (6.13) with the Monte Carlo method, m =

400, t = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.2 Convergence to (6.12) and (6.13) with the polynomial chaos method,

m = 400, t = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3 Norms of errors for dissipative and non-dissipative solutions. . . . . . 85

7.1 Relative simulation time using conservative variables and Roe vari-

ables, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

xiii

List of Figures

2.1 Haar wavelets, resolution levels 0,1,2. . . . . . . . . . . . . . . . . . . 13

2.2 Multiwavelets for Np = 2, Nr = 3. . . . . . . . . . . . . . . . . . . . . 15

5.1 Minimum λB for µ = exp(ξ). . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Solution statistics at t = 0.01 using stochastic Galerkin with p = 4 for

diffusion of a moving step function. . . . . . . . . . . . . . . . . . . . 41

5.3 Mean solution at t = 0.001 for diffusion of a moving step function, p = 4. 42

5.4 Mean solution for diffusion of a moving step function after 40 time

steps, two different Remesh. . . . . . . . . . . . . . . . . . . . . . . . . 43

5.5 Eigenvalues for order p = 3 Legendre polynomial chaos with 200 grid

points, µ(ξ) ∼ U [0, 0.1], v = 1. . . . . . . . . . . . . . . . . . . . . . . 45

5.6 Eigenvalues of the total operator Dtot for different orders of gPC and

different grid sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.7 Eigenvalues of the total operator for different orders of gPC. . . . . . 53

5.8 Limits on ∆t and max< (λDtot), lognormal viscosity. . . . . . . . . . . 57

5.9 Convergence with respect to the spatial discretization using stochastic

Galerkin and stochastic collocation. . . . . . . . . . . . . . . . . . . . 58

5.10 Approximate solution with p = 3 order of Legendre chaos. . . . . . . 59

5.11 Spatial convergence, lognormal viscosity. . . . . . . . . . . . . . . . . 60

5.12 Stochastic Galerkin and stochastic collocation as a function of the order

of gPC/number of quadrature points . . . . . . . . . . . . . . . . . . 61

5.13 Minimum and maximum viscosity for different orders of stochastic

Galerkin and stochastic collocation for two different distributions of

µ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

xiv

5.14 Number of iterations to steady-state for lognormal viscosity µ = c1 +

c2 exp(ξ) using stochastic Galerkin and stochastic collocation. . . . . 64

6.1 Exact solution u of the infinite order system as a function of x and ξ

at t = 0.5 for different orders of gPC. a = 1, b = 0.2. . . . . . . . . . . 74

6.2 Exact solution u of the truncated system as a function of x and ξ at

t = 0.5 for different orders of gPC. a = 1, b = 0.2. . . . . . . . . . . . 74

6.3 Expectation u0 as a function of x at t = 0.5 for different orders of gPC.

a = 1, b = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4 The first four gPC coefficients, t = 0.3, M = 5 and M = 3, m = 400. . 83

6.5 Dissipative solution on course grid (m = 200), computed for M = 3

and non-dissipative solution for M = 4. . . . . . . . . . . . . . . . . . 85

6.6 Convergence of the first chaos coefficients. Note the different scales in

the figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.7 M = 7. Convergence of the variance. Norm of the error relative to the

analytical variance (left) and error relative to the finest grid variance,

m = 800 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.8 Development of variance of the perturbed cosine wave. t = 0.5 for

M = 3, m = 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.9 Characteristics of the two perturbed cosine waves (Ex 1.1 and Ex 1.2)

for M = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.10 Variance of Ex. 1.1 and Ex. 1.2 for M = 1, calculated from w1, w2

using (6.43). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.11 Characteristics at t = 0.5, M = 1. . . . . . . . . . . . . . . . . . . . . 94

6.12 Expected value and variance at t = 0.5, M = 1. . . . . . . . . . . . . 95

6.13 Characteristics at t = 4 for M = 1. . . . . . . . . . . . . . . . . . . . 96

6.14 u0 (left) and u1 (right). t = 1. Complete set of data. . . . . . . . . . 98



6.17 u1 kept fixed at 0.2. t = 2. . . . . . . . . . . . . . . . . . . . . . . . . 100

6.18 u1 kept fixed at 0.2. t = 3. . . . . . . . . . . . . . . . . . . . . . . . . 100

xv

6.19 u1 kept fixed at 0.2. t = 5. . . . . . . . . . . . . . . . . . . . . . . . . 101

6.20 u1 extrapolated from the interior. t = 2. . . . . . . . . . . . . . . . . 102


6.22 u1 extrapolated from the interior. t = 5. The error is of the order 10−15.103

6.23 u0 is held fixed. t = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.24 u0 is held fixed. t = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.25 u0 is held fixed. t = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . 105




7.1 Convergence in space using the method of manufactured solutions,

Np = 10, Nr = 0 (Legendre polynomials). . . . . . . . . . . . . . . . . 122

7.2 Schematic representation of the initial setup for case 1 (left) and case

2 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.3 Initial w1 modes for case 2, first 8 basis functions. . . . . . . . . . . . 125

7.4 Decay in variance of velocity and energy as a function of the order

of expansion, polynomial order Np and resolution level Nr. Case 1,

t = 0.05, 280 spatial points restricted to x ∈ [0.4, 0.65]. Solution

obtained with the Roe variable scheme. . . . . . . . . . . . . . . . . . 126

7.5 Density as a function of x and ξ at t = 0.15. . . . . . . . . . . . . . . 128

7.6 Relative error in density, velocity, energy and Mach number at t = 0.15

for different shock strengths. . . . . . . . . . . . . . . . . . . . . . . . 129

8.1 Schematic representation of the solution of the two-phase problem. . . 138

8.2 Solution regions on the spatial mesh. . . . . . . . . . . . . . . . . . . 144

8.3 SBP 4-2-4, fixed proportion of SBP 2 points. Np = 8, Nr = 0 order of

multiwavelets (Legendre polynomials). . . . . . . . . . . . . . . . . . 154

8.4 Comparison of 2,2 norm of errors, three solution regions SBP 2-4-2

versus single region solved with SBP 2, t = 0.1. . . . . . . . . . . . . 155

8.5 Spatial convergence with three regions and two interfaces. t = 0.05.

Np = 8, Nr = 0 order of multiwavelets (Legendre polynomials). . . . . 155

xvi

8.6 Variances at t = 0.05, m = 400, fourth order SBP, single interface, and

HLL-MUSCL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.7 Convergence of the mean and variance of pressure with the order of

MW chaos, different orders of piecewise constant MW. . . . . . . . . 158

xvii

Chapter 1

Introduction

1.1 Uncertainty in physical systems

In many physical problems, data is limited in quality and quantity by variability,

bias in the measurements and by limitations to the extent measurements are possible

to perform. When we attempt to solve the problem at hand numerically, we must

account for those limitations, and in addition we must identify the possible limitations

of the numerical techniques and phenomenological models employed. Incomplete

understanding of the physical processes involved will add to the sources of possible

uncertainty in the models employed. In a general sense, we distinguish between errors

and uncertainty simply by saying that errors are recognizable deficiencies not due to

lack of knowledge, whereas uncertainties are potential and directly related to lack of

knowledge [70]. This definition clearly identifies errors as deterministic quantities

and uncertainties as stochastic in nature; uncertainty estimation and quantification

is, therefore, typically treated within a probabilistic framework.

Uncertainty quantification is also a fundamental step towards validation and cer-

tification of numerical methods to be used for critical decisions. Fields of application

of uncertainty quantification include but are not limited to turbulence, climatology

[81], turbulent combustion [83], flow in porous media [31, 18], fluid mixing [119] and

computational electromagnetics [15].

An example of the need for uncertainty quantification in applications related to

1

CHAPTER 1. INTRODUCTION 2

methods and problems studied here is the investigation of the aerodynamic stability

properties of an airfoil. Uncertainty in physical parameters such as structural fre-

quency and initial pitch angle, affect the probability of limit cycle oscillations. One

approach in particular, the polynomial chaos method, has been used to obtain a sta-

tistical characterization of the stability limits and to calculate the risk for system

failure [110, 9]; this approach will be studied in detail throughout this thesis.

The sources of uncertainty that we consider in this thesis are imprecise knowledge

of the input data, e.g. uncertainty due to finite sample sizes or measurement errors.

This results in numerical models that are subject to uncertainty in boundary or

initial conditions, model parameter values and even in the geometry of the physical

domain of the problem (input uncertainty). Uncertainty quantification in the sense

it will be used here is concerned with the propagation of input uncertainty through

the numerical model in order to clearly identify and quantify the uncertainty in the

output quantities of interest.

Without going into the details of how to transform a set of data into probability

distributions of the input variables [32], the starting point will be a partial differen-

tial equation formulation where parameters and initial and boundary conditions are

uncertain but determined in terms of probability distributions. Random variables are

used to parametrize the uncertainty in the input data. A spectral series representa-

tion, the generalized chaos series expansion, is then used to represent the solution to

the problem of interest.

The one-dimensional test problems that will be investigated here are evidently

subject to modeling error, should we like to use them as models of real-world phe-

nomena. For instance, we disregard viscous forces in the flow problems and ignore

reflections from the wall of the shock tube, treating it as a purely one-dimensional

problem. Thus, we do not account for epistemic uncertainty, i.e. uncertainty in the

physical and mathematical models themselves. In real-world problems, this would

be an important point. If the conceptual model is erroneous, for instance due to

an incompressibility assumption for a case of high Mach number flow, then there is

clearly no sense in a solution no matter the degree of accuracy of the representation

of variability in the input parameters [86].


There are several approaches to propagate the input uncertainty in numerical

simulations; the simplest one is the Monte Carlo method where a vast number of

simulations are performed to compute the output statistics. Conversely in the poly-

nomial chaos approach, the solution is expressed as a truncated series and only one

simulation is performed. The dimension of the resulting system of equations grows

with the number of the terms retained in the series (the order of the polynomial chaos

expansion) and the dimension of the stochastic input.

An increased number of Monte Carlo simulations implies a solution with better

converged statistics; on the other hand, in the polynomial chaos approach, one sin-

gle simulation is sufficient to obtain a complete statistical characterization of the

solution. However, the accuracy of this solution is dependent on the order of polyno-

mials considered, and therefore on the truncation in the polynomial chaos expansion.

Also, convergence requires the solution to be smooth with respect to the parameters

describing the input uncertainty [105].

Intrusive generalized chaos methods for nonlinear conservation laws have been

investigated in e.g. [100], where a reduced-cost Roe solver with entropy corrector was

presented, and in [99] with different localized representations of uncertainty in initial

functions and problem coefficients.

1.2 Outline

The aim of Chapter 2 and Chapter 3 of this thesis is to give a theoretical background

for the numerical and theoretical results to be presented in subsequent chapters. The

theory of spectral expansions of random fields is outlined in Chapter 2, followed by

an exposition of methods for the solution of PDEs with stochastic input in Chapter

3. Numerical discretization schemes are described in Chapter 4. As a motivation for

the use of generalized polynomial chaos methods as well as the numerical methods

of our choice, Chapter 5 introduces an advection-diffusion problem with a smooth

solution. For general nonlinear conservation laws, the solutions are non-smooth in the

deterministic case. In order to find suitable numerical methods and robust stochastic

representations for the corresponding stochastic Galerkin formulations, we analyze


the regularity of conservation laws with stochastic input conditions. In Chapter 6,

we investigate Burgers’ equation with uncertain boundary conditions in terms of

regularity. Next, we investigate Burgers’ equation in terms of the effect of incomplete

boundary conditions. This chapter serves to illustrate the method of imposition

of weak characteristic boundary conditions employed in all subsequent chapters. A

stochastic Galerkin method for the Euler equations combining robust representation

of input uncertainty with shock capturing methods is presented in Chapter 7. Finally,

in Chapter 8, we generalize the analysis of regularity to a two-phase flow problem.

Based on the spatial localization of smooth and non-smooth solution regions, we then

combine high-order methods with shock capturing methods in a hybrid scheme.

1.3 Accomplishments

The main part of this thesis is based on published articles and technical reports where

the author of this thesis is the first author. Chapter 5 is entirely based on [72] and

Chapter 6 contains results from [74] and [78]. Chapter 7 can be found in [77] and

Chapter 8 in [75].

In terms of original research, I have derived regularity properties of some stochastic

conservation laws and, for some cases, their stochastic Galerkin equivalents. The

analysis is to some extent based on the analysis of a special case in [17]. The extensive

analysis of regularity in [88] provided a theoretical framework that serves as a basis

for my generalizations to certain nonlinear problems.

The stability analysis using summation-by-parts operators would not have been

possible without the solid knowledge of professor Nordstrom. My contribution is to

extend the SBP framework to the stochastic Galerkin problems.

To the best of my knowledge, Roe’s variables have never been used in a stochastic

Galerkin formulation for the Euler equations. (In a deterministic setting, it does

not make sense to use them which explains why they probably have never been used

before, apart from, of course, the construction of the Roe average matrix for which

they were originally introduced by P. Roe).

All numerical results presented here are produced by scripts most of which I have


written from scratch. Exceptions include high order finite difference stencils from

collaborators at TDB, and eigenvalue decompositions and root finding, for which

I have used built-in functions in Matlab. Whenever results are presented that are

merely reproductions of other people’s work I have strived to indicate that and include

appropriate references.

Chapter 2

Random field representation

Nonlinear conservation laws subject to uncertainty are expected to develop solutions

that are discontinuous in the spatial as well as in the stochastic dimensions. In order

to allow piecewise continuous solutions to the problems of interest, we follow [25] and

broaden the concept of solutions to the class of functions equivalent to a function f ,

denoted Cf , and define a normed space that does not require its elements to be smooth

functions. Let (Ω,A, P ) be a probability space with event space Ω, and probability

measure P defined on the σ-field A of subsets of Ω. Let ξ = ξj(ω)Nj=1 be a set of

N independent and identically distributed random variables for ω ∈ Ω. We consider

u belonging to the space

L2(Ω,P) =

Cf |f measurable w.r.t.P ;

∫

Ω

f 2dP(ξ) <∞. (2.1)

The inner product between two functionals a(ξ) and b(ξ) belonging to L2(Ω,P) is

defined by

〈a(ξ)b(ξ)〉 =

∫

Ω

a(ξ)b(ξ)dP(ξ). (2.2)

This inner product induces the norm ‖f‖2L2(Ω,P) = 〈f 2〉.

Throughout the thesis, we will require that all random quantities are second-order

6

CHAPTER 2. RANDOM FIELD REPRESENTATION 7

random fields, i.e. any f(ξ) is subject to the constraint

‖f‖2L2(Ω,P) =

∫

Ω

f 2dP(ξ) <∞.

Spectral representations of random functionals aims at finding a series expansion in

the form

f(ω) =∞∑

k=0

fkψk(ξ(ω)),

where ψk(ξ)∞k=0 is the set of basis functions and fk∞k=0 is the set of coefficients to

be determined.

2.1 Karhunen-Loeve expansion

The Karhunen-Loeve expansion [48, 59], also known as proper orthogonal decompo-

sition or principal component analysis, provides a series representation of a random

field in terms of its spatial correlation (covariance kernel). Any second-order random

field f(x, ω) can be represented by the Karhunen-Loeve expansion,

f(x, ω) = f(x) +∞∑

k=1

ηk(ω)√λkφk(x),

where f(x) is the mean of f(x, ω), the random variables ηk are uncorrelated with

mean zero, and λk and φk are the eigenvalues and eigenfunctions of the covariance

kernel, respectively.

For random fields with known covariance structure, the Karhunen-Loeve expansion

is optimal in the sense of minimizing the mean-square error. The covariance function

of the output of a problem is in general not known a priori. However, Karhunen-

Loeve representations of the input data can often be combined with generalized chaos

expansions, to be presented in the next section.


2.2 Generalized chaos expansions

2.2.1 Generalized polynomial chaos expansion

The polynomial chaos (PC) framework based on series expansions of Hermite poly-

nomials of Gaussian random variables was introduced by Ghanem and Spanos [33]

and builds on the theory of homogeneous chaos introduced by Wiener in 1938 [108].

Any second order random field can be expanded as a generalized Fourier series in

the set of orthogonal Hermite polynomials, which constitutes a complete basis in the

Hilbert space L2(Ω,P) defined by (2.1). The resulting polynomial chaos series con-

verges in the L2(Ω,P) sense as a consequence of the Cameron-Martin theorem [11].

Although not limited to represent functions with Gaussian distribution, the polyno-

mial chaos expansion achieves the highest convergence rate for Gaussian functions.

Xiu and Karniadakis [115] generalized the polynomial chaos framework to the gener-

alized polynomial chaos (gPC) expansion, where random functions are represented by

any set of hypergeometric polynomials from the Askey scheme [5]. Hence, a function

with uniform distribution is optimally represented by Legendre polynomials that are

orthogonal with respect to the uniform measure, and a gamma distributed input by

Laguerre polynomials that are orthogonal with respect to the gamma measure etc.

The optimality of the choice of stochastic expansion pertains to the representation of

the input; the representation of the output of a nonlinear problem may well be highly

nonlinear as expressed in the basis of the input.

The Cameron-Martin theorem applies also to gPC with non-Gaussian random

variables, but only when the probability measure P(ξ) of the stochastic expansion

variable ξ is uniquely determined by the sequence of moments,

〈ξk〉 =

∫

Ω

ξkdP(ξ), k ∈ N0.

This is not always the case in situations commonly encountered; for instance, the log-

normal generalized chaos does not satisfy this property. Thus, there are cases when

the gPC expansion does not converge to the true limit of the random variable under


expansion [24]. However, lognormal random variables may be successfully represented

by gPC satisfying the determinacy of moments, e.g. Hermite polynomial chaos ex-

pansion. This motivates our choice to use Hermite polynomial chaos expansion to

represent lognormal viscosity in Chapter 5.

Consider a generalized chaos basis ψi(ξ)∞i=0 spanning the space of second order

(i.e. finite variance) random processes on this probability space. The basis functionals

are assumed to be orthonormal, i.e. they satisfy

〈ψiψj〉 = δij. (2.3)

Any second order random field u(x, t, ξ) can be expressed as

u(x, t, ξ) =∞∑

i=0

ui(x, t)ψi(ξ), (2.4)

where the coefficients ui(x, t) are defined by the projections

ui(x, t) = 〈u(x, t, ξ)ψi(ξ)〉, i = 0, 1, .... (2.5)

For notational convenience, we will not distinguish between u and its generalized

chaos expansion.

Independent of the choice of basis ψi∞i=0, we can express the mean and variance

of u(x, t, ξ) as

E(u(x, t, ξ)) = u0(x, t), Var(u(x, t, ξ)) =∞∑

i=1

u2i (x, t),

respectively. For practical purposes, (2.4) is truncated to a finite number M terms,

and we set

u(x, t, ξ) ≈M∑

i=0

ui(x, t)ψi(ξ). (2.6)

The number of basis functions M+1 is dependent on the number of stochastic dimen-

sions N and the order of truncation of the generalized chaos expansion. Assuming


the same order p of generalized chaos expansion for each stochastic dimension, the

relation M + 1 = (N + p)!/(N !p!) holds. Thus, high-dimensional problems tend to

become computationally infeasible and much effort has been put into alleviating the

computational cost through e.g. sparse-grid representations [90, 29]. In this thesis,

one stochastic dimension will be considered at the time, and the issue of the so-called

curse of dimensionality will not be further addressed.

The basis ψi∞i=0 is often a set of orthogonal polynomials. Given the two low-

est order polynomials, higher order polynomials can be generated by the recurrence

relation

ψn(ξ) = (anξ + bn)ψn−1(ξ) + cnψn−2(ξ),

where the coefficients an, bn, cn are specific to the class of polynomials.

The truncated chaos series (2.6) may result in solutions that are unphysical. An

extreme example is when a strictly positive quantity, say density, with uncertainty

within a bounded range is represented by a polynomial expansion with infinite range,

for instance Hermite polynomials of standard Gaussian variables. The Hermite series

expansion converges to the true density with bounded range in the limit M → ∞,

but for a given order of expansion, say M = 1, the representation ρ = ρ0 + ρ1H1(ξ)

results in negative density with non-zero probability since the Hermite polynomial

H1 takes arbitrarily large negative values. Similar problems may be encountered also

for polynomial representations with bounded support. Polynomial reconstruction of

a discontinuity in stochastic space leads to Gibbs oscillations that may yield negative

values of an approximation of a solution that is close to zero but strictly positive by

definition. Whenever discontinuities are involved, one should be careful with the use

of global polynomial representations.

Spectral convergence of the generalized polynomial chaos expansion is observed

when the solutions are sufficiently regular and continuous [115], but for general non-

linear conservation laws - such as in fluid dynamics problems - the convergence is

usually less favorable. Spectral expansion representations are still of interest for these

problems because of the potential efficiency with respect to brute force sampling meth-

ods, but special attention must be devoted to the numerical methodology used. For


some problems with steep gradients in the stochastic dimensions, polynomial chaos

expansions fail entirely to capture the solution [55]. Global methods can still give

superior overall performance, for instance Pade approximation methods based on ra-

tional function approximation [14], and hierarchical wavelet methods that are global

methods with localized support of each resolution level [52]. These methods do not

need input such as mesh refinement parameters and they are not dependent on the

initial discretization of the stochastic space. An alternative to polynomial expansions

for non-smooth and oscillatory problems is generalized chaos based on a localization

or discretization of the stochastic space [20, 79]. Methods based on stochastic dis-

cretization such as adaptive stochastic multi-elements [104] and stochastic simplex

collocation [109] will be described in some more detail in section 3.2.3. The robust

properties of discretized stochastic space can also be obtained by globally defined

wavelets, see [52, 54]. The next section outlines piecewise linear Haar wavelet chaos,

followed by an exposition about piecewise polynomial multiwavelet generalized chaos.

These classes of basis functions are robust to discontinuities.

2.2.2 Haar wavelet expansion

Haar wavelets are defined hierarchically on different resolution levels, representing

successively finer features of the solution with increasing resolution. They have non-

overlapping support within each resolution level, and in this sense they are localized.

Still, the Haar basis is global due to the overlapping support of wavelets belonging

to different resolution levels. Haar wavelets do not exhibit spectral convergence, but

avoid the Gibbs phenomenon.

Consider the mother wavelet function defined by

ψW (y) =

1 0 ≤ y < 12

−1 12≤ y < 1

0 otherwise,

(2.7)


Based on (2.7) we get the wavelet family

ψWj,k(y) = 2j/2ψW (2jy − k), j = 0, 1, ...; k = 0, ..., 2j−1,

Given the probability measure of the stochastic variable ξ with cumulative distribu-

tion function Fξ(ξ0) = P(ω : ξ(ω) ≤ ξ0), define the basis functions

Wj,k(ξ) = ψWj,k(Fξ(ξ))

Adding the basis function W0(y) = 1 in y ∈ [0, 1] and concatenating the indices j and

k into i = 2j + k so that Wi(ξ) ≡ ψWn,k(Fξ(ξ)), we can represent any random variable

u(x, t, ξ) with finite variance as

u(x, t, ξ) =∞∑

i=0

ui(x, t)Wi(ξ),

which is of the form (2.4). Figure 2.1 depicts the first eight basis functions of the

generalized Haar wavelet chaos.

2.2.3 Multiwavelet expansion

The main idea of multiwavelets is to combine the localized and hierarchical structure

of Haar wavelets with the convergence properties of orthogonal polynomials by com-

plementing the piecewise constant wavelets with piecewise polynomial wavelets. The

procedure of constructing these multiwavelets using Legendre polynomials follows the

algorithm in [4]. The same algorithm is outlined in [54] but with one step missing.

This step is included in appendix A.

An alternative to gPC expansions for non-smooth and oscillatory problems is gen-

eralized chaos based on a localization or discretization of the stochastic space [20, 79].

Methods based on stochastic discretization include adaptive stochastic multi-elements

[104] and stochastic simplex collocation [109]. The robust properties of discretized

stochastic space can also be obtained by globally defined wavelets, see [52]. In this

paper, we follow the approach of [54] and use piecewise polynomial multiwavelets


−1 −0.5 0 0.5 1

−2

−1

0

1

2

Resoluti on l eve l 0: W 0

−1 −0.5 0 0.5 1

−2

−1

0

1

2

Resoluti on l eve l 0: W 1

−1 −0.5 0 0.5 1

−2

−1

0

1

2

Resoluti on l eve l 1: W 2, W 3

−1 −0.5 0 0.5 1

−2

−1

0

1

2

Resoluti on l eve l 2: W 4, W 5, W 6, W 7

Figure 2.1: Haar wavelets, resolution levels 0,1,2.

(MW), defined on sub-intervals of [−1, 1]. The construction of a truncated MW basis

follows the algorithm in [4].

Wavelets are defined hierarchically on different resolution levels, representing suc-

cessively finer features of the solution with increasing resolution. They have non-

overlapping support within each resolution level, and in this sense they are localized.

Still, the basis is global due to the overlapping support of wavelets belonging to dif-

ferent resolution levels. Piecewise constant wavelets, denoted Haar wavelets, do not

exhibit spectral convergence, but avoid the Gibbs phenomenon in the proximity of

discontinuities in the stochastic dimension.

Starting with the space VNp of polynomials of degree at most Np defined on the

interval [−1, 1], the construction of multiwavelets aims at finding a basis of piecewise

polynomials for the orthogonal complement of VNp in the space VNp+1 of polynomials

of degree at most Np + 1. Merging the bases of VNp and that of the orthogonal

complement of VNp in VNp+1, we obtain a piecewise polynomial basis for VNp+1.

Continuing the process of finding orthogonal complements in spaces of increasing


degree of piecewise polynomials, leads to a basis for L2([−1, 1]).

We first introduce a smooth polynomial basis on [−1, 1]. Let Lei(ξ)∞i=0 be the

set of Legendre polynomials that are defined on [−1, 1] and orthogonal with respect to

the uniform measure. The normalized Legendre polynomials are defined recursively

by

Lej+1(ξ) =√

2j + 3

(√2j + 1

j + 1ξLej(ξ)−

j

(j + 1)√

2j − 1Lej−1(ξ)

),

Le0(ξ) = 1, Le1(ξ) =√

3ξ.

The set Lei(ξ)Npi=0 is an orthonormal basis for VNp . Double products are readily

computed from (2.3), and higher-order products are precomputed using numerical

integration.

Following the algorithm by Alpert [4] (see Appendix A), we construct a set of

mother wavelets ψWi (ξ)Npi=0 defined on the domain ξ ∈ [−1, 1], where

ψWi (ξ) =

pi(ξ) −1 ≤ ξ < 0

(−1)Np+i+1pi(ξ) 0 ≤ ξ < 1

0 otherwise,

(2.8)

where pi(ξ) is an ith order polynomial. By construction, the set of wavelets ψWi (ξ)Npi=0

are orthogonal to all polynomials of order at most Np, hence the wavelets are orthog-

onal to the set Lei(ξ)Npi=0 of Legendre polynomials of order at most Np. Based on

translations and dilations of (2.8), we get the wavelet family

ψWi,j,k(ξ) = 2j/2ψWi (2jξ − k), i = 0, ..., Np, j = 0, 1, ..., k = 0, ..., 2j−1.

Let ψm(ξ) for m = 0, ..., Np be the set of Legendre polynomials up to order Np, and

concatenate the indices i, j, k into m = (Np+1)(2j+k−1)+i so that ψm(ξ) ≡ ψWi,j,k(ξ)

for m > Np. With the MW basis ψm(ξ)∞m=0 we can represent any random variable


−1 −0.5 0 0.5 1−2

−1

0

1

2

3

Legen d r e p o l yn om i a l s

−1 −0.5 0 0.5 1−4

−2

0

2

4

Reso l u t i on l evel 0 , o r t h . comp l emen t

−1 −0.5 0 0.5 1−5

0

5

Reso l u t i on l evel 1

−1 −0.5 0 0.5 1−10

−5

0

5

10

Reso l u t i on l evel 2

Figure 2.2: Multiwavelets for Np = 2, Nr = 3. Resolution level 0 consistsof the first Np+1 Legendre polynomials and their orthogonal complement.Resolution level j > 0 contains (Np + 1)2j wavelets each. Each basisfunction is a piecewise polynomial of order Np.

u(x, t, ξ) with finite variance as

u(x, t, ξ) =∞∑

m=0

um(x, t)ψm(ξ),

which is of the form (2.4). In the computations, we truncate the MW series both in

terms of the piecewise polynomial order Np and the resolution level Nr. With the

index j = 0, ..., Nr, we retain P = (Np + 1)2Nr terms of the MW expansion.

The truncated MW basis is characterized by the piecewise polynomial order Np

and the number of resolution levels Nr, illustrated in Figure 2.2 for Np = 2 and

Nr = 3. As special cases of the MW basis, we obtain the Legendre polynomial basis

for Nr = 0 (i = j = 0), and the Haar wavelet basis of piecewise constant functions

for Np = 0.

Chapter 3

Polynomial chaos methods

In this chapter we review methods for formulating partial differential equations based

on the random field representations outlined in Chapter 2. These include the stochas-

tic Galerkin method, which is the predominant choice in this thesis, as well as other

methods that frequently occur in the literature. We also include methods that are

not polynomial chaos methods themselves but viable alternatives to these.

3.1 Intrusive methods

3.1.1 Stochastic Galerkin methods

The stochastic Galerkin method was introduced by Ghanem and Spanos in order

to solve linear stochastic equations [33]. It relies on a weak problem formulation

where the set of solution basis functions (trial functions) is the same as the space of

stochastic test functions. Consider a general conservation law defined on a spatial

domain Ωx with boundary Γx subject to initial and boundary conditions, given by

∂u(x, t, ξ)

∂t+∂f(u(x, t, ξ), ξ)

∂x= 0, x ∈ Ωx, t ≥ 0, (3.1)

LΓ(u, x, t, ξ) = g(t, ξ), x ∈ Γx, t ≥ 0, (3.2)

u = h(x, ξ), x ∈ Ωx, t = 0. (3.3)

16

CHAPTER 3. POLYNOMIAL CHAOS METHODS 17

where u is the solution and f is the flux. A weak approximation of 3.1 is obtained

by substituting the truncated gPC series of the solution u given by (2.6) into (3.1)

and projecting the resulting expression onto the subspace of L2(Ω,P) spanned by the

truncated basis ψi(ξ)Mi=0. The result is the stochastic Galerkin formulation of (3.1),

∂uk(x, t)

∂t+

∂

∂x

⟨f

(M∑

i=0

uiψi(ξ), ξ

), ψk

⟩=0, x ∈ Ωx, t ≥ 0, (3.4)

〈LΓ(u, x, t, ξ), ψk〉 = 〈g, ψk〉 , x ∈ Γx, t ≥ 0, (3.5)

〈u, ψk〉 = 〈h, ψk〉 , x ∈ Ωx, t = 0. (3.6)

for k = 0, ...,M , where the inner product 〈., .〉 is defined in (2.2). Although prevalent

in the literature, there are situations, even for linear problems, when it is essential not

to restrict the gPC approximations of all input quantities (e.g. material parameters)

to the same order M as the gPC representation of the solution. An example is

given in Section 5.1.2, where we show that the stochastic Galerkin formulation of an

advection-diffusion equation leads to an ill-posed problem unless an order at least

2M approximation of the diffusion parameter is used whenever an order M gPC

approximation is used to represent the solution.

The stochastic Galerkin formulation (3.4) is an extended deterministic system

of coupled equations. In general, it is different from the corresponding deterministic

problem and therefore needs to be solved using different numerical solvers. Compared

to sample-based methods where a number of decoupled equations are solved, some

of which will be described below, the number of coupled equations of the stochas-

tic Galerkin method may be significantly smaller, especially in the case of multi-

dimensional problems [118].

3.1.2 Semi-intrusive methods

Alternative approaches to generalized chaos methods have also been presented in

the literature. Abgrall et. al. [2, 3] developed a semi-intrusive method based on

a finite-volume like reconstruction technique of the discretized stochastic space. A


deterministic problem is obtained by taking conditional expectations given a stochas-

tic subcell, over which ENO constructions are used to reconstruct the fluxes in the

stochastic dimensions. This makes it particularly suitable for non-smooth probability

distributions, in contrast to gPC, where the convergence of requires the solution to

be smooth with respect to the parameters describing the input uncertainty [105].

3.2 Non-intrusive methods

An alternative to the polynomial chaos approach with stochastic Galerkin projec-

tion is to use multiple samples of solutions corresponding to some realizations of the

stochastic inputs. Such non-intrusive methods do not require modification of existing

codes but rely exclusively on repeated runs of the deterministic code which make

them computationally attractive.

3.2.1 Stochastic collocation methods

An alternative to gPC methods is the class of stochastic collocation methods, i.e.,

sampling methods with interpolation in the stochastic space, c.f. [113]. Several inves-

tigations of the relative performance of stochastic Galerkin and collocation methods

have been performed, c.f. [61, 8, 101]. The significant size of the stochastic Galerkin

system may lead to inefficient direct implementations compared to collocation meth-

ods and preconditioned iterative Krylov subspace methods. However, the use of

suitable techniques for large systems, such as preconditioners, may result in speedup

for the solution of stochastic Galerkin systems compared to multiple collocation runs

[101]. For high-dimensional problems where the collocation methods tend to become

prohibitively expensive, sparse grid adaptive methods have been suggested to alleviate

the computational cost [26].

Stochastic collocation takes a set of solutions u(j) evaluated at a set ξ(j) of

values of random input ξ, and constructs an interpolating polynomial from these

solution realizations [60, 113, 7]. A common choice of interpolation polynomials is

the set of Lagrange polynomials L(Mint)j (ξ)Mint

j=1 , defined by Mint points ξ(j)Mintj=1 ,


for which the polynomial interpolant becomes

Iu =

Mint∑

j=1

u(j)Lj(ξ). (3.7)

The distribution of the grid points ξ(j)Mintj=1 is implied by the measure P of ξ. For

instance, we choose ξ(j) to be the set of Gauss-Legendre quadrature points for the

case of uniformly distributed µ, and the set of Gauss-Hermite quadrature points for

the case of lognormal µ. The integral statistics of interest, such as moments, may

then be approximated by the corresponding quadrature rules. For instance, for some

quantity of interest 〈S(u)〉, we have

〈S(u)〉 ≈Mint∑

j=1

S(u(j))wj, (3.8)

where wj is the weight corresponding to the quadrature point ξ(j). The quadra-

ture points and weights can be computed through the Golub-Welsch algorithm [34].

Note that there is no need to find the Lagrange polynomials of (3.7) explicitly since

(Iu)(ξ(j)) = u(j) and we only need the values of Iu at the quadrature points in (3.8).

Stochastic collocation is similar to other non-intrusive methods such as pseu-

dospectral projection [83] and stochastic point collocation (stochastic response sur-

faces) [10], in that it relies on evaluating deterministic solutions associated with

stochastic quadrature points. The difference is the postprocessing step where quanti-

ties of interest are reconstructed by different means of numerical quadrature. Specif-

ically, in stochastic collocation, quantities of interest are computed directly without

representing the solutions as a gPC series. Pseudospectral projection, on the other

hand, involves the computation of the polynomial chaos coefficients of u through

numerical quadrature. Quantities of interest are then calculated as functions of the

polynomial chaos coefficients.

We note that properties of monotonicity, stiffness and stability of the PDE’s eval-

uated at the quadrature points is independent of the postprocessing step in which the

solution statistics are reconstructed. Therefore, the comparison between stochastic


Galerkin and stochastic collocation should be valid for a larger class of non-intrusive

methods, but we restrict ourselves to stochastic collocation for the numerical experi-

ments.

3.2.2 Spectral projection

Spectral projection, discrete projection or the pseudo-spectral approach [83, 111]

comprise a set of gPC based methods relying on deterministic solutions evaluated

at sampling points of the parameter domain. They are sometimes referred to as a

subgroup of the class of collocation methods [112]. Alternative spectral projection

approaches include weighted least squares formulations for determining the gPC co-

efficients (2.5) [43].

The integrals over the stochastic domain of the gPC projections defined by (2.5)

are approximated by sampling or employing numerical quadrature. For multiple

stochastic dimensions, sparse grids are attractive, e.g. Smolyak quadrature [49].

3.2.3 Stochastic multi-elements

In multi-element generalized polynomial chaos (ME-gPC), the stochastic domain is

decomposed into subdomains, and generalized polynomial chaos is applied element-

wise [104, 106]. Local orthogonal polynomial bases can be constructed numerically

using the Stieltjes procedure or the modified Chebyshev algorithm [27]. The stochastic

Galerkin method may be applied element-wise and in this sense the ME-gPC is an

intrusive method.

The multi-element framework allows the combination of refinement of the num-

ber of elements (h-refinement) and increasing the order gPC of each element (p-

refinement) [104].

Chapter 4

Spatial discretization

The problems investigated in this thesis can all be written as one-dimensional con-

servation laws,

ut + f(u)x = 0, 0 ≤ x ≤ 1, t ≥ 0, (4.1)

where u is the solution vector, and f is a flux function.

When solving (4.1) on a uniform grid, we will use two different classes of numerical

schemes. For smooth problems, we use high-order finite difference schemes, and for

non-smooth problems we apply shock capturing finite volume methods.

Summation by parts (SBP) is the discrete equivalent to integration by parts. SBP

operators are used for approximations of spatial derivatives. Their usefulness lies in

the possibility of expressing energy decay in terms of known boundary values, exactly

as in the continuous case [94, 64]. For smooth problems, one can often prove that the

numerical methods are stable and high-order accurate.

Despite the formal high-order accuracy of SBP operators, solutions with multiple

discontinuities are not well-captured. Instead, a more robust and accurate method is

the MUSCL scheme [103] or the HLL Riemann solver [42] with flux limiting, to be

described in Section 4.2.

21

CHAPTER 4. SPATIAL DISCRETIZATION 22

4.1 Summation-by-parts operators

In order to obtain stability of the semi-discretized problem for various orders of ac-

curacy and non-periodic boundary conditions, we use discrete operators satisfying a

summation-by-parts (SBP) property [50]. Instead of exact imposition of boundary

conditions, we enforce boundary conditions weakly through penalty terms, where the

penalty parameters are chosen such that the numerical method becomes stable.

The first and second derivative SBP operator were introduced in [50, 94] and

[13, 63], respectively. For the first derivative, we use the approximation ux ≈ P−1Qu,

where subscript x denotes partial derivative and Q satisfies

Q+QT = diag(−1, 0, . . . , 0, 1) ≡ B. (4.2)

Additionally, P must be symmetric and positive definite in order to define a

discrete norm. Operators of order 2n, n ∈ N, in the interior of the domain are

combined with boundary closures of order of accuracy n. It is possible to design

operators with higher order accuracy at the boundary, but this would require P to

have non-zero off-diagonal entries. We restrict ourselves to diagonal matrices P since

the proofs of stability to be presented in later sections rely on this assumption about

P .

For the approximation of the second derivative, we can either use the first deriva-

tive operator twice, or use uxx ≈ P−1(−M + BD)u, where M +MT ≥ 0, B is given

by (8.22), and D is a first-derivative approximation at the boundaries, i.e.,

D =1

∆x

d1 d2 d3 . . .

1. . .

1

. . . −d3 −d2 −d1

,

where di, i = 1, 2, 3, . . . , are scalar values leading to a consistent first-derivative

approximation at the boundaries.


4.1.1 Artificial dissipation operators

An artificial dissipation operator is a discretized even order derivative which is added

to the system to allow stable and accurate solutions to be obtained in the presence of

solution discontinuities. The artificial dissipation is designed to transform the global

discretization into a one-sided operator close to the shock location. Depending on

the accuracy of the difference scheme, this requires one or more dissipation operators.

All dissipation operators used here are of the form

A2k = −∆xP−1DTkBwDk, (4.3)

where P−1 is the diagonal norm of the first derivative as before, D is an approximation

of (∆x)k∂k/∂xk and Bw is a diagonal positive definite matrix. In most cases here,

Bw is replaced by a single constant βw. An appropriate choice of dissipation constant

results in an upwind scheme, suitable for problems where shocks evolve. For further

reading about the design of artificial dissipation operators we refer to [64].

4.2 Shock capturing methods

For finite volume methods on structured grids we partition the computational domain

into cells of equal size ∆x. Solution values uj are defined as cell averages of cell j,

and fluxes are defined on the edges of the cells.

4.2.1 MUSCL scheme

The MUSCL (Monotone Upstream-centered Schemes for Conservation Laws) scheme

was introduced in [103]. Let m be the number of spatial points and ∆x = 1/(m− 1)

and let u be the spatial discretization of u. The semi-discretized form of (4.1) is given

bydujdt

+Fj+1/2 − Fj−1/2

∆x= 0, j = 1, ...,m, (4.4)

where Fj+1/2 denotes the numerical flux function evaluated at the interface between

cells j and j + 1.


For the MUSCL scheme with slope limited states uL and uR, we take the numerical

flux function at the interface between cell j and cell j + 1

Fj+ 12

=1

2

(f(uL

j+ 12) + f(uR

j+ 12))

+1

2|Jj+ 1

2|(uLj+ 1

2− uR

j+ 12

), (4.5)

where J is an approximation of the flux Jacobian J = ∂f/∂u, from which is derived

the absolute value |Jj+ 12| given by

|Jj+ 12| = X

∣∣∣Λ(uj+ 12)∣∣∣X−1 =

1

2X∣∣∣Λ(uL

j+ 12) + Λ(uR

j+ 12)∣∣∣X−1. (4.6)

where Λ is a diagonal matrix with the eigenvalues of J and X is the eigenvector

matrix. J can be an average of the true Jacobian evaluated at the discretization

points or a Roe average matrix [87].

The left and right solution states are given by

uLj+ 1

2= uj + 0.5φ(rj)(uj+1−uj) and uR

j+ 12

= uj+1− 0.5φ(rj+1)(uj+2−uj+1)

respectively. The flux limiter φ(rj) takes the argument rj = (uj −uj−1)/(uj+1−uj).

As a special case, φ = 0 results in the first-order accurate upwind scheme. Second

order accurate and total variation diminishing schemes are obtained for φ that are

restricted to the region

φ(r) = 0, r ≤ 0,

r ≤ φ(r) ≤ 2r, 0 ≤ r ≤ 1,

1 ≤ φ(r) ≤ r, 1 ≤ r ≤ 2,

1 ≤ φ(r) ≤ 2, r ≥ 2,

φ(1) = 1,

as defined in [96]. The minmod, van Leer and superbee limiters that are used through-

out this thesis are all second order and total variation diminishing. For a more detailed

description of the MUSCL scheme, see e.g. [56].


4.2.2 HLL Riemann solver

As a simpler alternative to the MUSCL-Roe solver, we use the HLL (after Harten,

Lax and van Leer) Riemann solver introduced in [42] and further developed in [22].

Instead of computing the Roe average matrix needed for the Roe fluxes (7.8) and (7.9),

only the fastest signal velocities need to be estimated for the HLL solver. These signal

velocities SL and SR are the estimated maximum and minimum eigenvalues of the

Jacobian J = ∂f/∂u of the flux.

At the interface between the cells j and j + 1, the HLL flux is defined by

Fj+ 12

=

f(uLj+ 1

2

)if SL ≥ 0

SRf

(uLj+ 1

2

)−SLf

(uRj+ 1

2

)+SLSR

(uRj+ 1

2−uL

j+ 12

)

SR−SLif SL < 0 < SR

f(uRj+ 1

2

)if SR ≤ 0

.

In general, obtaining accurate eigenvalue estimates may be computationally costly.

However, for certain choices of stochastic basis functions in combination with known

eigenvalues of the deterministic system, we derive analytical expressions for the

stochastic Galerkin system eigenvalues [76].

The HLL-flux approximates the solution by assuming three states separated by

two waves. In the deterministic case, this approximation is known to fail in capturing

contact discontinuities and material interfaces of solutions to systems with more than

two waves [97]. For the Euler equations, the contact surface can be restored by

using the HLLC solver where three waves are assumed [98]. The stochastic Galerkin

system is a multi-wave generalization of the deterministic case, and similar problems

in capturing missing waves are expected. However, the robustness and simplicity

of the HLL-solver makes it a potentially more suitable choice compared to other

Riemann solvers that are theoretically more accurate, but also more sensitive to ill-

conditioning of the system matrix.

The HLL solver applied directly to the variable vector results in excessive numer-

ical diffusion [97], but by applying flux limiters in the same way as in the MUSCL

scheme for higher-order reconstruction, sharp features of the solution are recovered.

Chapter 5

The promises of gPC: linear

stochastic conservation laws

The aim of this chapter is to present accurate and stable numerical schemes for the

solution of a class of linear diffusive transport problems. The advection-diffusion

equation subject to uncertain viscosity with known statistical description is repre-

sented by a spectral expansion in the stochastic dimension. The gPC framework and

the stochastic Galerkin method are used to obtain an extended system which is an-

alyzed to find discretization constraints on monotonicity, stiffness and stability. A

comparison of stochastic Galerkin versus methods based on repeated evaluations of

deterministic solutions, e.g. stochastic collocation, is not the primary focus of this

chapter. However, we include a few examples on relative performance and numerical

properties with respect to monotonicity requirements and convergence to steady-state,

to motivate the use of stochastic Galerkin methods.

Special care is exercised to ensure that the stochastic Galerkin projection results

in a system with positive semi-definite diffusion matrix. The sign of the eigenvalues

of a pure advection problem is not a problem as long as the boundary conditions are

properly adjusted to match the number of ingoing characteristics, as shown in [37].

Unlike the case of stochastic advection, the sign of the eigenvalues of the diffusion

matrix of the advection-diffusion problem is crucial. A negative eigenvalue leads to

the growth of the solution norm and hence numerical instability. The source of the

26

CHAPTER 5. LINEAR STOCHASTIC CONSERVATION LAWS 27

growth is in the volume term, and no boundary condition can address that.

Advective-diffusive problems with uncertainty have been investigated by several

authors. Ghanem and Dham [31] considered a lognormal diffusion coefficient in a

multiphase porous medium problem. Le Maıtre et. al. investigated a set of Navier-

Stokes problems, resulting in coupled sets of advection-diffusion equations with un-

certain diffusion [53]. Wan et. al. investigated the advection-diffusion equation in

two dimensions with random transport velocity [107], and the effect of long-term time

integration of flow problems with gPC methods [105]. Xiu and Karniadakis studied

the Navier-Stokes equations with various stochastic boundary conditions [116], as well

as steady-state problems with random diffusivity [114]. We extend the work by pre-

vious authors by performing analysis of the numerical method used for the stochastic

Galerkin problem, e.g. investigating monotonicity and stability requirements and

convergence to steady-state.

The stochastic advection-diffusion equation and the stochastic Galerkin formu-

lation are presented in Section 5.1. Different basis functions and estimates of the

eigenvalues of the diffusion matrix are given in Section 5.2. We prove well-posedness

of the problem in Section 5.3 and monotonicity requirements for the solution are dis-

cussed in Section 5.4. In Section 5.5, we investigate the time-step limitations of the

numerical schemes using von Neumann analysis for a periodic case and summation-

by-parts operators for the nonperiodic case. We consider a spatially constant as well

as a spatially varying diffusion. Section 5.5 also includes analysis regarding the con-

vergence rate of the steady-state problem. Numerical results are then presented in

Section 5.6.

5.1 Problem definition

Let (Ω,F ,P) be a suitable probability space with the set of elementary events Ω and

probability measure P defined on the σ-algebra F . Let ξ(ω), ω ∈ Ω, be a random

variable defined on this space. Consider the following mixed hyperbolic-parabolic

stochastic PDE defined on (0, 1)× [0, T ] which holds P-almost surely in Ω,


∂u

∂t+ v

∂u

∂x=

∂

∂x

(µ(x, ξ)

∂u

∂x

), (5.1)

u(0, t, ξ) = g0(t, ξ),

∂u(x, t, ξ)

∂x|x=1 = g1(t, ξ),

u(x, 0, ξ) = uinit(x, ξ). (5.2)

Here the velocity v > 0 is a deterministic scalar and the diffusion µ(x, ξ) > µ0 > 0 is

a finite variance random field. As a special case of (5.1), we consider the case of µ(ξ)

being spatially constant, but with the same initial and boundary conditions.

In what follows, we approximate the stochastic solution u(x, t, ξ) using a gPC

expansion in the random space. We use the stochastic Galerkin method and compare

with the stochastic collocation method. Our objective is then to explore the stability,

stiffness and monotonicity requirements associated with the numerical solution of the

resulting coupled system of equations.

5.1.1 Uncertainty and solution procedure

We will consider the case where µ has a uniform marginal distribution and thus

bounded range, and the case where µ takes a lognormal distribution, a common

model in geophysics applications such as transport in porous media [19]. For other

distributions, we assume that the diffusion coefficient µ(ξ) has the cumulative distri-

bution function F . One may parameterize the uncertainty with a uniform random

variable ξ, defined on the interval [−1, 1] with constant probability density 0.5, de-

noted ξ ∼ U [−1, 1]. Then we get the expression

µ(ξ) = F−1

(ξ + 1

2

), (5.3)

which holds for general distributions F under the assumption that F−1 is defined.

For the cases of interest here, F−1 is a linear function in the case of a uniform µ. In

the case of a lognormal µ, we will alternatively use a different representation of µ in


terms of the Hermite polynomial chaos expansion in a Gaussian random variable.

In the context of a stochastic Galerkin solution of Eq. (5.1), we expand the

solution u(x, t, ξ) with respect to a gPC basis ψk(ξ)∞k=0. For optimal convergence,

under certain conditions [115], these polynomials are chosen to be orthonormal with

respect to the probability measure P ,

〈ψj, ψk〉 = δjk,

where δjk is the Kronecker δ. Possible choices of bases include Legendre polynomials

that are orthogonal with respect to the uniform measure and Hermite polynomials

that are orthogonal with respect to the Gaussian measure. These two sets of orthog-

onal polynomials are both used in the numerical experiments.

In the computations, we need to use a basis with finite cardinality. Hence, we

truncate the gPC basis ψk(ξ)∞k=0 to exactly represent polynomials up to order p,

up(x, t, ξ) =

p∑

k=0

uk(x, t)ψk(ξ), (5.4)

where ψk(ξ)pk=0 is the set of gPC basis functions of maximum order p.

5.1.2 Stochastic Galerkin projection

The unknown coefficients uk(x, t) are then computed through a Galerkin projection

onto the subspace spanned by the basis ψk(ξ)pk=0. Specifically, the truncated series

(5.4) is inserted into (5.1) and multiplied by each one of the basis functions ψk(ξ)pk=0.

The resulting expression is integrated with respect to the probability measure P over

the stochastic domain. This leads to a coupled linear system of deterministic PDE’s

of the form


∂uk∂t

+ v∂uk∂x

=

p∑

j=0

∂

∂x

(〈µψjψk〉

∂uj∂x

), k = 0, . . . , p, (5.5)

uk(0, t) = (g0)k, k = 0, . . . , p,

∂uk(x, t)

∂x|x=1 = (g1)k, k = 0, . . . , p,

uk(x, 0) = (uinit)k, k = 0, . . . , p,

where the orthogonality of the basis functions ψk(ξ)pk=0 has been used to cancel

terms. Here, (g0)k, (g1)k and (uinit)k are obtained by the projection of the left and

right boundary data and the initial function on basis polynomial ψk(ξ), k = 0, . . . , p.

In the sequel we use a compact notation to represent the system (5.5). Let u ≡(u0 u1 . . . up)

T be the vector of gPC coefficients in (5.4). Then the system (5.5) can

be equivalently written as

∂u

∂t+ V

∂u

∂x=

∂

∂x

(B(x)

∂u

∂x

), (5.6)

u(0, t) = g0(t),

∂u(x, t)

∂x|x=1 = g1(t),

u(x, 0) = uinit(x), (5.7)

where V = diag(v) and the matrix B is defined by

[B(x)]jk = 〈µ(x, ξ)ψjψk〉 j, k = 0, . . . , p. (5.8)

We will frequently refer to the case of spatially independent µ(ξ). Then, (5.6) can

be simplified to∂u

∂t+ V

∂u

∂x= B

∂2u

∂x2. (5.9)

With the gPC expansion of the diffusion coefficient, µ(x, ξ) =∑∞

k=0 µk(x)ψk(ξ),


(5.8) can be rewritten as

[B]ij = 〈µψiψj〉 =∞∑

k=0

µk(x) 〈ψiψjψk〉 , i, j = 0, . . . , p. (5.10)

For the basis functions that will be used in this Chapter, all triple (inner) products

〈ψiψjψk〉 satisfy

〈ψiψjψk〉 = 0, for k > 2p and i, j ≤ p. (5.11)

Explicit formulas for 〈ψiψjψk〉 for Hermite and Legendre polynomials can be found

in[6, 102]. Hence, using (5.11), (5.10) may be simplified to

[B]ij =

2p∑

k=0

µk(x) 〈ψiψjψk〉 , i, j = 0, . . . , p. (5.12)

The entries of B can thus be evaluated as finite sums of triple products that can

be computed exactly. Moreover, since [B]ij = 〈µψiψj〉 = 〈µψjψi〉 = [B]ji, it follows

that B is symmetric.

It is essential that the matrix B be always positive definite when it is derived from

a well-defined µ(ξ) > 0. This holds as a consequence of the following proposition. The

proof of the proposition follows closely that of the positive- (negative-) definiteness

of the advection matrix of Theorem 2.1 in [37] and Theorem 3.1 in [118]. However,

here we also emphasize the importance of a suitable polynomial chaos approximation

of B, since in this case negative eigenvalues would lead to instability of the numerical

method.

Proposition 1. The diffusion matrix B given by (5.12) derived from any µ(ξ) satis-

fying µ(ξ) ≥ 0 P-almost surely in Ω, has non-negative eigenvalues.


Proof. For any order p of gPC expansion and any vector u ∈ Rp+1,

uTBu =

p∑

i=0

p∑

j=0

uiuj

2p∑

k=0

〈ψiψjψk〉µk =

p∑

i=0

p∑

j=0

uiuj〈ψiψjµ〉 =

=

∫

Ω

(p∑

i=0

uiψi

)2

µ(ξ)dP(ξ) ≥ 0. (5.13)

Remark 1. The above proposition does not hold for the order p approximation µ(ξ) =∑p

k=0 µkψk(ξ). The second equality of (5.13) relies on substituting the gPC expansion

of µ of order 2p with the full gPC expansion of µ. This substitution is valid following

(5.11), but it would not be valid for the order p gPC approximation of µ. In the latter

case, the resulting B may have negative eigenvalues, thus, ruining the stability of the

discrete approximation of (5.6). Therefore, the 2p order of gPC expansion of µ is

crucial. Figure 5.1 illustrates this for the case of a lognormal µ(ξ) = exp(ξ) with

ξ ∼ N (0, 1).

0 2 4 6 8 10 12 14−1

−0.5

0

0.5

1

1.5

2

p

min λB

Order p

Order 2p

Figure 5.1: Minimum λB for µ = exp(ξ). Here [B(p)]ij =∑p

k=0〈ψiψjψk〉µkand [B(2p)]ij =

∑2pk=0〈ψiψjψk〉µk, respectively, and ψk(ξ) are the Hermite

polynomials.


The proof of Proposition 1 follows closely that of the positive- (negative-) definite-

ness of the advection matrix of Theorem 2.1 in [37]. However, here we also emphasize

the importance of a suitable polynomial chaos approximation of B, since in this case

negative eigenvalues would lead to instability of the numerical method.

5.1.3 Diagonalization of the stochastic Galerkin system

In order to reduce the computational cost, it is advantageous to diagonalize the

stochastic Galerkin systems whenever possible. If this is indeed the case, exact or

numerical diagonalization can be done as a preprocessing step, followed by the numer-

ical solution of p + 1 scalar advection-diffusion problems with different, but strictly

positive, viscosity µ((λB)j), where (λB)j are the eigenvalues of B, j = 0, . . . , p. The

system (5.6) can be diagonalized under certain conditions, which we elaborate on

next. Assuming, for a moment, that B(x) = WΛB(x)W T , i.e., that the eigenvectors

W of B(x) are not spatially dependent, then the system (5.6) can be diagonalized.

Multiplying (5.6) from the left by W T and letting u = W Tu, we get the diagonalized

system∂u

∂t+ V

∂u

∂x=

∂

∂x

(ΛB(x)

∂u

∂x

).

When the stochastic and space dependent components of µ(x, ξ) can be factorized

or only occur in separate terms of a sum, B(x) can be diagonalized. That is, for

general nonlinear functions f , g and h, and µ(x, ξ) = f(x)g(ξ) + h(x), we have

B(x) = f(x)WΛgWT + h(x) = W (f(x)Λg + h(x)I)W T = WΛB(x)W T ,

where ΛB(x) = f(x)Λg +h(x)I, Λg is a diagonal matrix, W is the eigenvector matrix

of the eigenvalue decomposition of [Bg]ij = 〈gψiψj〉, and I is the identity matrix. The

only requirement on f, g, and h is that the resulting µ(x, ξ) is positive for all ξ, and

bounded in the L2(Ω,P) norm.

Notice that the form µ(x, ξ) = f(x)g(ξ) + h(x) has a given distribution through-

out the domain, but not necessarily with the parameters of the distribution being


constant. For instance, with µ = c1(x) + c2(x) exp(ξ) and ξ ∼ N (0, 1), the viscos-

ity is lognormal for all x but with spatially varying statistics and diagonalization.

However, for the general case µ(x, ξ1, . . . , ξd) = exp(G(x, ξ1, . . . , ξd)), with G being a

multivariate Gaussian field, diagonalization is not possible.

For the general case of any empirical distribution with simultaneous spatial and

stochastic variation diagonalization is not possible. Then we solve the full stochastic

Galerkin system for which we perform analysis in the following sections. We also

present results on the diagonalizable case, since this admits a very direct comparison

to the stochastic collocation techniques, presented next.

5.2 The eigenvalues of the diffusion matrix B

In the analysis of the mathematical properties and the numerical scheme, e.g. well-

posedness, monotonicity, stiffness and stability, we will need estimates of the eigen-

values of B. We may express

B =∞∑

k=0

µkCk, (5.14)

where µk’s are the polynomial chaos coefficients of µ(ξ) and [Ck]ij = 〈ψiψjψk〉.

5.2.1 General bounds on the eigenvalues of B

Some eigenvalue estimates pertain to all gPC expansions, independent of the actual

choice of stochastic basis functions. For example, in cases where µ(ξ) is bounded

within an interval of the real line, the eigenvalues of the viscosity matrix B can

essentially be bounded from above and below by the upper and lower interval bound-

aries of possible values of µ, respectively. More generally, for any countable basis

ψk(ξ)∞k=0 of L2(Ω,P), by Theorem 2 of [93], it follows that there is a bound on the

set (λB)jpj=0 of the eigenvalues of B, given by

(λB)j ∈ conv(spect(µ(ξ))) = [µmin, µmax], (5.15)


where conv denotes the convex hull, and the spectrum spect of µ(ξ) is the essen-

tial range, i.e., the set of all possible values (measurable) µ can attain. For a more

general exposition and cases where µ is not confined to a convex region, we refer

the interested reader to [93]. In this paper, we only consider µ in intervals of finite

or infinite length (convex sets), and do not consider degenerate sets or single point

values. Following (5.15), for bounded µ such as uniformly distributed viscosity, the

eigenvalues (λB)j will be restricted to an interval for all orders p of gPC expansion.

We expect that the order of chaos expansion has a limited impact on system prop-

erties such as monotonicity and stiffness for these cases, as demonstrated in Section

5.5.3. For unbounded µ (e.g. lognormal distribution) there is no upper bound on the

eigenvalues of B and the system properties change with the order of gPC, also shown

in Section 5.5.3.

5.2.2 Legendre polynomial representation

When the viscosity µ is given by µ = µ0 + σξ, ξ ∼ U [−1, 1] and σ is a deterministic

scaling factor, only the first two Legendre polynomials are needed to represent µ

exactly, that is µ = µ0ψ0 + σ/√

3ψ1. Then, the stochastic Galerkin projection yields

a matrix B of the form

[B]jk = 〈µψjψk〉 = µ0I + µ1C1, j, k = 0, . . . , p,

where the eigenvalues of C1 are given by the Gauss-Legendre quadrature nodes scaled

by√

3. The scaling factor is due to the normalization performed to obtain unit-valued

inner double products of the Legendre polynomials. This result follows from the fact

that the eigenvalues of the matrix with (i, j) entries defined by 〈ξψiψj〉 are the same as

those of the Jacobi matrix corresponding to the three-term recurrence of the Legendre

polynomials. Thus, they are equal to the Gauss-Legendre quadrature nodes, see e.g.

[102, 35] for further details on this assertion.

The Gauss-Legendre nodes are located in the interval [−1, 1], from which it follows

that (λB)j ∈ [µ0 − σ, µ0 + σ]. Note that this holds exactly only for a uniformly

distributed µ; for non-uniform µ the polynomial expansion would result in a matrix


series representation of B of the form (5.14), where the matrices Ck are nonzero also

for k > 1.

5.2.3 Hermite polynomial representation

Representing the uncertainty of the input parameters with an orthogonal polynomial

basis whose weight function does not match the probability measure of the input

parameters may lead to poor convergence rates [116]. However, problems where the

inputs are functions of Gaussian variables may be represented by gPC expansions

in the Hermite polynomials with a weight function matching the Gaussian measure.

For instance, lognormal random processes can effectively be represented by Hermite

polynomial chaos expansion, see e.g., [30]. Let

µ(ξ) = c1 + c2eξ, c1, c2 ≥ 0, ξ ∼ N (0, 1). (5.16)

Then, the Hermite polynomial chaos coefficients of µ are given by

µj =c2e

1/2

√j!, j ≥ 1. (5.17)

The inner triple products of Hermite polynomials are given by

〈ψiψjψk〉 =

√i!j!k!

(s−i)!(s−j)!(s−k)!s integer, i, j, k ≤ s

0 otherwise,

(5.18)

with s = (i+ j + k)/2.

Applying Proposition 1 of Section 5.1.2 to the lognormal µ in (5.16), it follows

that the eigenvalues of B are bounded below by c1. The largest eigenvalue grows with

the order p of gPC expansion. Since the entries of B are non-negative due to (5.17)

and (5.18), by the Gershgorin’s circle theorem, the largest eigenvalue is bounded by

the maximum row (column) sum of B. This gives an estimate of the stiffness of

the problem, where a problem is loosely defined as stiff when its numerical solution


requires an excessively small time-step for stability.

5.3 Boundary conditions for well-posedness

A problem is well posed if a solution exists, is unique and depends continuously on

the problem data. Boundary conditions that lead to a bounded energy are necessary

for well-posedness. For hyperbolic stochastic Galerkin systems, boundary conditions

have been derived in [37] for the linear wave equation, and in [74] for the nonlinear

case of Burgers’ equation. Given the setting of (5.6) we derive the energy equation by

multiplying uT with the first equation in (5.6) and integrating over the spatial extent

of the problem. More specifically,

∫ 1

0

uT∂u

∂tdx+

∫ 1

0

uTV∂u

∂xdx =

∫ 1

0

uT∂

∂x

(B(x)

∂u

∂x

)dx, (5.19)

which can be compactly written as

∂‖u‖2

∂t+2

∫ 1

0

∂uT

∂xB(x)

∂u

∂xdx =

[uTV u− 2uTB(x)

∂u

∂x

]

x=0

−[uTV u− 2uTB(x)

∂u

∂x

]

x=1

.

(5.20)

Proposition 2. The problem (5.6) is well posed.

Proof. We consider homogeneous boundary conditions, i.e., let g0 = g1 = 0 in (5.7).

Notice that the right-hand-side of Eq. (5.20) is negative for the choice of boundary

conditions in (5.7), hence leading to a bounded energy norm of solution u in time.

Uniqueness follows directly from the energy estimate by replacing the solution by the

difference between two solutions u and v and noticing that the norm of the difference

is non-increasing with time, thus u ≡ v. The problem is parabolic with full-rank B

and the correct number of boundary conditions. This implies the existence of the

solution. Therefore, the problem (5.6) (and also (5.1)) is well posed.


5.4 Monotonicity of the solution

In this section we use a normal modal analysis technique [39] to derive necessary

conditions for the monotonicity of the steady-state solution of the system of equations

(5.9) with spatially constant, but random, viscosity. We provide these conditions for

second and fourth order discretization operators.

5.4.1 Second order operators

With standard second order central differences and a uniform grid, the semi-discrete

representation of (5.9) for the steady-state limit reads

Vui+1 − ui−1

2∆x= B

ui+1 − 2ui + ui−1

∆x2, (5.21)

where ui denotes the value of the vector of the discretized solution u at the grid point

i in space. This is a system of difference equations with a solution of the form

ui = yκi, (5.22)

for some scalar κ and vector y ∈ Rp+1 to be determined. By inserting (5.22) into

(5.21) we arrive at the eigen-problem

[∆x(κ2 − 1)

2V − (κ− 1)2B

]y = 0 (5.23)

whose non-trivial solution is obtained by requiring

det

(∆x(κ2 − 1)

2V − (κ− 1)2B

)= 0. (5.24)

The spectral decomposition of the symmetric positive definite matrix B, i.e., B =

WΛBWT , inserted into (5.24) leads to

v∆x(κ2j − 1)− (λB)j(κj − 1)2 = 0, j = 0, . . . , p. (5.25)


The solution to (5.25) is

κj = 1 or2 + θj2− θj

, j = 0, . . . , p, (5.26)

where θj = v∆x(λB)j

.

For a monotonic solution u, we must have κj ≥ 0 which demands a mesh such

that

Remesh = maxjθj ≤ 2. (5.27)

In the case of stochastic collocation, each realization will have a different mesh

Reynolds number Remesh based on the value of µ(ξ). In combination with the CFL

restriction on the time-step ∆t, this allows for larger time-steps for simulations cor-

responding to large values of µ(ξ), but forces small ones for small µ(ξ).

The importance of the mesh Reynolds number is illustrated in Figure 5.2. A step

function initially located at x = 0.2 is transported to the right and is increasingly

smeared by viscosity µ ∼ U [0.05, 0.15]. The mean value is monotonically decreasing,

but this property is clearly not preserved by numerical schemes not satisfying the

mesh Reynolds number requirement. It also has the effect of erroneously predicting

the location of the variance peaks.

When B can be diagonalized, the solution statistics are functions of linear combi-

nations of scalar advection-diffusion solutions with viscosity given by the eigenvalues

(λB)j. Then there is a local mesh Reynolds number (Remesh)j = θj for each eigenvalue

(λB)j, and a global mesh Reynolds number Remesh defined by (5.27). Remesh is de-

fined also for the cases when B cannot be diagonalized. If the global mesh Reynolds

number for the Galerkin system Remesh > 2, but the local mesh Reynolds number

(Remesh)j < 2 for some instances of the scalar advection-diffusion equation after di-

agonalization, the lack of monotonicity may not be obvious in the statistics, since

these are affected by averaging effects from all scalar solutions. Hence, the lack of

monotonicity of the mean solution is more obvious if (Remesh)j > 2 for all j = 0, . . . , p.

This is shown in Figure 5.3 with µ ∼ U [0.14, 0.16] for Remesh = 3 (and (Remesh)j > 2,

j = 0, . . . , p) and Remesh = 1, respectively.


Remark 2. The condition on the mesh Reynolds number is no longer present with

an upwind scheme, expressed as a central scheme with a certain amount of artificial

dissipation. To see this, let the diagonalized scheme with artificial dissipation be given

by

Vui+1 − ui−1

2∆x−ΛB

ui+1 − 2ui + ui−1

∆x2= α(ui+1 − 2ui + ui−1).

The choice α = v/(2∆x) leads to upwinding. With the ansatz (5.22), we get κj = 1

or κj = 1 + v∆x/(λB)j for j = 0, . . . . , p. This shows that the solution is oscillation

free independent of the mesh Reynolds number. However, the upwinding adversely

affects the accuracy of the solution.

5.4.2 Fourth order operators

With fourth order central differences, the semi-discrete representation of (5.9) for the

steady-state limit is given by

V−ui+2 + 8ui+1 − 8ui−1 + ui−2

12∆x= B−ui+2 + 16ui+1 − 30ui + 16ui−1 − ui−2

12∆x2.

(5.28)

Following the procedure of monotonicity analysis used for the second order operators

with the ansatz ui = yκi inserted in (5.28), we arrive at the eigen-problem

[(−κ4 + 8κ3 − 8κ+ 1)∆xV − (−κ4 + 16κ3 − 30κ2 + 16κ− 1)B

]y = 0. (5.29)

One may verify that κ = 1 is a root of (5.29), just as in the case of second order

central differences. Using the spectral decomposition of B and factoring out (κ− 1),

we obtain the third order equation

(1− θj)κ3j − (15− 7θj)κ

2j + (15 + 7θj)κj − (1 + θj) = 0, (5.30)

for j = 0, . . . , p. By Descartes’ rule of signs, (5.30) has only positive roots κj > 0 for

0 < θj < 1. For θj > 1, (5.30) has as at least one negative root. For θj = 1, (5.30)


0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

Mean

Num

Ref

(a) m = 40, Remesh = 14.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6x 10

−5 Variance

Num

Ref

(b) m = 40, Remesh = 14.

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

Mean

Num

Ref

(c) m = 300, Remesh = 1.9.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6x 10

−5 Variance

Num

Ref

(d) m = 300, Remesh = 1.9.

Figure 5.2: Solution statistics at t = 0.01 using stochastic Galerkinwith p = 4 for diffusion of a moving step function, u(x, t, ξ) =

ρ0erfc(

(x− (x0 + v(t+ τ)))/√

(4µ(ξ)(t+ τ)))

, µ(ξ) ∼ U [0.05, 0.15], ρ0 =

0.1, τ = 0.005, x0 = 0.2, and v = 1. Here, m denotes the number of spatialgrid points.


0.12 0.14 0.16 0.18 0.2 0.22 0.240.15

0.16

0.17

0.18

0.19

0.2

0.21

Mean

Num

Ref

(a) Remesh = 3, m = 70.

0.12 0.14 0.16 0.18 0.2 0.22 0.240.15

0.16

0.17

0.18

0.19

0.2

0.21

Mean

Num

Ref

(b) Remesh = 1, m = 200.

Figure 5.3: Mean solution at t = 0.001 for diffusion of a moving stepfunction, p = 4.

reduces to a second order equation with two positive roots. Hence, the monotonicity

condition κj ≥ 0 for the fourth order operators is equivalent to the mesh Reynolds

number bound

Remesh = maxjθj ≤ 1. (5.31)

Remark 3. The monotonicity analysis for sixth order operators can be performed

by following the method used for the fourth order ones. The mesh Reynolds number

monotonicity condition for sixth order operators is Remesh ≤ 23.

Figure 5.4 depicts an initial step function after 40 time steps, solved with second,

fourth and sixth order operators, respectively. The undershoots of the solutions tend

to increase with the order of the scheme which is in-line with the restriction on Remesh

that becomes more severe for higher order operators.

5.5 Stability of the semi-discretized problem

A numerical scheme is stable if the semi-discrete problem with homogeneous bound-

ary conditions leads to a bounded energy norm. A stable and consistent scheme


0.53 0.54 0.55 0.56 0.57 0.58 0.59

−0.01

0

0.01

0.02

0.03

0.04

x

2nd

4th

6th

(a) Remesh = 0.90.

0.53 0.54 0.55 0.56 0.57 0.58 0.59

−0.01

0

0.01

0.02

0.03

0.04

x

2nd

4th

6th

(b) Remesh = 1.90.

Figure 5.4: Mean solution for diffusion of a moving step function after 40time steps, p = 4, µ ∼ U [0.0095, 0.0195], m = 61 spatial points and twodifferent Remesh. The undershoot grows with the order of the operators.

converges by the Lax equivalence theorem. Our primary interest is the general case

of non-periodic boundary conditions, but the well-known periodic case with spatially

constant viscosity µ(ξ) is also included for comparison.

5.5.1 The initial value problem: von Neumann analysis

We consider the cases of second and fourth order accurate periodic versions of the

central finite difference operators in [63], and show that the amplification factors have

negative real parts, describing ellipses in the negative half-plane of the complex plane.

The generalization to higher order operators is straightforward.

Second order operators

Assuming spatially constant µ and diagonalizing (5.9), and using the standard central

difference discretization, for all k = 0, . . . , p, we get

∂uj∂t

+ vuj+1 − uj−1

2∆x= λk

uj+1 − 2uj + uj−1

(∆x)2. (5.32)


We assume periodic boundary conditions and use the Fourier ansatz uj = ueiα∆xj,

where α is the Fourier parameter. Then, with θk = v∆x/(2(λB)k), (5.32) becomes

∂u

∂t= −i v

∆x

eiα∆x − e−iα∆x

2iu + λk

eiα∆x − 2 + e−iα∆x

(∆x)2u =

= − v

∆x

[sin(α∆x)i+

2

θk(1− cos(α∆x))

]u. (5.33)

The coefficient of u in the right hand side of (5.33) is an expression of the form

f(ω) = c1 cos(ω)+ic2 sin(ω)+c3, i.e., the parametrization of an ellipse in the complex

plane. The real part is always non-positive due to the additive constant, so the

spectrum is an ellipse in the negative half-plane.

Fourth order operators

The fourth order semi-discretization is given by

∂uj∂t

+v−uj+2 + 8uj+1 − 8uj−1 + uj−1

12∆x= λk

−uj+2 + 16uj+1 − 30uj + 16uj−1 − uj−2

12(∆x)2.

(5.34)

Again using the Fourier ansatz, we have

∂u

∂t= i

v

6∆x

[ei2α∆x − e−i2α∆x

2i− 8

eiα∆x − e−iα∆x

2i

]u+

+λk

6(∆x)2

[−e

i2α∆x + e−i2α∆x

2+ 16

eiα∆x + e−iα∆x

2− 15

]u =

=

[iv

6∆x[sin(2α∆x)− 8 sin(α∆x)]− λk

3(∆x)2

[cos2(2α∆x) + 8(1− cos(α∆x))

]]u,

(5.35)

which again is an ellipse in the negative half-plane. This is illustrated in Figure 5.5,

showing the eigenvalues of the second and fourth order periodic spatial discretization

matrices Dper. Since Dper is applied to periodic functions, no special boundary treat-

ment is needed. Therefore, the entries of Dper are completely determined by the first


−0.5 −0.4 −0.3 −0.2 −0.1 0−8

−6

−4

−2

0

2

4

6

8x 10

−3

(∆x)2ℜ (λDper)

(∆x)

2ℑ(λ

Dper)

(a) Second order operators.

−0.5 −0.4 −0.3 −0.2 −0.1 0−8

−6

−4

−2

0

2

4

6

8x 10

−3

(∆x)2ℜ (λDper)

(∆x)

2ℑ(λ

Dper)

(b) Fourth order operators.

Figure 5.5: Eigenvalues for order p = 3 Legendre polynomial chaos with200 grid points, µ(ξ) ∼ U [0, 0.1], v = 1.

and second derivative approximations of (5.32) and (5.34), respectively. In Figure

5.5, the real part of the eigenvalues is denoted by <, and the complex part by =.

Each of the eigenvalues (λB)k, k = 0, 1, 2, 3, of B corresponds to one of the ellipses.

For uniformly distributed µ, the range of the eigenvalues is bounded, and increasing

the order of gPC does not increase the maximal eigenvalue significantly. Therefore,

the order of gPC expansion has a negligible impact on the time-step restriction in

this case.

For numerical stability, it is essential that the eigenvalues are all located in the

negative half-plane. In the next section, we perform stability analysis for the more

general case of an initial boundary value problem (with non-periodic boundary con-

ditions).

5.5.2 The initial boundary value problem

In order to obtain stability of the semi-discretized problem for various orders of ac-

curacy and non-periodic boundary conditions, we use discrete operators satisfying a

summation-by-parts (SBP) property [50].


Boundary conditions are imposed weakly through penalty terms, where the penalty

parameters are chosen such that the numerical method is stable. Operators of order

2n, n ∈ N, in the interior of the domain are combined with boundary closures of or-

der of accuracy n. For the advection-diffusion equation (5.1), this leads to the global

order of accuracy min(n + 2, 2n). We refer to [95] for a derivation of this result on

accuracy.

The first and second derivative SBP operator were introduced in [50, 94] and

[13, 63], respectively. For the first derivative, we use the approximation ux ≈ P−1Qu,

where subscript x denotes partial derivative and Q satisfies

Q+QT = diag(−1, 0, . . . , 0, 1) ≡ B. (5.36)


discrete norm. For the proof of stability of spatially varying viscosity µ(x, ξ), P must

be diagonal, so we will only use SBP operators leading to a diagonal P norm.

For the approximation of the second derivative, we can either use the first deriva-

tive operator twice, or use uxx ≈ P−1(−M + BD)u, where M +MT ≥ 0, B is given

by (8.22), and D is a first-derivative approximation at the boundaries, i.e.,

D =1

∆x

d1 d2 d3 . . .

1. . .

1

. . . −d3 −d2 −d1

,

where di, i = 1, 2, 3, . . . , are scalar values leading to a consistent first-derivative

approximation.

Data on the boundaries are imposed weakly through a Simultaneous Approxi-

mation Term (SAT), introduced in [12]. Let the matrices E0 = diag(1, 0, . . . , 0),

EN = diag(0, . . . , 0, 1) be used to position the boundary conditions, and let ΣI0, ΣV

0

and ΣVN be penalty matrices to be chosen for stability. Let ⊗ denote the Kronecker


product, of two matrices B and C by

B ⊗ C =

[B]11C . . . [B]1nC...

. . ....

[B]m1C . . . [B]mnC

.

The system (5.6) is discretized in space using SBP operators with the properties

described above. For the general case of spatially varying viscosity µ(x, ξ), first-

derivative operators will be successively applied to the viscosity term. An alternative,

not considered here, is to use the compact SBP operators for ∂/∂x(b(x)∂/∂x) with

b(x) > 0, developed in [62]. These operators have minimal stencil width for the

order of accuracy. We will first perform the stability analysis for the general case of

spatially varying viscosity. As a further illustration of the SBP-SAT framework, we

will then perform stability analysis for the special case of spatially constant viscosity

using compact second-derivative SBP operators.

Spatially varying viscosity

Consider the case of a spatially varying µ = µ(x, ξ), given by (5.6). Since µ depends

on x, we cannot write the semi-discretized version of B as a Kronecker product.

Instead, we introduce the block diagonal matrix

B = diag(B(x1), B(x2), . . . , B(xm)).

Note that B and the matrix (P−1 ⊗ I) commute, i.e.,

(P−1 ⊗ I)B = B(P−1 ⊗ I). (5.37)

Additionally, B is symmetric, positive definite, and block diagonal. The matrix

(P−1 ⊗ I)B is a scaling of each diagonal block B(xj) of B with the factor p−1jj > 0.

Thus, (P−1 ⊗ I)B is symmetric and positive definite. The numerical approximation


of (5.6) using SBP operators is given by

∂u

∂t+ (P−1Q⊗ V )u = (P−1Q⊗ I)B(P−1Q⊗ I)u

+ (P−1 ⊗ I)(E0 ⊗ ΣI0)(u− 0) + (P−1 ⊗ I)(QTP−1 ⊗ I)(E0 ⊗ ΣV

0 )(u− 0)

+ (P−1 ⊗ I)(EN ⊗ ΣVN)((P−1Q⊗ I)u− 0), (5.38)

where the first line corresponds to the discretization of the PDE, and the second and

third lines enforce the homogeneous boundary conditions weakly, here expressed as

(u− 0). Although the numerical experiments are performed with nonzero boundary

conditions, it is sufficient to consider the homogeneous case in the analysis of stability.

Proposition 3. The scheme in (5.38) with ΣVN = −B(xN), ΣV

0 = B(x1), and ΣI0 ≤

−V/2 is stable.

Proof. Multiplying (5.38) by uT (P ⊗ I) and replacing Q = EN −E0−QT in the first

term of the right-hand-side, we obtain

uT (P ⊗ I)∂u

∂t+

Advective term︷︸︸︷uT (Q⊗ V )u =

Viscous terms from PDE︷︸︸︷uT (EN ⊗ I)B(P−1Q⊗ I)u

−uT (E0 ⊗ I)B(P−1Q⊗ I)u− uT (QT ⊗ I)B(P−1 ⊗ I)(Q⊗ I)u︸︷︷︸Viscous terms from PDE

+ uT (E0 ⊗ ΣI0)u︸︷︷︸

Adv. penalty term

+ uT (QTP−1 ⊗ I)(E0 ⊗ ΣV0 )u︸︷︷︸

Left viscous penalty term

+ uT (EN ⊗ ΣVN)(P−1Q⊗ I)u︸︷︷︸

Right viscous penalty term

. (5.39)

The right viscous penalty term and the first viscous term from the PDE cancel if

we set ΣVN = −B(xN). Adding the transpose of the remaining terms of (5.39) to

themselves and using (5.37), we arrive at the energy equation


∂

∂t‖u‖2

P⊗I +

Advective boundary terms︷︸︸︷uT (EN ⊗ V )u− uT (E0 ⊗ V )u =

= −uT (E0 ⊗ I)B(P−1Q⊗ I)u− uT (QTP−1 ⊗ I)B(E0 ⊗ I)u︸︷︷︸Viscous terms from PDE

− 2uT (QT ⊗ I)B(P−1 ⊗ I)(Q⊗ I)u + 2uT (E0 ⊗ ΣI0)u︸︷︷︸

Adv. penalty term

+ uT (QTP−1 ⊗ I)(E0 ⊗ ΣV0 )u + uT (E0 ⊗ ΣV

0 )(P−1Q⊗ I)u︸︷︷︸Left viscous penalty terms

. (5.40)

The viscous terms from the PDE and the left viscous penalty terms cancel if we set

ΣV0 = B(x1). Pairing the second advective boundary term with the advective penalty

term for stability and choosing ΣI0 = −δV where δ ∈ R leads to

∂

∂t‖u‖2

P⊗I = uT0 (1−2δ)vu0−uTNvuN −2 [(Q⊗ I)u]T B(P−1⊗ I) [(Q⊗ I)u] . (5.41)

For δ ≥ 1/2, i.e., ΣI0 ≤ −V/2, the energy rate (5.41) shows that the scheme (5.38)

with variable B is stable, as the norm of u decays with time.

Spatially constant viscosity

For the case of spatially constant viscosity µ(ξ), we use compact second-derivative

SBP operators. We show that the choice of penalty matrices is similar to the case of

spatially varying viscosity µ(x, ξ) presented in the preceding section. The scheme is

given by

∂u

∂t+ (P−1Q⊗ V )u = (P−1(−M + BD)⊗B)u

+ (P−1 ⊗ I)(E0 ⊗ ΣI0)(u− 0) + (P−1 ⊗ I)(DT ⊗ I)(E0 ⊗ ΣV

0 )(u− 0)

+ (P−1 ⊗ I)(EN ⊗ ΣVN)((D ⊗ I)u− 0). (5.42)

Proposition 4. The scheme in (5.42) with the parameters ΣV0 = B, ΣV

N = −B, and


ΣI0 ≤ −V/2 is stable.

Proof. Multiplying (5.42) by uT (P ⊗ I) and using B = −E0 + EN , we get

uT (P ⊗ I)∂u

∂t+

Advective term︷︸︸︷uT (Q⊗ V )u =

=

Viscous terms from PDE︷︸︸︷−uT (M ⊗B)u− uT (E0D ⊗B)u + uT (END ⊗B)u +

Inviscid penalty term︷︸︸︷uT (E0 ⊗ ΣI

0)u

+ uT (DT ⊗ I)(E0 ⊗ ΣV0 )u︸︷︷︸

Left viscous penalty term

+ uT (EN ⊗ ΣVN)(D ⊗ I)u︸︷︷︸

Right viscous penalty term

. (5.43)

As in the case of variable viscosity, setting ΣV0 = B and ΣV

N = −B cancels the viscous

boundary terms of the ODE. Using the relation

uT (Q⊗ V )u = uT(

1

2(Q+QT )⊗ V

)u + uT

(1

2(Q−QT )⊗ V

)u

︸︷︷︸=0

=

=1

2uT ((−E0 + EN)⊗ V )u, (5.44)

we arrive at

uT (P ⊗ I)∂u

∂t= −uT (M ⊗B)u + uT (E0 ⊗ (V/2 + ΣI

0))u− uT (EN ⊗ V/2)u (5.45)

Finally, setting ΣI0 = −δV as in Section 5.5.2 and adding the transpose of (5.45) to

itself, we get the energy estimate

∂

∂t‖u‖2

(P⊗I) = uT0 (1− 2δ)vu0 − uTNvuN − uT ((M +MT )⊗B)u. (5.46)

Since M + MT and B are positive definite, the relation (5.46) with δ ≥ 1/2, i.e.

ΣI0 ≤ −V/2, proves that the scheme in (5.42) is stable.


5.5.3 Eigenvalues of the total system matrix

The semi-discrete scheme (5.42) is an ODE system of the form

∂u

∂t= Dtotu,

whose properties are determined by the complex-valued eigenvalues of the total system

matrix Dtot. The eigenvalues of Dtot must all have negative real parts for stability.

The utmost right lying eigenvalue determines the slowest decay rate, and thus the

speed of convergence to steady-state, see [65, 69]. The total spatial operator defined

by the scheme (5.42) with ΣV0 = B, ΣV

N = −B, and ΣI0 = −V/2 is given by the matrix

Dtot = (P−1 ⊗ I)(−(Q+ E0/2)⊗ V + (DTE0 − E0D −M)⊗B

). (5.47)

The location in the complex plane of the eigenvalues of Dtot depends on the dis-

tribution of µ, the spatial step ∆x, and the ratio between viscosity and advective

speed.

Figure 5.6 depicts the eigenvalues of Dtot for uniform µ(ξ) ∼ U [0, 0.04], v = 1,

different orders of polynomial chaos, and number of spatial grid points. The fourth

order SBP operators have been used and penalty coefficients are chosen according to

the stability analysis above. The eigenvalues all have negative real parts, showing that

the discretizations are indeed stable. Note that for an order of gPC expansion p, there

will be p+ 1 eigenvalues for each single eigenvalue of the corresponding deterministic

system matrix. The groups of p+1 eigenvalues are clustered around the corresponding

eigenvalue of the deterministic system matrix. When the range of possible viscosity

values (uncertainty) is increased, the spreading of the eigenvalues within each cluster

increases. When the mean of the viscosity is increased, the eigenvalues with non-zero

complex part will move away farther from the origin.

The change of location of the eigenvalues with increasing order of gPC expansion

gives an idea how the time-step restriction changes with the order of gPC. Figure

5.7 shows the eigenvalues of the total system matrix for uniform and lognormal µ

for first order (left) and fourth order (right) gPC. For the random viscosities to be


−0.2 −0.15 −0.1 −0.05

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

(∆x)2ℜ (λDtot)

(∆x)

2ℑ(λ

Dtot)

(a) p = 2, m = 20.

−0.2 −0.15 −0.1 −0.05 0

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

(∆x)2ℜ (λDtot)

(∆x)

2ℑ(λ

Dtot)

(b) p = 5, m = 20.

−0.2 −0.15 −0.1 −0.05 0

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

(∆x)2ℜ (λDtot

)

(∆x)

2ℑ(λ

Dtot)

(c) p = 2, m = 40.

−0.2 −0.15 −0.1 −0.05 0

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

(∆x)2ℜ (λDtot

)

(∆x)

2ℑ(λ

Dtot)

(d) p = 2, m = 80.

Figure 5.6: Eigenvalues of the total operator Dtot (including penaltyterms). Comparison of different orders of gPC (a) and (b), and differ-ent grid sizes (a), (c), and (d).


−0.5 −0.4 −0.3 −0.2 −0.1 0−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

(∆x)2ℜ (λDtot)

(∆x)

2ℑ(λ

Dtot)

lognormal µ

uniform µ

(a) p = 1.

−0.5 −0.4 −0.3 −0.2 −0.1 0−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

(∆x)2ℜ (λDtot)

(∆x)

2ℑ(λ

Dtot)

lognormal µ

uniform µ

(b) p = 4.

Figure 5.7: Eigenvalues of the total operator for m = 20 and differentorders of gPC. Here the viscosity µ has mean 〈µ〉 = 0.02 and varianceVar(µ) = 3.33× 10−5, and has uniform and lognormal distributions.

comparable, the coefficients are chosen such that the first and second moments of

the uniform and the lognormal µ match each other. For low-order polynomial chaos

expansions, the eigenvalues are close to each other and the systems are similar in

terms of stiffness. As the order of gPC expansion is increased, the scattering of the

eigenvalues of B resulting from lognormal µ increases (µ is unbounded). Hence, the

stochastic Galerkin system becomes stiffer with increasing order of gPC. The time-

step restriction for the uniform viscosity does not change significantly with the order

of gPC. The fourth order operators are a factor of approximately 1.5 stiffer than the

second-order operators. Here, we calculate stiffness as

ρstiff =max |λDtot |min |λDtot |

,

where |λDtot | denote the absolute values of the complex eigenvalues of the total spatial

operator Dtot.


5.5.4 Convergence to steady-state

As we let t→∞, the problem (5.6) with B(x) > 0 will reach steady-state, i.e., it will

satisfy ∂u/∂t = 0. This situation can be formulated as a time-independent problem

with solution u, that satisfies

V∂u

∂x=

∂

∂x

(B(x)

∂u

∂x

), (5.48)

u(x = 0) = g0,

∂u(x)

∂x|x=1 = g1.

By subtracting (5.48) from (5.6), we get the initial boundary value problem for the

deviation e = u− u from steady-state,

∂e

∂t+ V

∂e

∂x=

∂

∂x

(B(x)

∂e

∂x

), (5.49)

e(0, t) = 0,

∂e(x, t)

∂x|x=1 = 0,

e(x, 0) = uinit(x)− u(x) = e0(x), (5.50)

where it has been used that as t → ∞, the boundary data must be independent

of time and vanish. The problem (5.49) can be semi-discretized analogous to the

numerical schemes presented in Section 5.5.2. Thus, with Dtot defined in (5.47), the

aim is to solve the initial value problem

∂e

∂t= Dtote, t > 0, (5.51)

e = e0(x), t = 0, (5.52)

with the solution e(x, t) = e0(x) exp(Dtott). The largest real component of the eigen-

values of Dtot, denoted by max<(λDtot), must be negative; otherwise, the solution

will not converge to steady-state. The more negative max<(λDtot) is, the faster the

convergence to steady-state.

Although the boundary conditions may be altered in different ways to accelerate


the convergence to steady-state [69], we use the weak imposition of boundary con-

ditions described in Section 5.5.2 and compare the convergence to steady-state for a

diagonalizable stochastic Galerkin system with that of the stochastic collocation. The

number of iterations to reach convergence to steady-state depends on the size of the

time-step and the exponential decay of the solution, governed by the rightmost lying

eigenvalue of the total system matrix, max<(λDtot). For each stochastic quadrature

point of the advection-diffusion equation, there is a maximal time-step as well as a

maximal eigenvalue of the total system matrix. For stochastic Galerkin, each scalar

instance of the advection-diffusion equation corresponds to one of the eigenvalues

of B, and for stochastic collocation each instance corresponds to µ evaluated at a

stochastic quadrature point.

Explicit time integration together with various convergence acceleration tech-

niques such as residual smoothing, local time-stepping and multi-grids are the most

common methods for reaching steady-state in flow calculations [45, 47, 46]. In this

simplified case, explicit time integration with the maximum possible time step possi-

ble illustrates this scenario.

Figure 5.8 depicts the maximum time-step and the maximum eigenvalue of Dtot for

each one of the instances of advection-diffusion equations for different approximation

orders of the gPC and stochastic collocation. If diagonalization is possible, and for

sufficiently high orders of stochastic Galerkin, the scalar instances of the continuous

advection-diffusion equation with the most negative max<(λDtot) converge to steady-

state faster than the corresponding instances of stochastic collocation. However, the

severe time-step limit of stochastic Galerkin implies that a large number of time-

steps are needed to reach steady-state numerically with explicit time-stepping. It is

not clear from Figure 5.8 alone whether stochastic Galerkin or stochastic collocation

reaches steady-state numerically in the smaller number of time-steps. This will be

investigated in Section 5.6.2.

For non-diagonalizable stochastic Galerkin, the local bounds of Figure 5.8 on time-

steps and maximum eigenvalues no longer apply. Instead, the most severe local time-

step limit and eigenvalue will dominate the entire stochastic Galerkin system, with

deteriorating performance as a consequence. Stochastic collocation is still subject


to local time-step restrictions and local maximum eigenvalues, and is expected to

converge faster to steady-state than stochastic Galerkin.

A practical algorithm for steady-state calculations should be designed to be as

efficient as possible in terms of computational cost. For instance, one may use an

implicit/explicit scheme as devised in [118] for stochastic diffusion problems. What

we presented above is not an efficient algorithm for steady-state calculations, it is

rather an analysis of the properties of the semi-discrete system leading to convergence

to steady-state.

5.6 Numerical results

In the numerical examples of this Section, we use a fourth order Runge-Kutta method

for the time integration and the fourth order accurate SBP-SAT scheme in space. The

matrix operators can be found in [63]. The problem (5.1) with spatially independent

µ is solved for the initial function

u0(x, ξ) =ρ0√

4πµ(ξ)τexp

(−(x− (x0 + vτ))2

4µ(ξ)τ

), ρ0 > 0, x0 ∈ [0, 1], τ > 0,

for which the analytical solution at time t is given by

u(x, t, ξ) =ρ0√

4πµ(ξ)(t+ τ)exp

(−(x− (x0 + v(t+ τ)))2

4µ(ξ)(t+ τ)

). (5.53)

For the spatially varying µ(x, ξ), we employ the method of manufactured solutions

[85, 89] where we get the same solution as in the case of spatially constant µ(ξ) with

the aid of an appropriate source function s(x, t, ξ) in (5.1). The source function is

given by

s(x, t, ξ) =(x− (x0 + v(t+ τ)))(2µ(x, ξ)− µx(x, ξ)(x− (x0 + v(t+ τ))))µx(x, ξ)u

4µ2(x, ξ)(t+ τ)

+µ2x(x, t)

2µ(x, t)u. (5.54)


2 4 6 8 10−3

−2.5

−2

−1.5

−1

−0.5

x 10−3

Order of SC

(∆x)2 max ℜ (λD tot)

(a) max<(λDtot) for each quadrature point asa function of the order of stochastic collocation.

2 4 6 8 10−3

−2.5

−2

−1.5

−1

−0.5

x 10−3

Order of gPC

(∆x)2 max ℜ (λD tot)

(b) max<(λDtot) for the scalar advection-diffusion equations (one for each eigen-value (λB)) for different orders of stochasticGalerkin.

2 4 6 8 1010

−6

10−5

10−4

10−3

Order of SC

max ∆t

(c) Time-step limit for each quadrature pointof stochastic collocation.

2 4 6 8 1010

−6

10−5

10−4

10−3

Order of gPC

max ∆t

(d) Time-step limit for each scalar advectiondiffusion equation (diagonalizable system) ofstochastic Galerkin.

Figure 5.8: Convergence to steady-state depends on the limit on ∆t andmax< (λDtot). These quantities are plotted for lognormal viscosity µ(ξ) =0.02+0.05 exp(ξ), ξ ∼ N (0, 1). Stochastic collocation (left) and stochasticGalerkin (right).


51 101 201 40110

−8

10−7

10−6

10−5

10−4

10−3

m

Uniform µ

Mean SG

1st coeff SG

Var SG

Mean SC

Var SC

4th order decay

(a) Uniform µ ∈ (0.05, 0.15)

51 101 201 40110

−8

10−7

10−6

10−5

10−4

10−3

m

Lognormal µ

Mean SG

1st coeff SG

Var SG

Mean SC

Var SC

4th order decay

(b) Lognormal µ = 0.05 + 0.05 exp (ξ).

Figure 5.9: Convergence with respect to the spatial discretization us-ing stochastic Galerkin (SG) and stochastic collocation (SC). Plotted arenorms of the absolute errors in mean, first coefficient and variance withp = 12 order of generalized Legendre/Hermite chaos, and N = 13 quadra-ture points for stochastic collocation.

The stochastic reference solution (5.53) is projected onto the gPC basis functions

using a high-order numerical quadrature. The order N of the quadrature is chosen

sufficiently large so that the difference between two successive reference solutions of

order N−1 and N are several orders of magnitude smaller than the difference between

the solution from the numerical scheme and the reference solution.

Figure 5.9 illustrates the convergence as the spatial grid is refined for constant

order of gPC, p = 12 and N = 13 collocation points. For this high-order stochastic

representation, the theoretical fourth order convergence rate is attained for the mean

using stochastic Galerkin and stochastic collocation. For the variance, the stochas-

tic truncation error becomes visible for fine spatial meshes with lognormal µ, see

Figure 5.9 (b). There is no significant difference in performance between stochastic

collocation and stochastic Galerkin for this test case.


0.3

0.4

0.5

0.6

0.70.05

0.1

0.15

0

0.5

1

1.5

2

µ

Approximate u(x,ξ)

x

(a) Solution at t = 0.005, ∆x = 0.002

0.3

0.4

0.5

0.6

0.7

0.05

0.1

0.15

−0.4

−0.2

0

0.2

0.4

µ

Absolute error

x

(b) Error of the approximate solution.

Figure 5.10: Approximate solution with p = 3 order of Legendre chaos.

5.6.1 The inviscid limit

The theoretical results for the advection-diffusion problem are based on µ > 0. When

µ is arbitrarily close to 0 (but non-negative), the problem becomes nearly hyperbolic.

In the stochastic setting, this happens with non-zero probability whenever µ(ξ) ∈[0, c], c > 0. For small µ the mesh must be very fine, otherwise the mesh Reynolds

number requirement discussed in Section 5.4 will be violated. This is illustrated in

Figure 5.10 (numerical solution left and error right) for results obtained with fourth

order SBP operators, v = 1 and µ ∼ U [0.01, 0.19] . Note that the error is maximal

close to the inviscid limit of µ = 0.01. The solution (5.53) is a Gaussian in space for

any fixed value of ξ and t and varies exponentially in x with the inverse of µ. Thus,

spatial convergence requires a fine mesh for small µ. Deterioration of the convergence

properties for small µ is a well-known phenomenon for other problems with a parabolic

term, c.f. the Navier-Stokes equations in the inviscid limit.

Figure 5.11 shows the convergence in space for µ = 0.02 exp(ξ), ξ ∼ N (0, 1), us-

ing stochastic Galerkin (left) and stochastic collocation (right). When µ approaches

zero, the gradients become steeper, which requires finer meshes. The fourth order


51 101 201 40110

−6

10−5

10−4

10−3

10−2

10−1

m

Lognormal µ

Mean

1st coeff

Var

4th order decay

2.89

2.69

2.97

1.05

0.87

1.71

1.87 2.67

3.27

(a) Stochastic Galerkin with p = 9.

51 101 201 40110

−6

10−5

10−4

10−3

10−2

10−1

m

Lognormal µ

Mean

Var

4th order decay

1.87

2.90

3.32

0.88

1.80

2.89

(b) Stochastic collocation with 10 quadraturepoints.

Figure 5.11: Spatial convergence, lognormal viscosity, µ = 0.02 exp(ξ).T = 0.001 and τ = 0.005. Local numerical order of convergence indicatedin the plots.

convergence rate is not obtained for these coarse meshes. As long as the stochastic

basis is rich enough to represent the uncertainty, the choice of stochastic collocation

versus stochastic Galerkin has neither any significant effect on the rate of spatial con-

vergence, nor on the actual error. However, the number of stochastic basis functions

needed for a certain level of resolution increases as µ goes to zero; therefore, a simul-

taneous increase in spatial and stochastic resolution is necessary for convergence in

the inviscid limit.

The performance of stochastic Galerkin versus stochastic collocation depends on

the proximity to the inviscid limit. Figure 5.12 shows the convergence in the order of

gPC expansion (stochastic Galerkin) and the number of quadrature points (stochastic

collocation) for a fixed spatial grid. Two cases of lognormal µ are compared; one with

µmin = 0.2 and one with µmin = 0.01. For these cases, the stochastic Galerkin sys-

tem can be diagonalized, so the cost for stochastic Galerkin with an expansion order

p − 1 is equivalent to the cost of stochastic collocation with p quadrature points. If

the problem is highly diffusive, stochastic Galerkin is the more efficient method. If


2 4 6 8 10

10−8

10−6

10−4

10−2

Quadrature points/Bases

Error in variance

SG

SC

(a) Lognormal viscosity, µ = 0.2 + 0.01 exp(ξ).

2 4 6 8 10

10−4

10−3

10−2

10−1


Error in variance

SG

SC

(b) Lognormal viscosity, µ = 0.01 + 0.01 exp(ξ).

Figure 5.12: Stochastic Galerkin (SG) and stochastic collocation (SC) asa function of the order of gPC/number of quadrature points. Fixed meshof 201 spatial points.

the viscosity is close to zero with some non-zero probability, the difference in per-

formance decreases. Low viscosity sharpens the solution features. The effect of this

on the spatial convergence is seen in the low-viscosity case (µmin = 0.01) in Figure

5.12 (b), where the spatial truncation error becomes visible for high-order polynomial

chaos expansions. Due to the fixed number of spatial grid points, the convergence

rate decreases for high-order stochastic representations. With a sufficiently fine mesh,

one could show exponential convergence rate for any given order of stochastic repre-

sentation.

Both the stochastic collocation and diagonalizable stochastic Galerkin rely on a set

of scalar advection-diffusion problems, with the difference between the methods lying

in the choice of stochastic viscosity point values and the postprocessing used to obtain

statistics of interest. Figure 5.13 displays the difference in the range of the effective

values of µ for stochastic collocation and stochastic Galerkin (this corresponds to the

range of eigenvalues of B for stochastic Galerkin). From a purely numerical point

of view, stochastic Galerkin poses an additional challenge compared to stochastic


2 4 6 8 1010

−2

10−1

100

101

Order of gPC/SC

SG max λ

B

SG min λB

SC max µ(ξj)

SC min µ(ξj)

(a) Lognormal viscosity, µ = 0.01 + 0.2 exp(ξ).

2 4 6 8 1010

−2

10−1

100

101

Order of gPC/SC

SG max λ

B

SG min λB

SC max µ(ξj)

SC min µ(ξj)

(b) Lognormal viscosity, µ = 0.2 + 0.01 exp(ξ).

Figure 5.13: Minimum and maximum viscosity for different orders ofstochastic Galerkin (SG) and stochastic collocation (SC) for two differ-ent distributions of µ. This corresponds to the minimum and maximumλB for SG and to the minimum and maximum µ(ξ) for SC.

collocation in that a wider range of scales of diffusion must be handled simultaneously,

as shown in Figure 5.13. If we were to choose the eigenvalues of the matrix B as the

collocation points, the two methods would only differ in the postprocessing.

5.6.2 Steady-state calculations

Let the time of numerical convergence to steady-state be defined as the time Tss

when the discretized residual e satisfies ‖e‖2,∆x = (∆x∑m

i=1(e(xi, Tss))2)

1/2< tol,

where tol is a numerical tolerance to be chosen a priori. When µ is sufficiently large

so that diffusion is the dominating feature compared to advection, larger values of

the range of µ implies that Tss decreases, and steady-state is reached sooner. The

number of iterations to steady-state (i.e. the number of time-steps Tss/∆t) is inversely

proportional to ∆t. On the other hand, the limit on ∆t decreases with µ. Hence,

there is a trade-off in the number of iterations to steady-state between the size of the


time-step and the eigenvalues or quadrature point values of µ. In Figure 5.14, this

issue is explored for a lognormal µ = c1 + c2 exp(ξ) with different choices of c1 and c2.

From the previous analysis and Figures 5.8 and 5.13, we have observed how the

eigenvalues grow with the order p of gPC. For the most advection dominated case,

Figure 5.14 (a), the number of iterations grows superlinearly with the number of

quadrature points in the stochastic collocation approach. The same holds for up to

order p = 8 of stochastic Galerkin. For this case, the fastest convergence to steady-

state is obtained for stochastic collocation, where the range of possible µ values is

narrower than in the case of stochastic Galerkin, see Figure 5.13. In the more diffusive

case, i.e., Figure 5.14 (b), the relative performance for stochastic collocation versus

stochastic Galerkin is less pronounced. In the most diffusive case considered here, Fig-

ure 5.14 (c), the number of stochastic Galerkin iterations required to steady-state is a

sublinear function of the order of gPC. In this case, the largest eigenvalues of λB yield

advection-diffusion equations that converge within a relatively short time Tss, which

compensates for a severe time-step restriction. For these diffusive cases, stochas-

tic Galerkin is more efficient than stochastic collocation. The stochastic Galerkin

problem has been diagonalized to make the computational cost per iteration simi-

lar. In summary, Figure 5.14 shows that stochastic collocation converges faster than

stochastic Galerkin to steady-state for problems that are advection dominated or

moderately diffusive. For diffusion dominated flows, stochastic Galerkin converges

faster to steady-state compared to stochastic collocation.

5.7 Summary and conclusions

Summation-by-parts operators and weak boundary treatment have been applied to

a stochastic Galerkin formulation of the advection-diffusion equation. We have pre-

sented conditions for monotonicity for stochastic, but spatially constant, viscosity,

and stable schemes for the more general case of spatially varying and stochastic vis-

cosity.

Violation of the derived upper bound on the mesh Reynolds number may lead to

spurious oscillations, but it may also result in less obviously recognizable errors that


2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5x 10

4 Iterations


SG

SC

(a) µ = 0.02 + 0.005 exp(ξ).

2 4 6 8 100

1

2

3

4

5

6

7

8x 10

4 Iterations


SG

SC

(b) µ = 0.1 + 0.01 exp(ξ).

2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4x 10

5 Iterations


SG

SC

(c) µ = 2 + 0.2 exp(ξ).

Figure 5.14: Number of iterations to steady-state for different lognormalviscosity µ = c1 + c2 exp(ξ) using stochastic Galerkin and stochastic col-location. Here tol = 10−6.

are visible in different ways, e.g. as incorrect predictions of regions of large variation.

The limit on the mesh Reynolds number gets more severe for higher order operators.

In the case of spatially independent viscosity as well as spatially varying viscosity,

the advection-diffusion Galerkin system can be diagonalized under some conditions.

This results in a number of uncoupled systems and the numerical cost and perfor-

mance is very similar to non-intrusive methods such as pseudospectral projection and

stochastic collocation.

For diffusive problems, the stochastic Galerkin formulation leads to better ac-

curacy compared to stochastic collocation. For steady-state calculations, stochastic

collocation is faster for advection dominated cases and stochastic Galerkin is faster

for diffusive cases. When diagonalization of the viscosity matrix B is possible, the

problem should be solved in a non-intrusive way to reduce the computational cost.

SBP operators are suitable for smooth problems like the advection-diffusion equa-

tion investigated here, but many real-world flow problems contain regions of sharp

gradients or discontinuities. For these problems, one may use hybrid schemes con-

sisting of shock capturing methods in regions of strong variation, coupled through

weak interfaces with SBP schemes in smooth regions. This will be investigated in the


context of a two-phase problem in Chapter 8.

Chapter 6

Burgers’ equation and the

imposition of boundary conditions

The Burgers’ equation is an interesting and highly non-linear model problem. Al-

though the Burgers’ equation is of limited practical use in fluid mechanics applica-

tions, many results can be extended to other hyperbolic systems, such as the Euler

equations. In this chapter, a detailed uncertainty quantification analysis is performed

for the Burgers’ equation; we employ a spectral representation of the solution in the

form of polynomial chaos expansion. The equation is stochastic as a result of the un-

certainty in the initial and boundary values. Galerkin projection results in a coupled,

deterministic system of hyperbolic equations from which statistics of the solution can

be determined.

Previous uncertainty analyses have been performed with focus on the location of

the transition layer of a shock discontinuity arising in simulations of the Burgers’

equation with non-zero viscosity. Small one-sided perturbations imply large variation

in the location of the transition layer, so-called supersensitivity [117], which has been

shown to be a problem in deterministic as well as stochastic simulations. The results

from the polynomial chaos approach were accurate and the method was faster than

the Monte Carlo method [116, 117]. Burgers’ equation with a stochastic forcing term

has also been investigated and compared to standard Monte Carlo methods [44].

In this chapter we perform a fundamental analysis of the Burgers’ equation and

66

CHAPTER 6. BURGERS’ EQUATION AND BOUNDARY CONDITIONS 67

develop a numerical framework to study the effect of uncertainty in the boundary

conditions. With the assumption that the uncertainty of the boundary data has a

Gaussian distribution we allow the occurrence of unbounded solutions. Assuming

that the boundary data resemble the Gaussian distribution but are bounded to a suf-

ficiently large range does not alter the numerical results. Another reason for allowing

unbounded parameter range of the stochastic variable is that the problem becomes

more interesting from a mathematical point of view. Convergence is proved by a

suitable choice of functional space.

In order to ensure stability of the discretized system of equations, summation-by-

parts operators and weak imposition of boundary conditions [67, 68, 13] are used to

obtain energy estimates, as demonstrated in Chapter 5. The system is expressed in

a split form that combines the conservative and non-conservative formulation [66].

A particular set of artificial dissipation operators [64] and the simultaneous approx-

imation term (SAT) technique [12] for boundary treatment are used to enhance the

stability close to the shock. The discretization method is based on a fourth order cen-

tral difference operator in space and the fourth order Runge-Kutta method in time.

The summation-by-parts operators ensure stable solutions but the allowed time step

decreases with increasing gPC expansion as a result of the eigenvalues growing with

the order of the polynomial order (i.e. the size of the system).

An analytical solution is derived for a discontinuous and uncertain initial condi-

tion: the expectation and variance of the solution are shown to be smooth functions

while the coefficients of truncated polynomial chaos expansions are discontinuous.

An analysis of the characteristics of the truncated system also shows that the bound-

ary values are time-dependent and suggests a way of imposing accurate boundary

conditions.

In this chapter we also investigate to what extent low order approximations can

be used when appropriate high order boundary data (i.e. data with known high order

moments) are missing. Due to the lack of boundary data as well as to the compu-

tational cost of higher order polynomial chaos simulations, low-order approximations

with appropriate utilization of available data is a viable option. Because of the hyper-

bolic nature of the problem, information is traveling with finite but unknown speed


through the domain and will eventually affect the boundary.

By the convergence properties of the polynomial chaos series expansion, higher

order boundary terms are expected to decrease rapidly. On the other hand, although

small, these coefficients have a relatively large impact on the system eigenvalues

and might thus be crucial for accurate boundary treatment. In addition to this,

there are discontinuities in the stochastic dimension (we only assume one stochastic

dimension), which deteriorates the convergence. The net effect of the higher order

boundary coefficients is not clear and motivates the investigation of this chapter.

6.1 Polynomial chaos expansion of Burgers’ equa-

tion

Consider the inviscid Burgers’ equation in non-conservative form

∂u

∂t+ u

∂u

∂x= 0, 0 ≤ x ≤ 1. (6.1)

The solution u(x, t, ξ) is represented as a polynomial chaos series in the set of Hermite

polynomials ψi∞i=0 of a Gaussian variable ξ ∼ N (0, 1). The gPC series u(x, t, ξ) =∑∞

i=0 uiψi(ξ) is inserted into (6.1), which yields

∞∑

i=0

∂ui∂tψi(ξ) +

(∞∑

j=0

ujψj(ξ)

)(∞∑

i=0

∂ui∂x

ψi(ξ)

)= 0. (6.2)

A stochastic Galerkin projection is performed by multiplying (6.2) by ψk(ξ) for non-

negative integers k and integrating over the probability domain Ω with respect to

the Gaussian measure, i.e. with the weight function p(ξ) = exp(−ξ2/2)/√

2π. The

orthogonality of the basis polynomials then yields a system of deterministic equations.

By truncating the number of polynomial chaos coefficients to a finite number M , the

solution is projected onto a finite dimensional deterministic space. The result is a


symmetric system of equations,

∂uk∂t

+M∑

i=0

M∑

j=0

ui∂uj∂x〈ψiψjψk〉 = 0 for k = 0, 1, ...,M. (6.3)

For simplicity of notation, equation (6.3) can be written in matrix form as

ut + A(u)ux = 0 or ut +1

2

∂

∂x(A(u)u) = 0, (6.4)

where the matrix A(u) is defined by [A(u)]jk =∑M

i=0〈ψiψjψk〉ui.

6.1.1 Entropy and energy estimate for the M = 2 case

As an illustration, the 3 × 3 system given by (6.4) and truncation of the expansion

to M = 2 with a normalized Hermite polynomial basis is

u0

u1

u2

t

+

u0 u1 u2

u1 u0 +√

2u2

√2u1

u2

√2u1 u0 + 2

√2u2

u0

u1

u2

x

= 0,

Note that the matrix A(u) is symmetric. Let f = 12A(u)u denote the flux function

and introduce the entropy flux F = uTf −G, where

G =1

6u3

0 +1

2u0u

21 +

1

2u0u

22 +

√2

2u2

1u2 +

√2

3u3

2,

i.e.(∂G∂u

)T= f . Then, with the convex entropy function h = 1

2uTu, assuming

smoothness,

h(u)t + F (u)x = 0.

Anticipating the more general time-stability analysis of Section 6.4, we now consider

a semi-discretized formulation on an equidistant mesh with cell size ∆x and solution

uj = u(xj). With the straight-line parameterization uj(θ) = uj + θ(uj+1 − uj) and


flux fj+1/2 =∫ 1

0f(uj(θ))dθ, the semi-discrete formulation is given by

dujdt

+fj+1/2 − fj−1/2

∆x= 0

Multiplication by ∆xuTj and summing with respect to the grid index j, we obtain the

semi-discrete energy estimate

d

dt

(1

2∆x∑

j

uTj uj

)= ∆x

∑

j

uTjdujdt

= −∑

j

uTj

(fj+1/2 − fj−1/2

)=

∑

j

(uj+1 − uj)T fj+1/2 +B.T. =∑

j

Gj+1 −Gj +B.T. = B.T., (6.5)

which is an expression involving the boundary terms (B.T.) only. For the M = 2

case, we define

u2k,j+ 1

2

=u2k,j+1 + uk,juk,j+1 + u2

k,j

3, (6.6)

uk,j+ 12ul,j+ 1

2=

2uk,jul,j + uk,jul,j+1 + uk,j+1ul,j + 2uk,j+1ul,j+1

6, (6.7)

for k, l = 0, 1, 2. Then, the numerical flux function is given by

fj+ 12

=

12

(u2

0,j+ 12

+ u21,j+ 1

2

+ u22,j+ 1

2

)

u0,j+ 12u1,j+ 1

2+√

2u1,j+ 12u2,j+ 1

2

u0,j+ 12u2,j+ 1

2+√

22u2

1,j+ 12

+√

2u22,j+ 1

2

.

In Section 6.4, stability analysis is performed for the case of general order of expansion

M .

6.1.2 Diagonalization of the system matrix A(u)

For various purposes, such as analysis of well-posedness, design of dissipation opera-

tors and analysis of characteristics, the matrix A(u) is diagonalized. This is possible

for any u ∈ RM+1 since A(u) is always symmetric and thus has real-valued eigenvalues


and eigenvectors.

For any given u ∈ RM+1, let Λ denote a diagonal matrix with the eigenvalues λi of

A(u) on the main diagonal and let V be the matrix where the columns are the linearly

independent eigenvectors. Then A(u) = V ΛV T . Using the eigenvalue decomposition

and momentarily assuming a linearized Burgers’ equation, we obtain the diagonalized

system

wt + Λwx = 0,

where w = V Tu. Assuming non-zero eigenvalues, Λ can be split according to the sign

of its eigenvalues as Λ = Λ+ + Λ−. Introducing the split scheme into the system of

equations gives

wt + Λ+wx + Λ−wx = 0. (6.8)

This form will be used in the following sections.

6.2 Problem setup

In order to quantify the accuracy of the numerical methods, we need an analytical

solution to our problem. Consider the stochastic Riemann problem with an initial

shock location x0 ∈ [0, 1]

u(x, 0, ξ) =

uL = a+ p(ξ) if x < x0

uR = −a+ p(ξ) if x > x0

u(0, t, ξ) = uL, u(1, t, ξ) = uR

ξ ∈ N (0, 1),

(6.9)

As the most intuitive choice of polynomial chaos basis with regard to the boundary

uncertainty, the set of Hermite polynomials will be used. Here we will only consider

p(ξ) = bξ as a first order stochastic polynomial and a is a constant. By the Rankine-

Hugoniot condition, the shock speed is given by s = bξ, so for any bounded ξ the

shock location xs is

xs = x0 + tbξ.


The solution (for any bounded ξ) is given by

u(x, t, ξ) =

uL if x < x0 + tbξ

uR if x > x0 + tbξ

Since the analytical solution is known, the coefficients of the complete gPC ex-

pansion (M →∞) can be calculated for any given i, x and t. We have

ui(x, t) =

∫ ∞

−∞u(x, t, ξ)ψi(ξ)p(ξ)dξ = aδi0 + bδi1 − 2a

∫ ξs

−∞ψip(ξ)dξ, (6.10)

where we have defined ξs = (x − x0)/(bt) and p(ξ) = exp(−ξ2/2)/√

2π denotes the

Gaussian probability density function. Note that the limit of integration ξs(x, t) is

not a random variable itself. Using the recursion relation for normalized Hermite

polynomials

ψi(ξ) =1√i

(ξψi−1(ξ)− ψ′i−1(ξ)

),

(6.10) can be written

ui(x, t) = bδi1 + a

√2

iπψi−1(ξs)e

−ξ2s/2, (6.11)

for i ≥ 1. Differentiating (6.11) with respect to x and t results in

∂ui∂x

=∂ui∂ξs

∂ξs∂x

= −2aψi(ξs(x, t))p(ξs(x, t))1

bt,

and∂ui∂t

=∂ui∂ξs

∂ξs∂t

= 2aψi(ξs(x, t))p(ξs(x, t))x− x0

bt2,

from which it is clear that ui(x, t) is continuous in x and t for x ∈ [0, 1] and t > 0. (The

same is true for u0.) With an appropriate choice of initial function, the coefficients

would be continuous also for t = 0. For the treatment of a similar case of smooth

coefficients of a discontinuous solution, see [17].

The solution to the truncated problem will be compared to the expected value


and the variance of the analytical solution, given by

E(u) = a

(1− 2

∫ ξs(x,t)

−∞

e−ξ2/2

√2π

dξ

)(6.12)

and

V ar(u) = b2 + 4abe−ξ

2s/2

√2π

+ 4a2

∫ ξs

−∞

e−ξ2/2

√2π

dξ − 4a2

(∫ ξs

−∞

e−ξ2/2

√2π

dξ

)2

. (6.13)

These expressions can be generalized for different boundary conditions and polynomial

bases.

6.2.1 Regularity determined by the order of gPC

The solution of (6.1) is obtained through the evaluation of the gPC series with the

coefficients given by (6.10). Figure 6.1 shows the solution obtained by retaining only

the zeroth (a) and first (b) order gPC expansion terms. As a contrast to these low

order approximations, the true solution is discontinuous, as shown in (c). However,

due to the stochastic truncation error of the nonlinear problem, the finite order M

solution of the truncated stochastic Galerkin system (6.4) is not equal to the order

M solution of the original problem defined by the coefficients (6.10). The exact

solutions of the zeroth, first and second order stochastic Galerkin systems are shown

in Figure 6.2. Unlike the smooth coefficients of the original problem, the solutions

(and coefficients) of the truncated systems are discontinuous.


0

0.5

1 −2

0

2−2

0

2

ξx

(a) M = 0.

0

0.5

1 −2

0

2−2

0

2

ξx

(b) M = 1.

0

0.5

1 −2

0

2−2

0

2

ξx

(c) M =∞.

Figure 6.1: Exact solution u of the infinite order system as a function ofx and ξ at t = 0.5 for different orders of gPC. a = 1, b = 0.2.

0

0.5

1 −2

0

2−2

0

2

ξx

(a) M = 0.

0

0.5

1 −2

0

2−2

0

2

ξx

(b) M = 1.

0

0.5

1 −2

0

2−2

0

2

ξx

(c) M = 2.

Figure 6.2: Exact solution u of the truncated system as a function of xand ξ at t = 0.5 for different orders of gPC. a = 1, b = 0.2.

The dependence of smoothness on the order of gPC expansion is illustrated in

Figure 6.3, where the expectation is shown as a function of space for different orders

of gPC expansion and fixed time t = 0.5. For M = 0, 1, 2, there are, respectively,

1,2 or 3 solution discontinuities of the expectations. In Figure 6.3 (d), the M = 3

expectation appears to exhibit an expansion wave.


0 0.5 1

−1

−0.5

0

0.5

1

(a) M = 0.

0 0.5 1

−1

−0.5

0

0.5

1

(b) M = 1.

0 0.5 1

−1

−0.5

0

0.5

1

(c) M = 2.

0 0.5 1

−1

−0.5

0

0.5

1

(d) M = 3.

0 0.5 1

−1

−0.5

0

0.5

1

(e) M =∞.

Figure 6.3: Expectation u0 as a function of x at t = 0.5 for different ordersof gPC. a = 1, b = 0.2.

6.3 Well-posedness

A problem is well posed if the solution exists, is unique and depends continuously on

the problem data. An initial-boundary-value problem given by

ut +H(x, t, ∂∂xi

)u = F (x, t) x ∈ Ωphys t ≥ 0

u = f(x) x ∈ Ωphys t = 0

Lu = g(t) x ∈ Γphys t ≥ 0

(6.14)

is strongly well posed if the solution exists, is unique and is subject to the estimate

‖u‖2Ωphys

+

∫ t

0

‖u‖2Γphys

dτ ≤ Kceηct

(‖f‖2

Ωphys+

∫ t

0

‖F‖2Ωphys

+ ‖g‖2Γphys

dτ

). (6.15)

Kc and ηc are independent of F , f and g. See [38, 66] for more details.

The solution of (6.14) requires initial and boundary data. The data depend on the

expected conditions and the distribution of the uncertainty introduced; the stochastic

Galerkin procedure is again used to determine the polynomial chaos coefficients for

the initial and boundary values. In this section we will show that the truncated

system resulting from a truncated gPC expansion is well posed if correct boundary

conditions are given.


In the rest of this section, we assume u to be sufficiently smooth. Consider the

continuous problem in split form [84]:

ut + β∂

∂x

(A

2u

)+ (1− β)Aux = 0, 0 ≤ x ≤ 1.

Multiplication by uT and integration over the spatial domain Ωphys = [0, 1] yields

∫ 1

0

uTutdx+ β

∫ 1

0

uT∂

∂x

(A

2u

)dx+ (1− β)

∫ 1

0

uTAuxdx = 0.

Integration by parts gives

1

2

∂

∂t‖u‖2 = −β

2[uTAu]x=1

x=0 +β

2

∫ 1

0

uTxAudx− (1− β)

∫ 1

0

uTAuxdx. (6.16)

We choose β such thatβ

2− (1− β) = 0⇔ β =

2

3,

which is inserted into (6.16), yielding

∂

∂t‖u‖2 = −2

3[uTAu]x=1

x=0 =2

3

(wT0 (Λ+

0 + Λ−0 )w0 − wT1 (Λ+1 + Λ−1 )w1

). (6.17)

where A(u) has been diagonalized at the boundaries according to Sec. 6.1.2. Bound-

ary conditions are imposed on the resulting incoming characteristic variables which

correspond to Λ+ for x = 0 and Λ− for x = 1. On the left boundary, the conditions

are set such that

(w0)i = (V Tu(x = 0))i = (g0)i if λi > 0

and on the right boundary

(w1)i = (V Tu(x = 1))i = (g1)i if λi < 0.

The boundary norm is defined as

‖w‖2Γphys

= wTΛ+w − wTΛ−w = wT (Λ+ + |Λ−|)w = wT |Λ|w for x = 0, 1.


Inserting the boundary conditions and integrating Eq. (6.17) over time gives

‖u‖2Ωphys

+2

3

∫ t

0

‖w0‖2Γphys

+ ‖w1‖2Γphys

dτ ≤ ‖f‖2Ωphys

+4

3

∫ t

0

‖g0‖2Γphys

+ ‖g1‖2Γphys

dτ.

(6.18)

Since ‖w‖ ≤∥∥V T

∥∥ ‖u‖ ≤ C ‖u‖ for some C < ∞, the estimate (6.18) is in the form

of Eq. (6.15).

Remark 4. The assumption that u is smooth is actually true for an infinite number

of terms of the polynomial chaos expansion and t > 0.

6.4 Energy estimates for stability analysis

In order to ensure stability of the discretized system of equations, summation by

parts operators and weak imposition of boundary conditions [12, 67, 68, 13] are used

to obtain energy estimates. A particular set of artificial dissipation operators [64]

are used to enhance the stability close to the shock. Burgers’ equation has been

discretized with a fourth order central difference operator in space and a fourth order

Runge-Kutta method in time. For stability, artificial dissipation is added based on

the local system eigenvalues. The order of accuracy is not affected by the addition of

artificial dissipation.

The dominating error is instead due to truncation of the polynomial chaos ex-

pansion. General difficulties related to solving hyperbolic problems and nonlinear

conservation laws with spectral methods are discussed in [36, 73].

To obtain stability, we will use the so-called penalty technique [64] to impose

boundary conditions for the discrete problem [73]. Let E0 = (eij) where e11 = 1, eij =

0,∀i, j 6= 1 and En = (eij) where enn = 1, eij = 0, i, j 6= n. Define the block diago-

nal matrix Ag where the diagonal blocks are the symmetric matrices A(u(x)). With

penalty matrices Σ0 and Σ1 corresponding to the left and right boundaries respec-

tively, the discretized system can be expressed as

ut+Ag(P−1Q⊗I)u = (P−1⊗I)(E0⊗Σ0)(u−g0)+(P−1⊗I)(En⊗Σ1)(u−g1). (6.19)


Similarly, the conservative system in (6.4) can be discretized as

ut+1

2(P−1Q⊗I)Agu = (P−1⊗I)(E0⊗Σ0)(u−g0)+(P−1⊗I)(En⊗Σ1)(u−g1). (6.20)

Neither of the formulations (6.19) nor (6.20) will lead to an energy estimate. How-

ever, the non-conservative and conservative forms can be combined to get an energy

estimate by using the summation by parts property. A linear combination of the

conservative and the non-conservative form is used for the energy estimates, just as

in the continuous case. The split form is given by

ut + β1

2(P−1Q⊗ I)Agu + (1− β)Ag(P

−1Q⊗ I)u =

= (P−1 ⊗ I) [(E0 ⊗ Σ0)(u− g0) + (En ⊗ Σ1)(u− g1)] . (6.21)

Multiplication by uT (P ⊗ I) and then addition of the transpose of the resulting

equation yields

∂

∂t‖u‖2

(P⊗I) +β

2uT((Q⊗ I)Ag + Ag(Q

T ⊗ I))u+

+ (1− β)uT(Ag(Q⊗ I) + (QT ⊗ I)Ag

)u =

= 2uT (E0 ⊗ Σ0)(u− g0) + 2uT (En ⊗ Σ1)(u− g1). (6.22)

With the choice β = 2/3, the energy methods yields

∂

∂t‖u‖2

(P⊗I) =2

3

(uTx=0Aux=0 − uTx=1Aux=1

)+2uTx=0Σ0(ux=0−g0)+2uTx=1Σ1(ux=1−g1).

(6.23)

Restructuring (6.23) yields

∂

∂t‖u‖2

(P⊗I) = uTx=0(2

3A+ 2Σ0)ux=0−2uTx=0Σ0g0−uTx=1(

2

3A−2Σ1)ux=1−2uTx=1Σ1g1.

(6.24)

Stability is achieved by a proper choice of the penalty matrices Σ0 and Σ1. For that


purpose A is split according to the sign of its eigenvalues as

A = A+ + A− where A+ = V TΛ+V and A− = V TΛ−V. (6.25)

Choose Σ0 and Σ1 such that 23A+ + 2Σ0 = −2

3A+ ⇔ Σ0 = −2

3A+ and 2

3A− − 2Σ1 =

23A− ⇔ Σ1 = 2

3A−. We now get the energy estimate

∂

∂t‖u‖2

(P⊗I) = −2

3(ux=0 − g0)TA+(ux=0 − g0) +

2

3

[uTx=0A

−ux=0 + gT0 A+g0

]

− 2

3

[uT(x=1)A

+u(x=1) + gT1 A−g1

]+

2

3(u(x=1) − g1)TA−(u(x=1) − g1), (6.26)

which shows that the system is stable.

Remark 5. In the numerical calculations we use (6.20) for correct shock speed, see

[57].

In the analysis of well-posedness and stability above we have assumed that we

have perfect knowledge of boundary data but in practice it is rarely true. In practi-

cal calculations lack of data makes such analysis impossible and one has to rely on

estimates to assign boundary data. We will investigate the effect of that problem in

Section 6.9.

6.4.1 Artificial dissipation for enhanced stability

The complete difference approximation (6.20) augmented with artificial dissipation

of the form described in Section 4.1.1 is given by

(P ⊗ I)ut +1

2(Q⊗ I)Agu− (E0 ⊗ Σ0)(u− g1)− (En ⊗ Σ1)(u− g1) =

= −∆x∑

k

(DTk ⊗B)Bw,k(Dk ⊗ I)u, (6.27)

where Bw,k is a possibly non-constant weight matrix to be determined and k = 1, 2

for the fourth order accurate SBP operator.

Determining Bw,k in (6.27) requires estimates of the eigenvalues λj of A for j =


0, ...,M ; the largest eigenvalue is typically sufficient. For the system of equations

generated by polynomial chaos expansion of Burgers’ equation, max |λ| is not always

known. Since the only non-zero polynomial coefficients on the boundaries are u0

and u1 and since the polynomial chaos expansion converges in the L2(Ω,P) sense, a

reasonable approximation of the maximum eigenvalue of A is

|λ|max ≈ |u0|+M |u1| , (6.28)

where M is the order of polynomial chaos expansion. This estimate is justified by the

eigenvalue analysis performed in the next section as well as by computational results.

For the dissipation operators A2 and A4 in the simulations, we use

Bw,2 = diag

((|u0|+M |u1|)

6∆x

), Bw,4 = diag

((|u0|+M |u1|)

24∆x

). (6.29)

The second order dissipation operator is only applied close to discontinuities.

6.5 Time integration

The increase in simulation cost associated with higher order systems is due to a

number of factors. The size of the system depends both on the number of terms in

the truncated polynomial chaos expansion and the spatial mesh size.

For the Kronecker product A⊗B the relation

λi,jA⊗B = λiAλjB (6.30)

holds, where the indices i, j denotes all the eigenvalues of A and B respectively. This

enables a separate analysis of the eigenvalues corresponding to the polynomial chaos

expansion and the eigenvalues of the total spatial difference operator D. Assuming

constant coefficients, the maximum system eigenvalue is limited by

λmax ≤ (max λD)(max λA). (6.31)


The estimate (6.31) in combination with (6.28) will be used in order to obtain esti-

mates of the time step constraint.

6.6 Eigenvalue approximation

Analytic eigenvalues for the matrix A can only be obtained for a small number of

polynomial chaos coefficients and therefore approximations are needed. Even though

most eigenvalues of interest in this report can be calculated exactly for every particu-

lar case, a general estimate is of interest. The approximation of the largest eigenvalue

of the system matrix A is calculated from solution values on the boundaries, which are

the only values known a priori. For smooth solutions with boundary conditions where

the polynomial chaos coefficients ui are equal to 0 for i > 1, the higher order coeffi-

cients tend to remain small compared to lower order coefficients (strong probabilistic

convergence). For solutions where a shock is developing, higher order polynomial

chaos coefficients might grow and the approximation of the largest eigenvalue based

on boundary values is likely to be a less accurate estimate.

To get estimates of the eigenvalues, the system of equations can be written

ut +

(M∑

i=0

Ai(ui)

)ux = 0, (6.32)

where A(u) =∑M

i=0Aiui is a linear combination of the polynomial chaos coefficients.

The eigenvalue approximation used here is given by

maxλA = maxv∈RM+1

vT (∑Aiui)v

vTv≤

M∑

i=0

maxvi∈RM+1

vTi AivivTi vi

|ui| =∑

i

|ui|max |λAi | . (6.33)

Since A0(u0) = u0I, this approximation coincides with the exact eigenvalue for bound-

ary value with ui = 0 for i > 1. This can be seen by observing that if x1 is an

eigenvector with corresponding eigenvalue λ for the matrix A1 then A1x = λx and

(A1u1 + A0u0)x = u1λx+ u0Ix = (u1λ+ u0)x (6.34)


so u1λ+u0 and x1 are an eigenvalue-eigenvector pair of the matrix A = A0u0 +A1u1.

This shows that (6.28) is an appropriate eigenvalue approximation for problems where

only u0 and u1 are non-zero on the boundaries.

For a given boundary condition, the maximum eigenvalue of A0 corresponding to

the deterministic part of the condition does not change with increasing number of

polynomial chaos coefficients. However, the largest eigenvalue contribution from A1

grows with the number of polynomial chaos coefficients.

The eigenvalue approximation (6.28) is in general of the same order of magnitude

as the largest eigenvalue in the interior of the domain but might have to be adjusted

to remove all oscillations. The exact value is problem specific and an estimate based

on the interior values requires knowledge about the solution of the problem.

6.7 Efficiency of the polynomial chaos method

The convergence of the polynomial chaos expansion is investigated by measuring

the discrete Euclidean error norm of the variance and the expected value. For a

discretization with m spatial grid points, we have

‖εExp‖2 =1

m− 1

m∑

i=1

(E[u]m − E[uref ]m)2

and

‖εV ar‖2 =1

m− 1

m∑

i=1

(Var[u]m − Var[uref ]m)2

where uref denotes the analytical solution. Consider the model problem (6.9); the

problem is solved with the Monte Carlo method and gPC until time t = 0.3. Accuracy

(measured as the norm of the difference between the actual solution and the analytical

solution) and simulation cost are shown in Table 6.1 for the Monte Carlo method and

Table 6.2 for the gPC expansions.

For this highly non-linear and discontinuous problem, the polynomial chaos method

is more efficient than the Monte Carlo method with low accuracy requirements. The

convergence properties of these solutions are affected by the spatial grid size and the


0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x

E(u)

M=5M=3exact

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

u 1

M=5M=3exact

0 0.2 0.4 0.6 0.8 1

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x

u 2

M=5M=3exact

0 0.2 0.4 0.6 0.8 1

−0.3

−0.2

−0.1

0

0.1

0.2

x

u 3

M=5M=3exact

Figure 6.4: The first four gPC coefficients, t = 0.3, M = 5 and M = 3,m = 400.


N 10 50 100 400 1600‖εExp‖ 0.122 0.0374 0.0344 0.0257 0.0151‖εV ar‖ 0.127 0.0589 0.0426 0.0283 0.0189T (s) 240 1180 2390 9350 38460

Table 6.1: Convergence to (6.12) and (6.13) with the Monte Carlo method,m = 400, t = 0.3.

M 2 4 6 8‖εExp‖ 0.113 0.0544 0.0164 0.0150‖εV ar‖ 0.147 0.122 0.0409 0.0630T (s) 126 636 4180 10900

Table 6.2: Convergence to (6.12) and (6.13) with the polynomial chaosmethod, m = 400, t = 0.3.

accuracy of imposed artificial dissipation and no general conclusion of the relative

performances of the two methods will be drawn here. As will be further illustrated in

the section on analysis of characteristics, the solution coefficients of the truncated sys-

tem are discontinuous approximations to the analytical coefficients which are smooth.

Even though the gPC results do converge for this problem, the low order expansions

are qualitatively very different from the analytical solution, see for instance Figure

6.4. Also, note that excessive use of artificial dissipation is likely to produce a solution

closer to the analytical solution for lower order expansions. Note that, as expected

spatial grid refinement leads to convergence to the true solution of the truncated

system but does not get any closer to the analytical solution.

The use of artificial dissipation proportional to the largest eigenvalue makes the

solutions of large order expansions dissipative and spatial grid refinement is needed

for an accurate solution. This can be seen in Table 6.2, where the accuracy of the

variance decreases with large M .

6.7.1 Numerical convergence

The convergence of the computed polynomial chaos coefficients, the expected value

and the variance of the truncated system is investigated and comparisons to the


M 3 3 (dissipative) 4‖εExp‖ 0.0354 0.0173 0.0374‖εV ar‖ 0.0918 0.0370 0.0723

Table 6.3: Norms of errors for dissipative and non-dissipative solutions.

analytical solution derived in Section 6.2 are presented.

As mentioned earlier, the numerical results obtained for a small number of expan-

sion terms is expected to be a poor approximation to the analytical solution; this is

confirmed by the mesh refinement study reported in Figure 6.4 for M = 5. In this

particular application, the analytical solution admits continuous (smooth) coefficients

in spite of the discontinuous initial condition; on the other hand, the coefficients of

the truncated system are discontinuous.

Interestingly, the difference between the computed coefficients corresponding to

a finite gPC expansion (ui for i ≤ M) and the analytical (M = ∞) coefficients

indicates that a poorly resolved numerical solution with excessive dissipation might

be qualitatively closer to the analytical solution than a grid converged solution to the

truncated system. Figure 6.5 and Table 6.3 illustrates this phenomenon of illusory

convergence.

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x

E(u)

M=4M=3exact

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Var(u

)

M=4M=3exact

Figure 6.5: Dissipative solution on course grid (m = 200), computed forM = 3 and non-dissipative solution for M = 4.


The discrepancy between the truncated solution for M = 3 and the analytical

solution is also illustrated in Figure 6.6. The coefficients do not converge to the

analytical solution when the spatial grid is refined (Figure 6.6a, left). Instead the

coefficients converge numerically to a reference solution corresponding to a numerical

solution obtained with a large number of gridpoints (Figure 6.6a, right). For the 7th

order expansion, the solution is sufficiently close the the solution of the analytical

problem to exhibit spatial numerical convergence of the first four coefficients to the

analytical coefficients (Figure 6.6b).

100 200 400 8000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

m

Erro

r nor

m

100 200 400 8000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

m

Erro

r nor

m

||Err(u0)||

||Err(u1)||

||Err(u2)||

||Err(u3)||

||Err(u0)||

||Err(u1)||

||Err(u2)||

||Err(u3)||

(a) M = 3. Norm of the error relative to the analytical solution (left) and error relative to the finest

grid solution, m = 800 (right).


100 200 400 8000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

m

Erro

r nor

m

||Err(u0)||||Err(u1)||||Err(u2)||||Err(u3)||

100 200 400 8000

0.005

0.01

0.015

0.02

0.025

m

Erro

r nor

m

||Err(u0)||||Err(u1)||||Err(u2)||||Err(u3)||

(b) M = 7. Norm of the error relative to the analytical solution (left) and error relative to the finest

grid solution, m = 800 (right).

Figure 6.6: Convergence of the first chaos coefficients. Note the differentscales in the figures.

The variance calculated for M = 7 appears to converge to a function that is close

but not equal to the analytical variance given by (6.13), see Figure 6.7.

100 200 400 8000

0.05

0.1

0.15

0.2

0.25

m

Erro

r nor

m

100 200 4000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

m

Erro

r nor

m

||Var(unum)−Var(uexact)|| ||Var(unum)−Var(ufine grid)||

Figure 6.7: M = 7. Convergence of the variance. Norm of the errorrelative to the analytical variance (left) and error relative to the finestgrid variance, m = 800 (right).


6.8 Theoretical results and interpretation

6.8.1 Analysis of characteristics: disturbed cosine wave

In this section, the characteristics of the stochastic Burgers’ equation with M = 1

(truncated to 2×2 system) will be investigated to give a qualitative measure of the

time development of the solution. The system is given by

(u0

u1

)

t

+

(u0 u1

u1 u0

)(u0

u1

)

x

= 0. (6.35)

With w1 = u0 + u1 and w2 = u0 − u1, (6.35) can be diagonalized and rewritten

(w1

w2

)

t

+

(w1 0

0 w2

)(w1

w2

)

x

= 0. (6.36)

Equation (6.36) is the original Burgers’ equation for w1, w2 and the shock speeds

are given by

sw1 =[f(w1)]

[w1]=w1R + w1L

2= u0 + u1 (6.37)

and

sw2 =[f(w2)]

[w2]=w2R + w2L

2= u0 − u1 (6.38)

where we have introduced the mean over the shock, ui = (uiL + uiR)/2. Double

brackets [ ] denotes the jump in a quantity over a discontinuity. Similarly to (6.37)

and (6.38), with the non-diagonalized system in conservation form, the propagation

speeds of discontinuities in u0, u1 are given by

su0 =[(u2

0 + u21)/2]

[u0]= u0 + u1

[u1]

[u0](6.39)

and

su1 =[u0u1]

[u1]= u0 + u1

[u0]

[u1]. (6.40)

The analysis of characteristics w1 and w2 describes the behavior and emergence of


discontinuities in the coefficients u0 and u1 of the truncated system. However, the

coefficients of the solution to the problem given by the infinite gPC expansion are

smooth (except for t = 0 for the Riemann problem). Diagonalization of large systems

is not feasible but we can obtain expressions for the shock speeds of the coefficients.

For instance, the expression (6.39) for the shock speed in u0 can be generalized for

gPC expansions of order M as

su0 =M∑

i=0

ui[ui]

[u0](6.41)

In the assumption that only one Gaussian variable ξ is introduced, and the un-

certainty is (linearly) proportional to ξ only a limited number of different values of

the correlation coefficient between the left and right state can occur. Since we are

also assuming the same model for the left and right state uncertainties, only a few

combinations of covariance matrices describing their correlation are realizable. With

the assumptions made here, the dependence between the two states is determined by

the correlation coefficient ρLR, which for these cases is either 1 or −1.

Ex 1.1

u(x, 0, ξ) =

uL = 1 + σξ x < x0

uR = −1− σξ x < x0

u(x, 0) = cos(πx)(1 + σξ)

ξ ∼ N (0, 1)

ρLR = −1

Ex 1.2

u(x, 0, ξ) =

uL = 1 + σξ x < x0

uR = −1 + σξ x < x0

u(x, 0) = cos(πx) + σξ

ξ ∼ N (0, 1)

ρLR = 1

The problems are similar in terms of expected value and variance at the boundary,

but the difference in correlation between the left and right states completely changes

the behavior over time. The difference in initial variance in the interior of the domain

has only a limited impact on the time dependent difference between the solutions;


this has been checked by varying the initial functions. Note that Ex 1.1 is included

to show the importance of the sign of the stochastic variable, but is a special case of

a more general phenomenon of superimposition of discontinuities exhibited by Ex 1.2

and further explained and analyzed below. Figure 6.8 shows the two cases at time

t = 0.5 for M = 3. We use σ = 0.1 and x0 = 0.5.

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

E(u)E(u)+/−Std(u)

0 0.2 0.4 0.6 0.8 10.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

Variance

(a) Ex 1.1. Symmetric boundary conditions.

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

E(u)E(u)+/−Std(u)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Variance

(b) Ex 1.2. Constant initial variance.

Figure 6.8: Development of variance of the perturbed cosine wave. t = 0.5for M = 3, m = 400.


To explain the differences between the solutions depicted in Figure 6.8, we turn to

analysis of the characteristics for the truncated system with M = 1. The polynomial

chaos coefficients of the boundaries are given by

u0 = 1

u1 = 0.1

x = 0,

u0 = −1

u1 = −0.1

x = 1 (Ex 1.1)

and

u0 = 1

u1 = 0.1

x = 0,

u0 = −1

u1 = 0.1

x = 1 (Ex 1.2)

respectively.

Note that with more polynomial chaos coefficients included, the higher order co-

efficients are zero at the boundaries. The expected boundary values as well as the

boundary variance are the same for Ex 1.1 and Ex 1.2. In order to relate the con-

cepts of characteristics with expected value and variance, we will use the fact that

the expected value at each point is the average of the characteristics,

E(u) = u0 =w1 + w2

2(6.42)

and that the variance depends on the distance between the characteristics,

Var(u) = u21 =

(w1 − w2

2

)2

. (6.43)

To explain the qualitative differences between the two cases Ex 1.1 and Ex 1.2, con-

sider the decoupled system (6.36). The boundary values for u0 and u1 are inserted into

the characteristic variables w1 and w2; discontinuities emerge when the characteristics

meet.

For Ex 1.1 we have w1(x = 0) = −w1(x = 1) and w2(x = 0) = −w2(x = 1).

Inserting these values in (6.37) and (6.38) gives the shock speeds sw1 = sw2 = 0,

corresponding to two stationary shocks (of different magnitude) at x = 0.5, which

can be seen in Figure 6.9a. Inserting the characteristic values (can be evaluated in


Figure 6.9) into Eq. (6.43) result in uniform variance except around the discontinuity,

Figure 6.10a. Since the characteristic solution is propagating from the boundaries,

this interval shrinks with time and collapses at x = 0.5.

In Ex. 1.2, the characteristics are w1(x = 0) = 1.1 > −w1(x = 1) = 0.9 and

w2(x = 0) = 0.9 < −w2(x = 1) = 1.1. Evaluating (6.37) and (6.38) when the

characteristics cross yields sw1 = 0.1 and sw2 = −0.1. The discontinuity when the

characteristics meet will then split and propagate as two moving shocks in u0 and

u1, located equidistantly from the mid-point x = 0.5. In w1 and w2 there will still

be a single shock. The shock speeds are given by the expressions (6.37)-(6.40). The

vertical gap between the characteristics at x = 0.5 in Figure 6.9b corresponds to the

variance peak at this location in 6.8b.

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

w1w2

(a) Ex 1.1. The variance is undefined at x =

0.5.

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

w1w2

(b) Ex 1.2. The variance peaks at x = 0.5. w1

is left-going and w2 is right-going.

Figure 6.9: Characteristics of the two perturbed cosine waves (Ex 1.1 andEx 1.2) for M = 1.


0 0.2 0.4 0.6 0.8 10.005

0.01

0.015

0.02

Var(u

(a) Ex 1.1. The variance is constant except

around the discontinuity.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Var(u)

(b) Ex 1.2. The variance is maximal at the

shock location and spreads towards the bound-

aries.

Figure 6.10: Variance of Ex. 1.1 and Ex. 1.2 for M = 1, calculated fromw1, w2 using (6.43).

The system used for analysis of characteristics is truncated to M = 1, but the

conclusions about the qualitative behavior holds for higher order systems. Including

more polynomial chaos coefficients would result in additional shocks of different mag-

nitude and speed. Observe the qualitative similarities between the solutions in Figure

6.9 and Figure 6.8. Regardless of the truncation of polynomial chaos coefficients, the

variance approaches 0 at the shock location in Ex 1.1. At the shock location in Ex 1.2

the variance reaches a maximum that will spread towards the boundaries and cancel

the discontinuity. The observation that the same boundary and initial expected value

and variance can give totally different solutions indicates that knowledge about the

polynomial chaos coefficients is required to obtain a unique solution.

The analysis of characteristics further shows that the problem could be parti-

tioned into several phases of development, depending on the speeds of the character-

istics. Consider again the boundary conditions of Ex 1.1 and Ex 1.2 but now assume

u(x, 0) = 0 for x ∈ (0, 1). The solution for M = 1 before the characteristics meet


is shown in Figure 6.11. With more polynomial chaos coefficients, the sharp edges

in the solution will disappear. At time t = 0.5, the solutions to the two problems

are still similar, with two variance peaks at the shocks that are traveling towards the

middle of the domain.

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

w1w2

(a) Ex 1.1. Boundary conditions: u(0, t) =

(1, 0.1, 0, ...); u(1, t) = (−1,−0.1, 0, ...).

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

w1w2

(b) Ex 1.2. Boundary conditions: u(0, t) =

(1, 0.1, 0, ...); u(1, t) = (−1, 0.1, 0, ...).

Figure 6.11: Characteristics at t = 0.5, M = 1.

For comparison, Figure 6.12 shows the expected value and variance calculated

from the characteristics in Figure 6.11.


0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

E(u)E(u)+/−Std(u)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Variance

(a) Ex 1.1. Symmetric boundary conditions: u(0, t) = (1, 0.1, 0, ...); u(1, t) = (−1,−0.1, 0, ...).

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

E(u)E(u)+/−Std(u)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Variance

(b) Ex. 1.2. Boundary conditions: u(0, t) = (1, 0.1, 0, ...); u(1, t) = (−1, 0.1, 0, ...).

Figure 6.12: Expected value and variance at t = 0.5, M = 1.

Asymptotically in time, the symmetric problem (Ex. 1.1) will result in a stationary

shock. The variance will equal the initial boundary variance except for a peak at the

very location of the shock. The boundary conditions are independent of time. This


property is illustrated in Figure 6.13a, where the solution has reached steady state.

The time development of the solution of Ex 1.2 is not consistent with the station-

ary boundary conditions stated in the problem formulation. The characteristics are

transported from one boundary to the other (see Figure 6.13b), thus changing the

boundary data. The boundary conditions of Ex 1.2 must therefore be time-dependent

(and can be calculated exactly from (6.10) for this example). Unlike the continuously

varying boundary conditions of the full polynomial chaos expansion problem, the

boundary conditions for the truncated system of Figure 6.13b will change discontin-

uosly from the initial boundary condition to zero at the moment the characteristics

reach the boundaries. In a general hyperbolic problem, the imposition of correct time-

dependent boundary conditions might become one of the more significant problems

with the gPC method. A detailed investigation is necessary to identify an approach

to specify time-dependent stochastic boundary data, especially for the higher order

moments. Special non-reflecting boundary conditions will be required. In the case

studied here, analytical boundary conditions have been derived and can be correctly

imposed for any time and order of chaos expansions.

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

w1w2

(a) Ex 1.1. Characteristics have reached

steady-state.

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

w1w2

(b) Ex 1.2. w1 is right-going, w2 left-going.

Figure 6.13: Characteristics at t = 4 for M = 1.


6.9 Dependence on available data

For M = 1, the system (6.4) can be diagonalized with constant eigenvectors and we

get an exact solution to the truncated problem. With a and b as in the problem setup

(section 6.2), the analytical solution for the 2× 2-system (x ∈ [0, 1]) is given by

(u0

u1

)=

(a, b)T if x < x0 − bt(0, a+ b)T if x0 − bt < x < x0 + bt

(−a, b)T if x > x0 + bt

for 0 ≤ t < x0

b

(0, a+ b)T for t > x0

b

(6.44)

We expect different numerical solutions depending on the amount of available bound-

ary data. We will assume that the boundary data are known on the boundary x = 1

and investigate three different cases for the left boundary x = 0 corresponding to

complete set of data, partial information about boundary data and no data available,

respectively. For all cases, we will solve a system of the form

(u0

u1

)

t

+1

2

[(u0 u1

u1 u0

)(u0

u1

)]

x

= 0. (6.45)

with boundary data

(u0

u1

)

x=−1

=

(g0(t)

g1(t)

);

(u0

u1

)

x=1

=

(h0(t)

h1(t)

).

6.9.1 Complete set of data

The boundary conditions are

u(0, t) =

(a, b)T 0 ≤ t < x0

b

(0, a+ b)T t > xb

(6.46)

Consider a = 1, b = 0.2. Both u0 and u1 are known at x = 0 and the two ingoing

characteristics are assigned the analytical values. The system satisfies the energy


estimate (6.26) and is stable. Figure 6.14, 6.15 and 6.16 show the solution at time

t = 1, t = 2 and t = 3 respectively.

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

Mean

x

u0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Standard deviation

x

u1

Figure 6.14: u0 (left) and u1 (right). t = 1. Complete set of data.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Standard deviation

x

u1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Standard deviation

x

u1


6.9.2 Incomplete set of boundary data

Without a complete set of boundary data, the time-dependent behavior of the solution

will be hard to predict. Here we assume that the boundary conditions at x = 1 is

u = (−1, 0.2) as before (Equation (6.46)) and consider different ways of dealing with

unknown data at x = 0. The initial function is the same as in the analytical problem

above, i.e.

(u0(x, 0), u1(x, 0))T =

(a, b)T if x < x0

(−a, b)T if x > x0

u1 unknown at x = 0, guess u1

First assume that u0 is known and u1 is unknown and put u1 = 0.2 at the boundary for

all time. This problem setup lead to an energy estimate and stability. There are two

ingoing characteristics at t = 0. u0 at x = 0 changes with the boundary conditions of

the analytical solution as given by (6.46). The time development follows the analytical

solution at first (Figure 6.17) but eventually becomes inconsistent with the boundary

conditions (Figure 6.18 and Figure 6.19)


0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

Mean

x

u0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Standard deviation

x

u1

Figure 6.17: u1 kept fixed at 0.2. t = 2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Standard deviation

x

u1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Standard deviation

x

u1


u1 unknown at x = 0, extrapolate u1

Now, the extrapolation g1 = (u1)1 is used to assign boundary data to the presumably

unknown coefficient u1. This case does not lead to stability using the energy method.

As long as the analytical boundary conditions do not change, the numerical solution

follows the analytical solution as before, see Figure 6.20. After t = 2.5 the character-

istics have reached the opposite boundaries and the error grows (Figure 6.21) before

reaching the steady state (Figure 6.22).


0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

Mean

x

u0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Standard deviation

x

u1

Figure 6.20: u1 extrapolated from the interior. t = 2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

Standard deviation

x

u1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

Standard deviation

x

u1

Figure 6.22: u1 extrapolated from the interior. t = 5. The error is of theorder 10−15.

u0 unknown at x = 0, guess u0

Next we assume that the boundary data for u0 is unknown. This case leads to an

energy estimate and stability. The same analysis is carried out for u0 as was done

for u1 in the preceding section. First u0 at x = 0 is held fixed for all times. Figure

6.23 and Figure 6.24 show the solution before and after the true characteristics reach

the boundaries. Note that the solution after a long time is not coincident with the

analytical solution and that the boundary conditions are not satisfied (Figure 6.25).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

Standard deviation

x

u1

Figure 6.23: u0 is held fixed. t = 2.

0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Mean

x

u0

0 0.2 0.4 0.6 0.8 11.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Standard deviation

x

u1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Standard deviation

x

u1


u0 unknown at x = 0, extrapolate u0

The data for u0 can instead be extrapolated from the interior of the domain. The

extrapolation g0 = (u0)1 is used, see Figures 6.26, 6.27 and 6.28. This case does not

lead to stability using the energy method. Note that the solution after a long time is

very close to the analytical solution (Figure 6.28).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Standard deviation

x

u1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.05

1.1

1.15

1.2

1.25

1.3

Standard deviation

x

u1


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Mean

x

u0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.05

1.1

1.15

1.2

1.25

1.3

Standard deviation

x

u1


6.9.3 Discussion of the results with incomplete set of data

The results in the preceding section are interesting and surprising. First of all, ex-

cellent results at steady state (for long time) are obtained using the extrapolation

technique. This is probably due to the fact that only one bounadry condition is

needed at the left boundary for t > 2.5. Also, by guessing data of the mean value


and the variance, equally poor results are obtained. The higher order modes might

be very important. The order of the error obtained here indicate that appropriate

approximation of the higher order terms is as important as guessing the expectation

to get accurate results.

6.10 Summary and conclusions

The stochastic Galerkin method has been presented for Burgers’ equation with stochas-

tic boundary conditions. Stable difference schemes are obtained by the use of artifi-

cial dissipation, difference operators satisfying the summation by parts property and

a weak imposition of characteristic boundary conditions.

A number of mathematical properties of the deterministic Burgers’ equation hold

for the hyperbolic problem that results from the Galerkin projection of the truncated

gPC expansions. The system is symmetric and a split form combining conservative

and non-conservative formulations is used to obtain an energy estimate. The trun-

cated linearized problem is shown to be well-posed. The system eigenvalues cannot be

computed analytically and this makes the choice of the time step difficult; moreover,

this affects the accuracy of the methods since the dissipation operators are eigenvalue

dependent. An eigenvalue estimate is provided.

Even though the solution of the Burgers’ equation is discontinuous for a par-

ticular value of the uncertain (stochastic) variable, the polynomial chaos coefficient

functions are in general continuous for the Riemann problems investigated. The so-

lution coefficients of the truncated system are discontinuous and can be treated as a

superimposition of a finite number of discontinuous characteristic variables. This has

been shown explicitly for the 2×2-case. The discontinuous coefficients converge with

the number of polynomial chaos coefficients to continuous functions.

Examples have shown the need to provide time-dependent boundary conditions

that might include higher order moments. Stochastic time-dependent boundary con-

ditions have been derived for the Burgers’ equation.

An increasing number of polynomial chaos modes and use of extra boundary data


give solutions that are qualitatively different to the cruder approximation. How-

ever useful for a qualitative description of the dynamics of the hyperbolic system,

the approximation error due to truncation of the infinite polynomial chaos series is

dominating the total error.

As shown in Table 6.3, excessive use of artificial dissipation can give a numerical

solution that more closely resembles the solution of the original problem compared to

a solution where a small amount of dissipation within the order of accuracy is used to

preserve the discontinuities of the truncated solution. Clearly, only the latter method

could be justified from a theoretical point of view.

In many problems, sufficient data is not available to specify the correct number of

variables. Unknown boundary values can then be constructed by extrapolation from

the interior or by simply guessing the boundary data. We have investigated these two

possible cases and for this specific problem the extrapolation technique was superior.

It was also found that missing data for the expectation were not more serious than

the lack of data for the higher mode (approximating the variance). This casts new

light on the data requirement for higher order expansions.

In a general hyperbolic problem, the imposition of correct time-dependent bound-

ary conditions will probably prove to be one of the more significant problems with

the stochastic Galerkin method. A detailed investigation is necessary to find ways

around the lack of time-dependent stochastic boundary data, especially for the higher

moments. Most likely, special non-reflecting boundary conditions must be developed.

Chapter 7

A robust intrusive formulation of

the Euler equations

7.1 Introduction

In many nonlinear applications of the stochastic Galerkin method, truncation of the

generalized chaos expansion leads to non-unique formulations of the systems of equa-

tions. For instance, cubic products between stochastic quantities a, b and c, are rep-

resented as products of truncated approximations a, b and c, but the pseudospectral

multiplication operator ∗, to be explicitly defined in a later section, is not associative,

i.e. (a ∗ b) ∗ c 6= a ∗ (b ∗ c). Similar problems are investigated in more detail in [51].

The need to introduce stochastic expansions of inverse quantities, or square-roots

of stochastic quantities of interest, adds to the number of possible different ways to

approximate the original stochastic problem. This leads to ambiguity of the problem

formulation. We present a method where this ambiguity is avoided since no auxiliary

quantities are needed.

Poette et. al. [80] used a nonlinear projection method to bound the oscillations

close to stochastic discontinuities by polynomial chaos expansion of the entropy vari-

ables obtained from a transformation of the conservative variables. Each time step is

complemented by a functional minimization to obtain the entropy variables needed

to update the solution vector. We will present a variable transformation method for

109

CHAPTER 7. EULER EQUATIONS 110

the Euler equations that may appear similar in scoop at first sight, but it relies on a

different kind of variable transformation and not on kinetic theory considerations. We

do not suggest a variable transformation for general conservation laws, but a formu-

lation that specifically targets the solution of the Euler equations with uncertainty in

the variables. It is in fact less complicated than a direct polynomial chaos expansion

of the conservative variables.

7.2 Euler equations with input uncertainty

Consider the 1D Euler equations, in non-dimensional form given by

ut + f(u)x = 0, 0 ≤ x ≤ 1, t > 0, (7.1)

where the solution and flux vector are given by

u =

ρ

ρv

E

, f =

ρv

ρv2 + p

(E + p)v

,

where ρ is density, v velocity, E total energy and pressure p. A perfect gas equation

of state is assumed, and energy and pressure are related by

E =p

γ − 1+

1

2ρv2,

where γ is the ratio of the specific heats. For the numerical method, we need the flux

Jacobian, given by

∂f

∂u=

0 1 012(γ − 3)v2 (3− γ)v γ − 1

12(γ − 1)v3 − vH H − (γ − 1)v2 γv

,

with the total enthalpy H = (E + p)/ρ.

We scale the physical variables to get the dimensionless variables ρ = ρ′/ρ′ref ,


E = E ′/(γp′ref ), p = p′/(γp′ref ) and v = v′/a′ref where a′ = (γp′/ρ′)1/2 and the

subscript ref denotes a reference state.

7.2.1 Formulation in Roe variables

Roe [87] introduced the variables

w =

w1

w2

w3

=

ρ1/2

ρ1/2v

ρ1/2H

.

The flux and the conservative variables are given by

f(w) =

w1w2

γ−1γw1w3 + γ+1

2γw2

2

w2w3

, u = g(w) =

w21

w1w2

w1w3

γ+ γ−1

2γw2

2

.

Then

g(w)t + fx(w) = 0 (7.2)

is equivalent to (7.1). The flux Jacobian in the Roe variables is given by

∂f

∂w=

w2 w1 0γ−1γw3

γ+1γw2

γ−1γw1

0 w3 w2

.

7.2.2 Stochastic Galerkin formulation of the Euler equations

Define the pseudo-spectral product u ∗ v of order P = P (Np, Nr) by

(u ∗ v)k =P∑

i=0

P∑

j=0

uivj〈ψiψjψk〉, k = 0, ..., P,


where

〈ψiψjψk〉 =

∫

Ω

ψi(ξ)ψj(ξ)ψk(ξ)dP .

Alternatively, using matrix notation, we can write the spectral product as u ∗ v =

A(u)v, where

[A(u)]jk =P∑

i=0

ui〈ψiψjψk〉. (7.3)

We will need the pseudo-spectral inverse q−∗, defined as the solution of q ∗ q−∗ = 1,

and the pseudo-spectral square root, defined as the solution q∗/2 of q∗/2 ∗ q∗/2 = q,

where the spectral expansion of the quantity of interest q is assumed to be known.

For more details, see [21].

Let uP denote the vector of coefficients of the MW expansion of u of order P =

P (Np, Nr). P may take the same value for two distinct pairs of (Np, Nr) but this

ambiguity in notation will not matter in the derivation of the numerical method

so for brevity we use only P in the superscripts. The Euler equations represented

by the conservative formulation (7.1) can be written as an augmented system, after

stochastic Galerkin projection,

uPt + fP (uP )x = 0, (7.4)

where

uP =

uP1

uP2

uP3

=

[(u1)0, ..., (u1)P ]T

[(u2)0, ..., (u2)P ]T

[(u3)0, ..., (u3)P ]T

, f

P (uP ) =

uP2

(uP1 )−∗ ∗ uP2 ∗ uP2 + pP

(uP3 + pP ) ∗ uP2 ∗ (uP1 )−∗

.

with pP = (γ − 1)(uP3 − (uP1 )−∗ ∗ uP2 ∗ uP2 /2). The cubic products of (7.4) are ap-

proximated by the application of two third-order tensors, instead of one fourth-order

tensor. That is, we replace (a ∗ b ∗ c)l =∑

ijk〈ψiψjψkψl〉aibjck by the approximation

(a ∗ b ∗ c)l ≈ ((a ∗ b) ∗ c)l.For the Roe variable formulation, the stochastic Galerkin projection of (7.2) gives


the system

gP (wP )t + fP (wP )x = 0, (7.5)

where

gP (wP ) =

wP1 ∗ wP1wP1 ∗ wP2

wP1 ∗wP3γ

+ γ−12γwP2 ∗ wP2

, f

P (wP ) =

wP1 ∗ wP2γ−1γwP1 ∗ wP3 + γ+1

2γwP2 ∗ wP2

wP2 ∗ wP3

.

The flux Jacobian for the stochastic Galerkin system in the Roe variables is given by

∂fP

∂wP=

A(wP2 ) A(wP1 ) 0(P+1)×(P+1)

γ−1γA(wP3 ) γ+1

γA(wP2 ) γ−1

γA(wP1 )

0(P+1)×(P+1) A(wP3 ) A(wP2 )

. (7.6)

As P → ∞, the formulations (7.4) and (7.5), as well as any other consistent

formulation, are equivalent. However, P is assumed to be small (< 20), and truncation

and conditioning of the system matrices will play an important role for the accuracy

of the solution.

7.3 Numerical method

As our main numerical method we will use the MUSCL (Monotone Upstream-centered

Schemes for Conservation Laws) scheme introduced in [103].

7.3.1 Expansion of conservative variables

Let m be the number of spatial points and ∆x = 1/(m− 1) and let uP be the spatial

discretization of uP . The semi-discretized form of (7.4) is given by

duPjdt

+F Pj+1/2 − F P

j−1/2

∆x= 0, j = 1, ...,m, (7.7)


where F Pj+1/2 denotes the numerical flux function evaluated at the interface between

cells j and j + 1.

For the MUSCL scheme with slope limited states uL and uR, we take the numerical

flux

F Pj+ 1

2=

1

2

(fP (uL

j+ 12) + fP (uR

j+ 12))

+1

2|(JPc )j+ 1

2|(uLj+ 1

2− uR

j+ 12

), (7.8)

where the Roe average JPc is the pseudo-spectral generalization of the standard Roe

average of the deterministic Euler equations, i.e.

JPc (v,H) =

0P×P IP×P 0P×P12(γ − 3)A(v)2 (3− γ)A(v) (γ − 1)IP×P

12(γ − 1)A(v)3 − A(v)A(H) A(H)− (γ − 1)A(v)2 γA(v)

where

v = (ρ−∗/2L + ρ

−∗/2R ) ∗ (ρ

∗/2L ∗ vL + ρ

∗/2R ∗ vR),

and

H = (ρ∗/2L ∗HL + ρ

∗/2R ∗HR) ∗ (ρ

−∗/2L + ρ

−∗/2R ).

The computation of v and H require the spectral square root ρ∗/2 and its inverse,

that are computed solving a nonlinear and a linear system, respectively.

Further details about the formulation of the Roe average matrix are given in [99].

The scheme is a direct generalization of the deterministic MUSCL scheme. Flux

limiters are applied componentwise to all MW coefficients in sharp regions. For a

more detailed description of the MUSCL scheme and flux limiters, see e.g. [56], and

for application to the stochastic Burgers’ equation [71].

7.3.2 Expansion of Roe’s variables

Let wP denote the spatial discretization of wP . The semi-discretized form of (7.5) is

given by

∂gP (wPj )

∂t+F Pj+1/2 − F P

j−1/2

∆x= 0, j = 1, ...,m,


with the numerical flux function

Fj+ 12

=1

2

(fP (wL

j+ 12) + fP (wR

j+ 12))

+1

2|JPj+ 1

2|(wLj+ 1

2−wR

j+ 12

), (7.9)

where JP = JP (wP ) is the Roe matrix for the stochastic Galerkin formulation of the

Euler equations in Roe’s variables, to be derived below.

Each time step provides the update of the solution vector gPj = gP (wPj ), j =

1, ...,m, from which we can solve for wP to be used in the update of the numerical

flux. This involves solving the nonlinear systems

A(wP1,j)w

P1,j = gP1,j, j = 1, ...,m, (7.10)

for wP1,j, and then using wP

1,j to solve the linear (P + 1)× (P + 1)-systems

A(wP1,j)w

P2,j = gP2,j, j = 1, ...,m,

for wP2,j, and

A(wP1,j)W

P3,j = γgP3,j −

γ − 1

2A(wP

2,j)wP2,j, j = 1, ...,m,

for wP3,j.

The system (7.10) is solved iteratively with a trust-region-dogleg algorithm1.

Starting with the value of the previous time-step as initial guess, few iterations are

required (typically 2-3). (The same method is used to solve for spectral square roots

in the conservative variable formulation.)

7.3.3 Stochastic Galerkin Roe average matrix for Roe vari-

ables

The Roe average matrix JP is given as a function of the Roe variables w = (w1, w2, w3)T ,

where each wi is a vector of generalized chaos coefficients. It is designed to satisfy

1This is the default algorithm for fsolve in Matlab. For more details, see [82].


the following properties:

(i) JP (wL, wR)→ ∂fP

∂w

∣∣∣w=w′

as wL, wR → w′.

(ii) JP (wL, wR)× (wL − wR) = fP (wL)− fP (wR), ∀wL, wR

(iii) JP is diagonalizable with real eigenvalues and linearly independent eigenvectors.

In the standard approach introduced by Roe and commonly used for deterministic

calculations, the conservative variables are mapped to the w variables, which are then

averaged.

In the deterministic case, we have

fL − fR = J(wL, wR)× (wL − wR), (7.11)

where

J(wL, wR) =

w2 w1 0γ−1γw3

γ+1γw2

γ−1γw1

0 w3 w2

.

Overbars denote arithmetic averages of assumed left and right values of a variable,

i.e.

wj =wLj + wRj

2, j = 1, 2, 3.

It is a straightforward extension of the analysis by Roe in [87] to show properties

(i) and (ii) for the Roe variables, without mapping to the conservative variables. To

prove (iii) we note that there exists an eigenvalue decomposition

J = V DV −1, (7.12)

where

V =

w1

w3

w1

w3−w1

w3

w2−√w2

2+8w1w3γ(γ−1)

2γw3

w2+√w2

2+8w1w3γ(γ−1)

2γw30

1 1 1

, (7.13)


D =

w2(1+2γ)−√

8w1w3γ(γ−1)+w22

2γ0 0

0w2(1+2γ)+

√8w1w3γ(γ−1)+w2

2

2γ0

0 0 w2

. (7.14)

The eigenvalues of J are real and distinct, so property (iii) is also satisfied.

Now consider the stochastic Galerkin formulation, i.e. assume that the wi’s are

vectors of generalized chaos coefficients. The stochastic Galerkin Roe average matrix

JP for the Roe variables formulation is a generalization of the mapping (7.11), i.e. of

the matrix J . We define

JP (wL, wR) = JP (w) =

A(w2) A(w1) 0P×Pγ−1γA(w3) γ+1

γA(w2) γ−1

γA(w1)

0P×P A(w3) A(w2)

, (7.15)

where the submatrix A(wj) is given by (7.3) and w = (wL + wR)/2.

Proposition 5. Property (i) is satisfied by (7.15).

Proof. With wL = wR = w′, JP (wL, wR) = JP (w′, w′) = ∂fP

∂wP

∣∣∣w=w′

by (7.6).

Proposition 6. Property (ii) is satisfied by (7.15).

Proof.

JP (wL, wR)× (wL − wR) =1

2

(JP (wL) + JP (wR)

)(wL − wR) =

=1

2JP (wL)wL − 1

2JP (wR)wR = fP (wL)− fP (wR), (7.16)

where the last equality follows from the fact that the stochastic Galerkin generaliza-

tions of the Euler equations are homogeneous of degree 1.

To prove (iii), we will need the following proposition.

Lemma 1. Let A(wj) (j = 1, 2, 3) be defined by (7.3) and A(wj) = QΛjQT be an

eigenvalue decomposition with constant eigenvector matrix Q and assume that Λ1 and

Λ3 are non-singular. Then the stochastic Galerkin Roe average matrix JP has an

eigenvalue decomposition JP = XΛPX−1 with a complete set of eigenvectors.


Proof. We will use the Kronecker product ⊗, defined for two matrices B (of size

m× n) and C by

B ⊗ C =

b11C . . . b1nC...

. . ....

bm1C . . . bmnC

.

The eigenvalue decompositions of each (P + 1)× (P + 1) matrix block of (7.15) have

the same eigenvector matrix Q, hence we can write

JP = (I3 ⊗Q)J(I3 ⊗QT ) (7.17)

where

J =

Λ2 Λ1 0(P+1)×(P+1)

γ−1γ

Λ3γ+1γ

Λ2γ−1γ

Λ1

0(P+1)×(P+1) Λ3 Λ2

.

By assumption, I3 ⊗ Q is non-singular, and it remains to show that J has distinct

eigenvectors. Let

S = diag(Λ1Λ−13 ,√

(γ − 1)/γΛ1/21 Λ

−1/23 , I(P+1)×(P+1)).

By assumption, Λ1 and Λ3 are invertible, so S and S−1 exist. We have

JS ≡ S−1JS =

Λ2

[γ−1γ

Λ1Λ3

]1/2

0P×P[γ−1γ

Λ1Λ3

]1/2γ−1γ

Λ2

[γ−1γ

Λ1Λ3

]1/2

0P×P

[γ−1γ

Λ1Λ3

]1/2

Λ2

. (7.18)

Clearly, JS is symmetric and has the same eigenvalues as J and JP . Hence, JS has

an eigenvalue decomposition JS = Y ΛPY T . Then,

J = SY ΛPY TS−1 = SY ΛP (SY )−1. (7.19)


Combining (7.17) and (7.19), we get

JP = [(I3 ⊗Q)SY ]ΛP [(I3 ⊗Q)SY ]−1.

Setting X = (I3 ⊗ Q)SY , we get the eigenvalue decomposition JP = XΛPX−1. By

assumption, S and Y are non-singular, and we have that

det (X) = det ((I3 ⊗Q)SY ) 6= 0,

which proves that X is non-singular, and thus JP has a complete set of eigenvectors.

Proposition 7. Property (iii) is satisfied by (7.15).

Proof. Lemma 1 shows that since the eigenvalue matrix ΛP is also the eigenvalue

matrix of the symmetric matrix JS defined in (7.18), the eigenvalues are all real.

Lemma 1 also shows that the eigenvectors are distinct.

The conditions in Lemma 1 are true for certain basis functions assuming moderate

stochastic variation, but it can not be guaranteed for every case, and it certainly does

not hold for pathological cases with e.g. negative density. The requirement of non-

singularity of Λ1,Λ3 is not very restrictive since it amounts to excluding unphysical

behavior, for instance naturally positive quantities taking negative values with non-

zero probability. The assumption of constant eigenvectors of the matrix A holds for

Haar wavelets (i.e. multiwavelets with Np = 0), for all orders P + 1 = 2Nr , with Nr ∈N. See appendix B.1 for a proof sketch. Expressions for the first constant eigenvalue

decompositions are included in appendix B.2 for Haar wavelets and piecewise linear

multiwavelets. The eigenvectors of A for P + 1 = 1, 2, 4, 8 are shown to be constant,

but we do not give a proof that this is true for piecewise linear multiwavelets of any

order P .

Remark 6. The Roe variable scheme has been outlined under the implicit assump-

tion of uncertainty manifest in the variables, e.g. initial and boundary condition

uncertainty. However, situations such as uncertainty in the adiabatic coefficient γ


may be treated in a similar way, although it would result in additional pseudo-spectral

products. Pseudo-spectral approximations of (γ − 1)/γ and (γ + 1)/γ could then be

precomputed to sufficient accuracy.


We use the method of manufactured solutions to verify the second order convergence

in space of a smooth problem using the MUSCL scheme with Roe variables. We

then introduce two test cases for the non-smooth problem; case 1 with an initial

function that can be exactly represented by two Legendre polynomials, and case 2

with slow initial decay of the MW coefficients in both Np and Nr. The errors in

computed quantities of interest (here variances) as functions of the order of MW

are investigated. Qualitative results are then presented to indicate the behavior we

can expect for the convergence of two special cases of MW, namely the Legendre

polynomials and Haar wavelet basis, respectively. Robustness with respect to more

extreme cases (density close to zero leading to high Mach number) is demonstrated

for the Roe variable formulation for a supersonic case where the conservative variable

method breaks down. Finally, we perform a comparative study of the computational

time for the formulation in conservative variables and the formulation in Roe variables.

7.4.1 Spatial convergence

The MUSCL scheme with appropriate flux limiters is second order accurate for smooth

solutions. Since the Euler solution in general becomes discontinuous in finite time,

the method of manufactured solutions [85, 89] is used to solve the Euler equations

with source terms for a known smooth solution. The smooth solution is inserted into

the Euler equations (7.1) and results in a non-zero right-hand side that is used as a

source function. In order to test the capabilities of the method, we choose a solution

that varies in space, time and in the stochastic dimension and with time-dependent

boundary conditions. It is designed to resemble a physical solution with non-negative


density and pressure. The solution is given by

ρ

v

p

=

ρ0 + ρ1 tanh(s(x0 − x+ t+ σξ))

tanh(s(x0 + v0 − x+ t+ σξ)) + tanh(−s(x0 − v0 − x+ t+ σξ))

p0 + p1 tanh(s(x0 − x+ t+ σξ))

.

The parameters are set to ρ0 = p0 = 0.75, ρ1 = p1 = x0 = 0.25, v0 = 0.05, s = 10,

σ = 0.1 and ξ ∈ U [−1, 1].

We measure the error in the computed u(x, t, ξ) in the L2(Ω,P) norm and the

discrete `2 norm,

∥∥uP − u∥∥

2,2≡∥∥uP − u

∥∥`2,L2(Ω,P)

=

(∆x

m∑

i=1

∥∥uP (xi, t, ξ)− u(xi, t, ξ)∥∥2

L2(Ω,P)

)1/2

=

(∆x

m∑

i=1

∫

Ω

(uP (xi, t, ξ)− u(xi, t, ξ))2dP(ξ)

)1/2

=

≈(

∆xm∑

i=1

q∑

j=1

(uP (xi, t, ξ(j)q )− u(xi, t, ξ

(j)q ))2w(j)

q

)1/2

, (7.20)

where a q-point quadrature rule with points ξ(j)q qj=1 and weigths w(j)

q qj=1 was used

in the last line to approximate the integral in ξ. The Gauss-Legendre quadrature is

used here since the solution is smooth in the stochastic dimension.

Figure 7.1 depicts the spatial convergence in the ‖.‖2,2 norm of the error in density,

velocity and energy. An order (Np, Nr) = (10, 0) basis is used to represent the uncer-

tainty. The solution dynamics is initially concentrated in the left part of the spatial

domain. By the time of t = 0.4, it has moved to the right and has begun to exit the

spatial domain, so the time snapshots of Figure 7.1 summarizes the temporal history

of the spatial error decay. The theoretical optimal convergence rate for the MUSCL

scheme with the van Leer flux limiter is obtained for all times and all quantities.


21 41 81 161

10−4

10−3

10−2

10−1

m

∥∥ρP − ρ∥∥2 ,2

∥∥vP − v∥∥2 ,2

∥∥EP − E∥∥2 ,2

ord. 2 dec .

(a) t = 0.05.

21 41 81 161

10−4

10−3

10−2

10−1

m

∥∥ρP − ρ∥∥2 ,2

∥∥vP − v∥∥2 ,2

∥∥EP − E∥∥2 ,2

ord. 2 dec .

(b) t = 0.1.

21 41 81 161

10−4

10−3

10−2

10−1

m

∥∥ρP − ρ∥∥2 ,2

∥∥vP − v∥∥2 ,2

∥∥EP − E∥∥2 ,2

ord. 2 dec .

(c) t = 0.2.

21 41 81 161

10−4

10−3

10−2

10−1

m

∥∥ρP − ρ∥∥2 ,2

∥∥vP − v∥∥2 ,2

∥∥EP − E∥∥2 ,2

ord. 2 dec .

(d) t = 0.4.

Figure 7.1: Convergence in space using the method of manufactured solu-tions, Np = 10, Nr = 0 (Legendre polynomials).

7.4.2 Initial conditions and discontinuous solutions

We consider (7.1) with two different initial functions on the domain [0, 1]. Since

the analytical solution of Sod’s test case is known for any fixed value of the input

parameters, the exact stochastic solution can be formulated as a function of the


stochastic input ξ. Exact statistics can be computed by numerical integration over ξ.

As case 1, assume that the density is subject to uncertainty, and all other quantities

are deterministic at t = 0. The initial condition for (7.1) is given by

u(x, t = 0, ξ) =

uL = (1 + σξ, 0, 2.5/γ)T x < 0.5

uR = (0.125(1 + σξ), 0, 0.25/γ)T x > 0.5

where we assume ξ ∈ U [−1, 1], γ = 1.4 and the scaling parameter σ = 0.5. This

is a simple initial condition in the sense that the first two Legendre polynomials are

sufficient to represent the initial function exactly. As case no 2, we consider (7.1)

subject to uncertainty in the initial shock location. Let

u(x, t = 0, ξ) =

uL = (1, 0, 2.5/γ)T x < 0.5 + ση

uR = (0.125, 0, 0.25/γ)T x > 0.5 + ση

where we assume γ = 1.4 and the scaling parameter σ = 0.05. Here, η takes a

triangular distribution, which we parameterize as a nonlinear function in ξ ∈ U [−1, 1],

i.e.

η(ξ) = (−1 +√ξ + 1)1−1≤ξ≤0(ξ) + (1−

√1− ξ)10<ξ≤1(ξ),

where the indicator function 1A of a set A is defined by 1A(ξ) = 1 if ξ ∈ A and zero

otherwise. For case 2, exact representation of the initial function requires an infinite

number of expansion terms in the MW basis. Figure 8.1 depicts the shock tube setup

for the two cases, with dashed lines denoting uncertain parameters. We will also

investigate another version of case 2, where the right state density is significantly

reduced to obtain a strong shock.


0 x0 1

ρL

ρR

2σ

?

6

0.25σ ?6

2σ-

0 x0 1

ρL

ρR

Figure 7.2: Schematic representation of the initial setup for case 1 (left)and case 2 (right).

7.4.3 Initial conditions and resolution requirements

For case 2, it should be noted that although the initial shock position can be exactly

described by the first two terms of the Legendre polynomial chaos expansion, this

is not the case for the initial state variables. In fact, for the the polynomial chaos

expansions of the density, momentum and energy, the error decay only slowly with

the number of expansion terms. Thus, unless a reasonably large number of expansion

terms are retained, the stochastic Galerkin solution of case 2 will not be accurate

even for small times.

The Legendre coefficients at small times display an oscillating behavior that be-

comes sharper with the order of the coefficients. The wavelet coefficients exhibit

peaks that get sharper with the resolution level, and require a fine mesh. Figure

7.3 shows the initial Legendre coefficients and the initial Haar wavelets for case 2.

The numerical method has a tendency to smear the chaos coefficients, resulting in

under-predicting the variance. The increasing cost of using a larger number of basis

functions is further increased by the need for a finer mesh to resolve the solution

modes.


0.3 0.4 0.5 0.6 0.7−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(w1)0

(w1)1

(w1)2

(w1)3

(w1)4

(w1)5

(w1)6

(w1)7

(a) Legendre polynomials

0.3 0.4 0.5 0.6 0.7−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(w1)0

(w1)1

(w1)2

(w1)3

(w1)4

(w1)5

(w1)6

(w1)7

(b) Haar wavelets

Figure 7.3: Initial w1 modes for case 2, first 8 basis functions.

7.4.4 Convergence of multiwavelet expansions

For moderate simulation times, the numerical solution on a sufficiently fine spatial

mesh converges as the order of MW expansion increases by increasing the polynomial

degree Np or the resolution level Nr. Figure 7.4 shows the decay in the error of the

variance of velocity and energy as a function of Np and Nr. For well-behaved cases

like these, one may freely choose between increasing Np and Nr, in order to increase

the accuracy of the solution of the quantity of interest.


0

1

2

3

1

2

3

0

0.01

0.02

0.03

0.04

NrN

p

(a) Case 1,∥∥V ar(vP )− V ar(v)

∥∥2.

0

1

2

3

1

2

3

0

0.002

0.004

0.006

0.008

0.01

NrN

p

(b) Case 1,∥∥V ar(EP )− V ar(E)

∥∥2.

0

1

2

3

1

2

3

0

0.01

0.02

0.03

0.04

NrN

p

(c) Case 2,∥∥V ar(vP )− V ar(v)

∥∥2.

0

1

2

3

1

2

3

0

0.01

0.02

0.03

0.04

NrN

p

(d) Case 2,∥∥V ar(EP )− V ar(E)

∥∥2.

Figure 7.4: Decay in variance of velocity and energy as a function of theorder of expansion, polynomial order Np and resolution level Nr. Case 1,t = 0.05, 280 spatial points restricted to x ∈ [0.4, 0.65]. Solution obtainedwith the Roe variable scheme.

For longer simulation times or more extreme cases, e.g. supersonic flow, high-

order polynomial representation (increasing Np) may not lead to increased accuracy,


but instead breakdown of the numerical method. Next, we study the qualitative

properties of the MW representation of case 1 and case 2 for two extreme cases:

Legendre polynomials (Nr = 0) and piecewise constant Haar wavelets (Np = 0).

Figure 7.5 shows the density surface in the x− ξ-plane of case 1 and case 2 at t =

0.15 based on exact solution evaluations, and computed with Legendre polynomials

and Haar wavelets. The computed solution with Legendre polynomial reconstruction

captures essential features of the exact solution, but the use of global polynomials

cause oscillations downstream of the shock.

With Haar wavelets, there are no oscillations downstream, as in the Legendre

polynomials case. However, the 8 ’plateaus’ seen in figure 7.5 (e) corresponds to the

8 basis functions. When the order of wavelet chaos expansion increases, the number

of plateaus increases, and the solution converges to the exact solution.

From Figure 7.5 it is clear that the effect of the choice of multiwavelet basis to

some extent depends on the problem at hand. The Haar wavelets yield numerical

solutions that are free of oscillations but converge only slowly. Oscillations around

discontinuities in stochastic space should be expected when a polynomial basis is used

and may lead to severe problems when variables attain unphysical values, e.g. when

the oscillations downstream of the shock leads to negative density. Thus, more robust

multiwavelets are required for problems with stronger shocks, as we demonstrate

below.

7.4.5 Robustness

The stochastic Galerkin method applied to the Roe variables gives a more robust

method than the conservative variables formulation. Figure 7.6 shows the relative

errors of the solution in the 2, 2 norm for modified versions of case 2 with stronger

shocks, ρL = 1, and a range of right state densities, ρR = 2−k, k = 3, .., 8 for 8

basis wavelets. This corresponds to Mach numbers up to Ma = 2.0. Figure 7.6

also includes the relative error of the Mach number to verify that the cases solved

for were reasonable close to the supersonic range they model. For this problem, the

conservative variable formulation was unstable except for the original subsonic case 2


00.5

1 −1

0

10

0.5

1

1.5

ξx

(a) Exact solution, case 1.

00.5

1 −1

0

10

0.2

0.4

0.6

0.8

1

ξx

(b) Exact solution, case 2.

00.5

1 −1

0

10

0.5

1

1.5

ξx

(c) Lege. polyn. (Np, Nr) = (8, 0), case 1.

00.5

1 −1

0

10

0.2

0.4

0.6

0.8

1

ξx

(d) Lege. polyn. (Np, Nr) = (8, 0), case 2.

00.5

1 −1

0

10

0.5

1

1.5

ξx

(e) Haar wavelets (Np, Nr) = (0, 3), case 1.

00.5

1 −1

0

10

0.2

0.4

0.6

0.8

1

ξx

(f) Haar wavelets (Np, Nr) = (0, 3), case 2.

Figure 7.5: Density as a function of x and ξ at t = 0.15.


(ρR = 0.125). Thus, although no clear explanation is known to us, it seems that the

Roe variable formulation is more suitable for problems where robustness is an issue.

Legendre polynomials are not suitable for this problem. As seen in Figure 7.5 (c)

and (d), the solution is oscillatory in the right state close to the shock. If the right

state density is small, as in this supersonic case, such oscillations cause the density

to be very close to zero, or even negative. This leads to an unphysical solution and

breakdown of the numerical method.

−3 −4 −5 −6 −7 −80

0.02

0.04

0.06

0.08

0.1

0.12

log2(ρR)

‖ρP−ρ‖2 ,2

‖ρ‖ 2 ,2

‖vP−v‖2 ,2

‖v‖ 2 ,2

‖EP−E‖2 ,2

‖E‖ 2 ,2

‖MaP−Ma‖2 ,2

‖Ma‖ 2 ,2

Figure 7.6: Relative error in density, velocity, energy and Mach numberat t = 0.15 for different shock strengths. m = 300 spatial points, 8 Haarwavelets (Np = 0, Nr = 3).

7.4.6 Computational cost

For two stochastic Galerkin systems of order P = (Np + 1)2Nr − 1 and P ′ = (N ′p +

1)2N′r − 1 where P = P ′ but Np 6= N ′p, N

′r 6= Nr, the size of the problem and the

computational cost is the same. Although the different bases could possibly result in

properties that make them very different in e.g. the number of iterations required to

solve the nonlinear matrix problems, no such tendency was observed. The numerical


experiments yield very similar computational costs for the cases tested.

In order to compare the computational cost of the Roe variable expansion with

that of the conservative expansion, a similar experimental setup is used for both

methods. Sufficiently small test cases are run in order not to exceed the cache limit

which would slow down the simulation time for fine meshes and bias the result. We

used test case 1 for short simulation times.

In the experiments, the same time step has been used for the different variable

expansions, since the stability limit is very similar. Table 7.1 displays the relative

simulation time of the two different variable expansions for increasing number of Haar

wavelets (P = P + 1 = 2Nr , Np = 0). One time unit is defined as the time for the nu-

merical simulation of a single deterministic problem using the same numerical method

with similar input conditions, discretization and time step. The higher computational

cost for the conservative variable formulation is due to the need to compute inverse

quantities and cubic spectral products. The Roe variable formulation only requires

the solution of the nonlinear system for the square root of the density and quadratic

flux function evaluations. The relative benefit of the Roe variable expansion decreases

with the order of wavelet expansion. This is due to the increasing cost of forming

spectral products that dominates the total cost for high-order expansions.

Order of MW P = 2 P = 4 P = 8 P = 16

Time Roe variables 14 16 26 60

Time cons. variables 410 457 534 643

Table 7.1: Relative simulation time using conservative variables and Roevariables, respectively. One time unit is defined as the simulation timeof a single deterministic problem with the same time-step as for the MWcases.


7.5 Conclusions

An intrusive formulation of the stochastic Euler equations based on Roe variables is

presented. A Roe average matrix for the standard MUSCL-Roe scheme with Roe

variables is derived, and we prove that it satisfies the conditions stated by Roe under

certain conditions.

The Legendre polynomial basis exactly represents the input uncertainty in our

first test case, but it leads to oscillations around the discontinuity in stochastic space.

The Haar wavelets do not represent the input uncertainty exactly in either test case,

but are more robust to discontinuities.

The Roe variable formulation is robust for supersonic problems where the con-

servative variable formulation fails, but only for localized basis functions of the gen-

eralized chaos representation. For global Legendre polynomials, the discontinuities

in stochastic space lead to oscillations and unphysical behavior of the solution and

numerical instability. Wavelet functions are more robust in this respect, and do not

yield oscillations around discontinuities in stochastic space.

The Roe variable formulation leads to speedup compared to the conservative vari-

able formulation. The relative speedup decreases with the order of generalized chaos

since the total computational cost for high-order expansions is no longer dominated

by spectral inversion and square root calculations. Instead, the main cost lies in the

formation of spectral product matrices. However, for low order multiwavelet expan-

sions, the speedup is significant.

We demonstrate the need for robust flux functions by presenting cases where the

standard MUSCL-Roe flux fails to capture the solution. The design of a robust

numerical method is also highly dependent on the choice of the stochastic basis. The

Haar wavelets are not only more robust than Legendre polynomials for representation

of discontinuities in stochastic space, but also admit the proof of existence of a Roe

matrix and more specifically the hyperbolicity of the stochastic Galerkin formulation.

This implies that the truncated problem mimics the original problem - a desirable

feature.

If the representation of the initial function has not converged, the solution at


future times can not be accurate. Case 2 illustrates the need to find a representation

of uncertainty with fast decay of the coefficients of the generalized chaos expansion.

An alternative to more accurate representation of the input uncertainty is to combine

the intrusive Roe variable formulation presented here with multi-element methods,

for instance in the manner presented in [99].

Chapter 8

A hybrid scheme for two-phase flow

A stochastic two-phase problem in one spatial dimension is investigated as a first

step towards developing an intrusive method for shock-bubble interaction with generic

uncertainty in the input parameters. So et al [91] investigated a two-dimensional two-

phase problem subject to uncertainty in bubble deformation and contamination of the

gas bubble, based on the experiments in [40]. The eccentricity of the elliptic bubble

and the ratio of air-helium of the bubble were assumed to be random variables, and

quantities of interest were obtained by numerical integration in the stochastic range

(stochastic collocation). Previous work on uncertainty quantification for multi-phase

problems include petroleum reservoir simulations with stochastic point collocation

where deterministic flow solvers are evaluated at stochastic collocation points [58]

and Karhunen-Loeve expansions combined with perturbation methods [16].

We assume uncertainty in the location of the material interface, which requires a

stochastic representation of all flow variables. Stochastic quantities are represented

as generalized chaos series, that could be either global as in the case of generalized

polynomial chaos [115], or localized, see e.g. [20]. For robustness, we use a general-

ized chaos expansion with multiwavelets to represent the solution in the stochastic

dimension [79]. It should be noted that this basis is global, so the method is fully in-

trusive. However, the basis is hierarchically localized in the sense that multiwavelets

belonging to the same resolution level are grouped into families with non-overlapping

support. These features makes it suitable for approximating discontinuities in the

133

CHAPTER 8. A HYBRID SCHEME FOR TWO-PHASE FLOW 134

stochastic space without the oscillations that occur in the case of global polynomial

bases.

The stochastic Galerkin method is applied to the stochastic two-phase formulation,

resulting in a finite-dimensional deterministic system that shares many properties

with the original deterministic problem. The regularity properties of the stochastic

problem are essential in the design of an appropriate numerical method. Chen et al

studied the steady-state inviscid Burgers’ equation with a source term [17]. We used

a similar approach for the inviscid Burgers’ equation with uncertain boundary condi-

tions and also analyzed the regularity of low-order stochastic Galerkin approximations

of the problem [74]. Schwab and Tokareva analyzed regularity of scalar hyperbolic

conservation laws and a linearized version of the Euler equations with uncertain ini-

tial profile [88]. In this Chapter, we analyze smoothness of the stochastic two-phase

problem.

The stochastic Galerkin problem is hyperbolic. This generalized and extended

two-phase problem is solved with a hybrid method coupling the continuous phase

region with the discontinuous phase region through a numerical interface. The non-

smooth region is solved with the HLL-flux, MUSCL-reconstruction in space, and

fourth order Runge-Kutta integration in time. The minmod flux limiter is employed

in the experimental results displayed below.

Finite-difference operators in summation-by-parts (SBP) form are used for the

high-order spatial discretization. A symmetrized problem formulation that generalizes

the energy estimates in [28] for the Euler equations is used for the stochastic Galerkin

system. The coupling between the different solution regions is performed with a weak

imposition of the interface conditions through an interface using a penalty technique

[13]. A fourth order Runge-Kutta method is used for the integration in time.


8.1 Two-phase flow problem

We assume two phases with volume fractions α and β = 1−α on the domain x ∈ [0, 1],

governed by the advection equation

∂

∂tα + v′(x, t)

∂

∂xα = 0, (8.1)

where we let v′(x, t) = v(x, t) be the advective velocity obtained from the conservative

Euler system below. The Euler equations determine the conservation of masses αρα

and βρβ, momentum ρv, and total energy E of the two phases through

∂u

∂t+∂f

∂x= 0, (8.2)

where

u =

αρα

βρβ

ρv

E

, f =

αραv

βρβv

ρv2 + p

(E + p)v

. (8.3)

We assume that the pressure p is given by the perfect gas equation of state for two

phases

p = (γ − 1)

(E − 1

2ρv2

), γ =

1αγα

+ βγβ

,

where γ denotes the ratio of specific heats. The total density is given by ρ = αρα+βρβ.

Note that the sum of the first and second equations of (8.2) is the standard mass

conservation of the Euler equations. Thus, an equivalent formulation is the Euler

equations supplemented with an extra mass conservation equation for one of the

phases α and β.

We investigate the Riemann problem defined by the initial conditions

(α, αρα, βρβ, ρv, E)T =

(1, 1, 0, 0, 2.5)T x < x0 + ξ

(0, 0, 0.125, 0, 0.25)T x > x0 + ξ, (8.4)


where ξ is a parametrization of the measured or modeled uncertainty in the initial

membrane location. Despite the seemingly simple nature of the initial condition, the

MW series of the initial condition has an infinite number of non-zero terms. Thus,

stochastic truncation error is an issue already at t = 0.

The stochastic Galerkin formulation of the two-phase problem is obtained by

multiplying (8.1) and (8.2) by each one of the basis functions ψi(ξ), and integrating

with respect to the probability measure P over the range of ξ. Initial functions are

obtained by projection of (8.4) onto the basis functions ψi(ξ). The MW expansion is

truncated to M + 1 terms and we get the systems for the MW coefficients

∂

∂tαk +

M∑

i=0

M∑

j=0

vi∂

∂xαj〈ψiψjψk〉 = 0, k = 0, ...,M, (8.5)

βk = δk0 − αk, , k = 0, ...,M, (8.6)

and

∂

∂t

(αρα)k

(βρβ)k

(ρv)k

Ek

+∂

∂x

∑Mi=0

∑Mj=0(αρα)ivj〈ψiψjψk〉∑M

i=0

∑Mj=0(βρβ)ivj〈ψiψjψk〉∑M

i=0

∑Mj=0(ρv)ivj〈ψiψjψk〉+ pk∑M

i=0

∑Mj=0(Ei + pi)vj〈ψiψjψk〉

= 0, k = 0, ...,M. (8.7)

MW expansions for e.g. the pressure can be updated from the MW of the conservative

variables, and then be inserted into the fluxes. We use a pseudo-spectral approxima-

tion of high-order stochastic products for the pressure update. In the computation of

e.g. the order M product z(ξ) =∑M

k=0 zkψk of three stochastic variables a(ξ), b(ξ),


c(ξ), for k = 0, . . . ,M , we use the approximation

zk = (a(ξ)b(ξ)c(ξ))k =

⟨(M∑

i=0

aiψi(ξ)

)(M∑

j=0

bjψj(ξ)

)(M∑

l=0

clψl(ξ)

)ψk(ξ)

⟩

=M∑

i=0

M∑

j=0

M∑

l=0

〈ψiψjψkψl〉 aibjcl ≈M∑

i=0

M∑

m=0

〈ψiψmψk〉 aiM∑

j=0

M∑

l=0

〈ψjψlψm〉 bjcl︸︷︷︸

(bc)Mm

≡ (a ∗ (b ∗ c))k, (8.8)

where the pseudo-spectral product z = a ∗ b of order M is defined by

zMk = (a ∗ b)k =M∑

i=0

M∑

j=0

〈ψiψjψk〉aibj, k = 0, . . . ,M. (8.9)

In matrix notation, we can express this as

zM = A(aM)bM , (8.10)

where zM = (z0, . . . , zM)T is the vector of MW coefficients of z and [A(aM)]j+1,k+1 =∑M

i=0〈ψiψjψk〉ai. By successively applying (8.10), we obtain approximations of a

range of stochastic functions including polynomials, square roots and inverse quanti-

ties.

8.2 Smoothness properties of the solution

8.2.1 Analytical solution

The exact solutions to (8.1) and (8.2) subject to (8.4) can be determined analytically,

and are discontinuous for all times. The advection problem (8.1) with v independent

of x and t has the solution

α(x, t) = α0(x− vt),


which is to be interpreted in the weak sense here since it is discontinuous for all t when

α0 is chosen to be a step function. The conservation law (8.2) is a straightforward

extension of the Sod test case for shock tube problems and its exact piecewise smooth

solution can be found in [92]. The solution consists of five distinct smooth regions

(denoted u(L), u(exp), u(2), u(1) and u(R)), and the discontinuities may be found at the

interfaces between the different regions. Assume that the initial interface location is

xs0 = x0 + ξ as given in (8.4). We can then express the deterministic solution for any

fixed ξ as a piecewise smooth solution, separated by the four spatial points

x1(t, ξ) = x0 + ξ −√γpLρLt (8.11)

x2(t, ξ) = x0 + ξ +

(v2 −

√γp2

ρ2

)t (8.12)

x3(t, ξ) = x0 + ξ + v2t (8.13)

x4(t, ξ) = x0 + ξ +Mst, (8.14)

where Ms is the Mach number of the shock.

6t

-xx4x3x2x1 xs0

\\\\\\\\

cc

cc

ccc

cct′

0

u(L) u(exp) u(2) u(1) u(R)

6t

-ξξ1ξ2ξ3ξ4 ξs0

JJJJJJJJ

BBBBBBBB

#########t′

0

u(R) u(1) u(2) u(exp) u(L)

Figure 8.1: Schematic representation of the solution of the two-phase prob-lem. Solution regions in the x − t space for a fixed ξ (left), and solutionregions in ξ − t space for a fixed x (right).

Any given value of ξ will determine the location of the different regions of piecewise

continuous solutions, so the true stochastic solution can be expressed as a function of

ξ and the variables of the true deterministic solution. In the x-t-ξ-space, all solution

discontinuities are defined by triplets (x, t, ξ) satisfying (8.11) - (8.14). The solution

regions are depicted in Figure 8.1 (left) for any fixed value of ξ.


For any point x, the solution regions can be defined as functions of ξ and t. This is

shown in Figure 8.1 (right), where the points in the stochastic dimension separating

the different solutions regions are given by

ξ1(x, t) = x− x0 +

√γpLρLt (8.15)

ξ2(x, t) = x− x0 −(v2 −

√γp2

ρ2

)t (8.16)

ξ3(x, t) = x− x0 − v2t (8.17)

ξ4(x, t) = x− x0 −Mst. (8.18)

The solution can be written

u(x, t, ξ) = u(L)1ξ1≤ξ + u(exp)(x− ξ)1ξ2≤ξ≤ξ1 + u(2)1ξ3≤ξ≤ξ2 + u(1)1ξ4≤ξ≤ξ3

+ u(R)1ξ≤ξξ4 (8.19)

where the indicator function 1A of a set A is defined by 1A(ξ) = 1 if ξ ∈ A and

zero otherwise.

Note that if the range of ξ is bounded, some solution states may not occur with

non-zero probability for an arbitrary x. The situation shown in Figure 8.1 (right)

requires a sufficiently large range of ξ, or, equivalently, that x is sufficiently close to

x0. The expression (8.19) is always true however.

8.2.2 The stochastic modes

The solutions of (8.1) and (8.2) for fixed values of ξ are discontinuous, but the stochas-

tic modes (multiwavelet coefficients) are continuous. To see this, we proceed from the

solution (8.19) to derive exact expressions for the stochastic modes. We assume that

the probability measure P has a probability density p. The kth mode uk is given by

the projection of (8.19) on ψk(ξ),


uk(x, t) =

∫

Ω

u(x, t, ξ)ψk(ξ)p(ξ)dξ = u(L)

∫ ∞

ξ1

ψk(ξ)p(ξ)dξ

+

∫ ξ1

ξ2

u(exp)(x− ξ)ψk(ξ)p(ξ)dξ + u(2)

∫ ξ2

ξ3

ψk(ξ)p(ξ)dξ + u(1)

∫ ξ3

ξ4

ψk(ξ)p(ξ)dξ

+ u(R)

∫ ξ4

−∞ψk(ξ)p(ξ)dξ. (8.20)

The density p and multiwavelet ψk are at least piecewise continuous functions, so by

(8.20) uk ∈ C0. Now assume that the parametrization ξ of the uncertainty in the

location of x0 has a probability density p ∈ Cs(R) for some degree of regularity s ∈ N.

There exists a set ψi∞i=1 of polynomials that are orthogonal with respect to p. With

this choice of basis functions, we may differentiate (8.20) with respect to x,

∂

∂xuk = −u(L)ψk(ξ1)p(ξ1) + uexp(x− ξ1)ψk(ξ1)p(ξ1)− uexp(x− ξ2)ψk(ξ2)p(ξ2)

+

∫ ξ1

ξ2

u′(exp)(x− ξ)ψk(ξ)p(ξ)dξ + u(2)ψk(ξ2)p(ξ2)− u(2)ψk(ξ3)p(ξ3)

+ u(1)ψk(ξ3)p(ξ3)− u(1)ψk(ξ4)p(ξ4) + u(R)ψk(ξ4)p(ξ4), (8.21)

where we used that ∂ξi/∂x = 1, i = 1, 2, 3, 4. In fact, uk(x, t) as given by (8.20) is

s+ 1 times differentiable in x or t for t > 0 and uk ∈ Cs+1.

Remark 7. Note that the smoothness of uk in x and t ultimately depends on the

smoothness of p and the choice of basis functions ψi∞i=0, which are all functions of

ξ. In contrast, for any fixed value of ξ, the solution u(x, t, ξ) is discontinuous in the

spatial and temporal dimensions, no matter the smoothness of p and ψi∞i=0.

8.2.3 The stochastic Galerkin solution modes

We investigated the smoothness properties of the stochastic modes of the original

problems problem (8.1) and (8.2) above, but in all actual computations we solve the


modified stochastic Galerkin approximation (8.5)-(8.7). For low-order MW approxi-

mations (small M), the smoothness properties are very different from those derived

above. For instance, the M = 0 approximation is the deterministic two-phase problem

with its characteristic discontinuous solution profile. First order gPC approximations

using a group of orthogonal polynomials and multiwavelets results in linear combi-

nations of deterministic two-phase problems. In terms of regularity, these problems

are clearly equivalent to the deterministic problem. Higher order gPC approxima-

tions result in large nonlinear stochastic Galerkin problems that in general cannot

be diagonalized into a set of deterministic two-phase problems. Due to their non-

linear nature, we expect these problems to develop discontinuities. However, it is a

reasonable assumption that the solution converges to the solution of (8.2). Hence,

we assume that the discontinuities get weaker with the order of gPC expansion so

that high-order MW approximations have regularity properties that approach the

smoothness properties of the analytical stochastic modes.

We have analyzed smoothness of the particular problem of uncertain initial loca-

tion of the shock in the Riemann problem (8.4). An essential feature of the analysis

is that for t > 0, the locations of the discontinuities become stochastic. If this were

not the case, the gPC coefficients would not be smooth. Thus, for any given set

of initial conditions, smoothness should be analyzed in order to determine about an

appropriate numerical method.

In order to solve (8.5)-(8.7) numerically for arbitrary order M of MW expansion

(that may vary in space depending on the smoothness of the solution), we need shock-

capturing methods that can account for the discontinuities that are expected due to

the stochastic truncation. In regions away from the discontinuities, the solution is at

least as smooth as the corresponding deterministic problem and high-order methods in

combination with smooth polynomial stochastic basis functions are more suitable. In

the next section, we present a method which combines high-order and shock-capturing

methods for the stochastic Galerkin systems.


8.3 Numerical method

The computational domain is divided into regions of smooth behavior of the solution,

and regions of sharp variation. At this stage, these are assumed to be known a priori

and do not change with time. However, the methodology may be extended to time-

dependent regions, see [23]. A fourth-order Runge-Kutta method is used for the time

integration.

8.3.1 Summation-by-parts operators

The smooth regions are discretized using a high-order finite difference method based

on SBP operators. Boundary conditions are imposed weakly through penalty terms,

where the penalty parameters are chosen such that the numerical method is stable.

Operators of order 2n, n ∈ N, in the interior of the domain are combined with

boundary closures of order of accuracy n.

The first derivative SBP operator was introduced in [50, 94]. Let u denote the

uniform spatial discretization of u. For the first derivative, we use the approximation

ux ≈ P−1Qu, where subscript x denotes partial derivative and Q satisfies

Q+QT = diag(−1, 0, . . . , 0, 1) ≡ B. (8.22)


discrete norm. For the proof of stability, P must be diagonal.

8.3.2 HLL Riemann solver

In the non-smooth regions, MUSCL-type flux limiting [103] is used for the reconstruc-

tion of the left and right states of the conservative fluxes and the advection of the

volume fractions. For the conservative problem (8.2), we employ the HLL Riemann


solver introduced by Harten et. al. [42],

Fj+ 12

=

f(uLj+ 1

2

) if SL ≥ 0

SRf

(uLj+ 1

2

)−SLf

(uRj+ 1

2

)+SLSR

(uRj+ 1

2−uL

j+ 12

)

SR−SLif SL < 0 < SR

f(uRj+ 1

2

) if SR ≤ 0

,

where S denotes the fastest signal velocities. These are taken as estimates of the

maximum and minimum eigenvalues of the Jacobian of the flux. In the deterministic

case, the eigenvalues of the Jacobian are known analytically, so the method is inex-

pensive. For the stochastic Galerkin system, analytical expressions are not available,

and numerical approximations of the eigenvalues are used instead. In general, ob-

taining accurate eigenvalue estimates may be computationally costly. However, for

the piecewise constant and piecewise linear multiwavelet expansion, we have explicit

expressions for the system eigenvalues due to the constant eigenvectors of the inner

triple product matrices A given by (8.10), see [77].

The HLL-flux approximates the solution by assuming three states separated by

two waves. In the deterministic case, this approximation is known to fail in capturing

contact discontinuities and material interfaces [97]. The stochastic Galerkin system is

a multiwave generalization of the deterministic case, and similar problems in capturing

missing waves are expected. However, by applying flux limiters (minmod) in the

same way as in the MUSCL scheme with Roe flux, sharp features of the solution are

recovered.

The HLL-flux and MUSCL reconstruction is applied to solve the conservative

problem (8.7). The (standard) MUSCL scheme is used to solve the advection problem

(8.5) in the regions where the solution is expected to be non-smooth.

8.3.3 Hybrid scheme

Numerical interfaces can be designed for stable coupling of problems solved separately

using SBP operators. The MUSCL scheme can be rewritten in SBP operator form

with an artificial dissipation term [1] and can therefore be coupled with other schemes


using SBP operators [23]. The coupling requires the artificial dissipation to be zero

at the interface in order to enable energy estimates.

The computational domain is divided into a left smooth solution region and a right

non-smooth solution region that are weakly coupled with an interface. The leftmost

lying part of the right region is a transition region where a second order one-sided

SBP scheme is applied that transitions into the HLL-MUSCL scheme. In this way,

there is a stable coupling between the high-order SBP scheme of the left domain and

the second order SBP scheme of the transition region. Numerical dissipation within

the order of the scheme is added to the regions where SBP operators are used. Figure

8.2 schematically depicts the hybrid scheme, applied to two spatial grids and coupled

with an interface.

× × × × × ×SBP high order -

× × × × × × × × × × × ×SBP 2nd order - HLL-MUSCL -

Figure 8.2: Solution regions on the spatial mesh.

An energy estimate for the continuous problem

We will analyze stability for two solution regions coupled by an interface. However,

we start with the continuous problem on a single domain. In order to do this, we sym-

metrize the two-phase problem. We assume the existence of a convex entropy function

S(uM), i.e. the Hessian ∂2S/∂uMi ∂uMj is positive definite. (Note that convexity as

defined here does not allow for zero eigenvalues of the Hessian). Then, by [41], there

exists a variable transformation wM(uM) = ∂S/∂uM such that f(wM) = f(uM) and

HwMt + JwwMx = 0,

where wM denotes the vector of MW coefficients of the order M approximation of the

transformed variables, and the inverse Hessian H = ∂uM/∂wM = (∂2U/∂ui∂uj)−1

and Jacobian Jw = ∂f/∂w are symmetric matrices. Due to convexity H is positive

definite and thus defines a norm. As in the case of the Euler equations, the two-phase


equations are homogeneous of degree τ , which implies

HwM = τuM and JwwM = τ fM . (8.23)

We will use the canonical splittings

uMt =τ

1 + τuMt +

1

1 + τHwMt , fMx =

τ

1 + τfMx +

1

1 + τJww

Mx .

To obtain an energy estimate for the continuous and stability for the semi-discrete

problem, the stochastic Galerkin formulation of the two-phase problem must be ho-

mogeneous. To show that this holds under the assumption that the corresponding

deterministic problem is homogeneous and some additional assumptions, we consider

a deterministic problem that is homogeneous of degree τ . Let

J(u)u = τf(u), (8.24)

with solution u ∈ Rn, Jacobian J ∈ Rn×n and flux f ∈ Rn for a system of n equa-

tions. Now assume that the problem satisfying (8.24) is subject to uncertainty in

the parameters or in the input conditions. Let Jij denote the (i, j) entry of J which

can be expressed as a truncated MW expansion Jij =∑M

k=0(Jij)kψk. The stochastic

Galerkin Jacobian JM corresponding to J consists of n× n submatrices, each of size

(M + 1)× (M + 1). Let JMij be the (i, j) submatrix of JM , defined by

[JMij ]lm = 〈ψlψmJij〉 =

M∑

k=0

(Jij)k〈ψkψlψm〉, i, j = 1, . . . , n, l,m = 0, . . . ,M.

(8.25)

The stochastic Galerkin flux vector of MW coefficients fM = ((f1)0, . . . , (f1)M , . . . ,

(fn)0, . . . , (fn)M)T is a nonlinear function and for an arbitrary order M basis of mul-

tiwavelets, it is not uniquely defined. To see this, with the pseudo-spectral product

∗ defined in (8.9), in general

(a ∗ b) ∗ c 6= a ∗ (b ∗ c)


for MW approximations of stochastic functions a(ξ), b(ξ), c(ξ), each one truncated

to some order M . This implies that the definition of the stochastic Galerkin flux

fM depends on the order in which pseudo-spectral operations are performed when

evaluating fM . Hence, it is not uniquely defined. We may now either restrict ourselves

to MW bases where the order of pseudospectral operations does not matter e.g.

Haar wavelets, or, we may restrict the order in which pseudo-spectral operations

are performed so as to make sure that mathematical properties of interest - e.g.

homogeneity - are satisfied. We take the latter approach and define the order M

approximation of f through its MW coefficients by

(fi)k ≡1

τ

n∑

j=1

(Jij ∗ uj)k, i = 1, . . . , n, k = 0, . . . ,M, (8.26)

which is consistent with the deterministic homogeneous problem. Note that relation

(8.26) is essentially just a restriction on the order of pseudo-spectral operations in the

calculation of f . It stipulates that f must be defined in terms of the approximation

of J . Clearly, the approximation of J should also be as close to the true (i.e. inifinite

order MW expansion) J as possible. However, for the energy estimates that require

homogeneity of the stochastic Galerkin formulation, we only need to satisfy (8.26).

Proposition 8. Assume that the deterministic problem (8.24) holds and for a con-

sistent pseudo-spectral approximation JM of J , let the stochastic Galerkin flux fM

be given by the MW coefficients as defined in (8.26). Then the stochastic Galerkin

formulation of order M is also homogeneous of degree τ , i.e. it satisfies

JM(uM)uM = τfM(uM), (8.27)

where uM = ((u1)0, . . . , (u1)M , . . . , (un)0, . . . , (un)M)T ∈ Rn(M+1) and

fM = ((f1)0, . . . , (f1)M , . . . , (fn)0, . . . , (fn)M)T ∈ Rn(M+1).

Proof. Using the notation (8.10) for the pseudo-spectral product ∗, by (8.25) the (i, j)

submatrix of JM can be written

JMij = A(JMij ), i, j = 1, . . . , n,


where JMij = ((Jij)0, . . . , (Jij)M)T . Thus, we have

JM =

A(JM11 ) . . . A(JM1n)...

. . ....

A(JMn1) . . . A(JMnn)

.

By the relation (8.26), any subvector fMi = ((fi)0, . . . , (fi)M)T of the total flux vector

of MW coefficients fM can be written

fMi =1

τ

n∑

j=1

A(JMij )uMj , i = 1, . . . , n.

Then, considering the ith row of submatrices,

[JMuM ]i =n∑

j=1

JMij u

Mj =

n∑

j=1

A(JMij )uMj = τfMi , i = 1, . . . , n,

which is equal to (8.27).

Remark 8. The Jacobian entries Jij are nonlinear functions so JM may also not be

uniquely defined due to the possible ambiguity in the pseudo-spectral approximations.

For the proof, we have only restricted fM depending on JM but we have not defined

JM uniquely by specifying the order of pseudo-spectral operations of its calculation.

We will now derive an energy estimate for the continuous symmetrized formulation

of the stochastic Galerkin Euler equations in split form,

τ

1 + τuMt +

1

1 + τHwMt +

τ

1 + τfMx +

1

1 + τJwwx = 0. (8.28)

Under the conditions of Proposition 8, multiply (8.28) by (1 + τ)(wM)T and integrate


over the physical domain. We get

τ

∫ 1

0

(wM)TuMt dx+

∫ 1

0

(wM)T HwMt dx+ τ

∫ 1

0

(wM)T fMx dx+

∫ 1

0

(wM)TJwwMx dx =

=

∫ 1

0

((wM)T (HwM)t + (wM)T HwMt

)dx+

∫ 1

0

((wM)T (Jww

M)x + (wM)TJwwMx

)dx =

=d

dt

∥∥wM∥∥H

+ [(wM)TJwwM ]10 = 0, (8.29)

where the first equality follows from (8.23). The generalized energy estimate (8.29)

is a straightforward stochastic Galerkin generalization of the one given for the deter-

ministic problem in [28].

Stability in a single domain

Next we consider the semi-discrete problem and start with a single domain. The

stability analysis is a direct generalization of the stability of the symmetrized Euler

equations in [28]. We define the flux and the Jacobian under the conditions of Proposi-

tion 8 which implies that the stochastic Galerkin system is homogeneous. Let uM and

wMdenote the spatial discretizations of uM and wM , respectively, on a mesh consisting

of m equidistant grid points. Let E1 = diag(1, 0, . . . , 0) and Em = diag(0, . . . , 0, 1).

The semi-discretized scheme is

τ

1 + τuMt +

1

1 + τHwM

t +τ

1 + τ(P−1Q⊗ I)fM(wM) +

1

1 + τJw(P−1Q⊗ I)wM

= (P−1E1 ⊗ Σw1 )(wM − g1) + (P−1Em ⊗ Σw

m)(wM − gm) (8.30)

where H is block diagonal with each diagonal block equal to H evaluated at the

spatial points. Σw1 and Σw

m are penalty matrices to be determined and g1 and gm

are vectors where only the entries corresponding to the left and right boundary are

allowed non-zero values. We assume a diagonal norm P , so (P ⊗ I)H = H(P ⊗I). Also, Jw commutes with (P ⊗ I). In order to show stability, we may assume

homogeneous boundary conditions g1 = gm = 0. Multiplying (8.30) from the left by


(1 + τ)(wM)T (P ⊗ I) and using the homegeneity properties of (8.23) yields

d

dt‖wM‖2

(P⊗I)H + (wM)T(

(Q⊗ I)Jw + Jw(Q⊗ I))

wM

= (1 + τ)(wM)T1 Σw1 wM

1 + (1 + τ)wTmΣw

mwMm (8.31)

Add the transpose of (8.31) to itself and use the SBP relation (8.22)

d

dt

∥∥wM∥∥2

(P⊗I)H = wT1

(Jw(wM

1 ) + (1 + τ)Σw1

)wM

1

+ (wMm )T

(−Jw(wM

m ) + (1 + τ)Σwm

)wMm (8.32)

The scheme is stable with the penalties

Σw1 = −δ1J

+w (wM

1 ), Σwm = δmJ

−w (wM

m ), δ1, δm ≥1

1 + τ.

Remark 9. The stability analysis above follows that in [28]; for the case M = 0 the

analysis is in fact identical. We show here that the analysis in [28] generalizes to

the stochastic Galerkin formulation of order M of multiwavelet expansion under the

conditions of Proposition 8.

Stability at the interface

Now consider a problem with two domains connected by an interface. By ignoring

the imposition of boundary conditions, the semi-discrete systems of the left and right

domains are given by

τ

1 + τ(uML )t+

1

1 + τH(wM

L )t+τ

1 + τ(P−1

L QL⊗I)fM(wML )+

1

1 + τJw(P−1

L QL⊗I)wML

= (P−1L Em ⊗ Σw

L)(wMm,L −wM

1,R), (8.33)


and

τ

1 + τ(uMR )t+

1

1 + τJu(w

MR )t+

τ

1 + τ(P−1

R QR⊗I)fM(wMR )+

1

1 + τJw(P−1

R QR⊗I)wMR

= (P−1R E0 ⊗ Σw

R)(wM1,R −wM

m,L), (8.34)

respectively. We follow the procedure of section 8.3.3. Multiplying (8.33) from the

left by (1 + τ)(wML )T (PL ⊗ I) and using the homogeneity identity (8.23), we have

d

dt

∥∥wML

∥∥2

(PL⊗I)H+ (wM

L )T (QL ⊗ I)JwwML + (wM

L )T Jw(QL ⊗ I)wML

= (1 + τ)wMm,LΣw

L(wMm,L −wM

1,R) (8.35)

Adding the transpose of (8.35) to itself, neglecting the outer boundaries and perform-

ing similar operations on (8.34), we get

d

dt

(∥∥wML

∥∥2

(PL⊗I)H+∥∥wM

R

∥∥2

(PR⊗I)H

)+(wM

m,L)TJw(wMm,L)wM

m,L−(wM1,R)TJw(wM

1,R)wM1,R

= (1 + τ)(wMm,L)TΣw

L(wMm,L −wM

1,R) + (1 + τ)(wM1,R)TΣw

R(wM1,R −wM

m,L). (8.36)

Assuming symmetric ΣwL and Σw

R, we get the stability condition

[wMm,L

wM1,R

]T [−Jw(wM

m,L) + (1 + τ)ΣwL −1+τ

2(Σw

L + ΣwR)

−1+τ2

(ΣwL + Σw

R) Jw(wM1,R) + (1 + τ)Σw

R

][wMm,L

wM1,R

]≤ 0

(8.37)

Being in the smooth domain we assume Jw(wMm,L) = Jw(wM

1,R) = J and obtain stabil-

ity with

ΣwL =

1

1 + τJ − θ, Σw

R = − 1

1 + τJ − θ,

where θ is a positive semi-definite matrix. This is completely analogous to the penal-

ties derived in the constant advection problem presented in [23].

The penalties derived in the stability analysis apply to the entropy variables w

but in the numerical experiments we use a conservative formulation for correct shock

speed and employ the conservative variables u. Therefore we need to transform the


penalties to the conservative variables. Assuming that the solution is smooth and

H(wMm,L) = H(wM

1,R), we rewrite the interface terms

ΣwL(wM

m,L −wM1,R) =

1

1 + τJ(wM

m,L −wM1,R)− θ(wM

m,L −wM1,R)

=1

1 + τ

(fM(wM

m,L)− fM(wM1,R))− θτ

(H−1(wM

m,L)uMm,L − H−1(wM1,R)uM1,R

)

=1

1 + τ

(fM(uMm,L)− fM(uM1,R)

)− θ

(uMm,L − uM1,R

)=

(1

1 + τJu − θ

)(uMm,L − uM1,R

)

= ΣuL(uMm,L − uM1,R), (8.38)

where

Ju =∂fM

∂uM

∣∣∣∣x=xint

,

and

ΣuL =

1

1 + τJu − θ, (8.39)

and θ = τθH−1 is a positive semi-definite matrix for τ > 0 since H is positive definite

and θ is positive semi-definite. Similarly, we get the right penalty matrix

ΣuR = − 1

1 + τJu − θ. (8.40)

Conservation at the interface

In order to show conservation over the interface, we want to mimic the continuous

case where we multiply the conservative formulation by a smooth function φ, integrate

by parts to get

∫ xint

0

φutdx+

∫ 1

xint

φutdx =

∫ xint

0

φxf(u)dx+

∫ 1

xint

φxf(u)dx+B.T., (8.41)

where B.T. denotes outer boundary terms. In (8.41) no interface terms are present.

Consider the semi-discrete scheme

(uML )t + (P−1L QL ⊗ I)fM(uML ) = (P−1

L Em ⊗ ΣuL)(uMm,L − uM1,R) (8.42)


(uMR )t + (P−1R QR ⊗ I)fM(uMR ) = (P−1

R E1 ⊗ ΣuR)(uM1,R − uMm,L), (8.43)

Multiplying from the left by φTL(PL⊗ I) and φTR(PR⊗ I), respectively, where φL and

φL are discretized smooth functions satisfying φm,L = φ1,R = φI , we get

φTL(PL ⊗ I)(uML )t + φTR(PR ⊗ I)(uMR )t = (DLφL)T (PL ⊗ I)fM(uML )

+ (DRφR)T (PR ⊗ I)fM(uMR ) +B.T.

+ φTI[(uMm,L − uM1,R)(Σu

L − ΣuR)− fM(uMm,L) + fM(uM1,R)

](8.44)

The semi-discrete formulation (8.44) mimics the continuous expression (8.41) if we

choose ΣuL and Σu

R such that

(uMm,L − uM1,R)(ΣuL − Σu

R)− fM(uMm,L) + fM(uM1,R) = 0.

We assume Jw(wMm,L) = Jw(wM

1,R) = J . Then, the interface terms cancel with the

choice ΣuL −Σu

R = J , which is consistent with the condition for stability given by the

penalties (8.39) and (8.40) and τ = 1.


The exact solution of the test problem is known analytically for any given value of the

stochastic variable ξ. Thus, we can obtain the exact statistics to arbitrary accuracy

by averaging the exact Riemann solutions over a large number of realizations of ξ. In

the numerical experiments, we will assume ξ ∼ U [−0.05, 0.05], where U denotes the

uniform distribution. For the numerical solutions, we use SBP operators that can be

found in [63].


8.4.1 Convergence of smooth solutions

The method of manufactured solutions is used to impose a smooth time dependent

solution of the two-phase problem through a source term. We consider the manufac-

tured solution defined by

α = α0 + α1 tanh(s(x0 − x+ t+ ξ))

β = β0 + β1 tanh(−s(x0 − x+ t+ ξ))

v = tanh(s(v0 + x0 − x+ t+ ξ)) + tanh(−s(−v0 + x0 − x+ t+ ξ))

p = p0 + p1 tanh(s(x0 − x+ t+ ξ)),

with s = 15, v0 = 0.03, α0 = α1 = β0 = β1 = 0.5, p0 = 0.75, p1 = 0.25. We take

ρα = 1 and ρβ = 0.125. We measure the error in the L2(Ω,P) norm and the discrete

`2 norm,

∥∥uM − u∥∥

2,2≡∥∥uM − u

∥∥`2,L2(Ω,P)

=

(∆x

m∑

i=1

∥∥uM(xi, t, ξ)− u(xi, t, ξ)∥∥2

L2(Ω,P)

)1/2

=

(∆x

m∑

i=1

∫

Ω

(uM(xi, t, ξ)− u(xi, t, ξ))2dP(ξ)

)1/2

=

≈(

∆xm∑

i=1

q∑

j=1

(uM(xi, t, ξ(j)q )− u(xi, t, ξ

(j)q ))2w(j)

q

)1/2

, (8.45)

where a q-point quadrature rule with points ξ(j)q qj=1 and weigths w(j)

q qj=1 was used

in the last line to approximate the integral in ξ. The Gauss-Legendre quadrature is

used here since the solution is smooth in the stochastic dimension.

Figure 8.3 (a) shows the spatial convergence when the proportion of low-order

and high-order points remains constant. The low-order scheme dominates the error,

so the overall convergence rate is second order. In regions of fourth order operators,

the error levels are lower and therefore the local accuracy higher compared to the

regions of second order operators, see Figure 8.3 (b). This is further illustrated in

Figure 8.4 where a similar problem with sharp gradients in the middle of the domain


39 79 159 319

10−4

10−3

10−2

m

∥∥ρM − ρ∥∥2 ,2

∥∥vM − v∥∥2 ,2

∥∥EM − E∥∥2 ,2

ord. 2 dec .

(a) 2,2 norm of errors for smooth solution, t =0.05.

0.2 0.3 0.4 0.5 0.6 0.7 0.8−4

−3

−2

−1

0

1

2

3

4

5

6x 10

−4

x

4th

ord.

2nd

ord

(b) Error in mean density, t = 0.1.

Figure 8.3: SBP 4-2-4, fixed proportion of SBP 2 points. Np = 8, Nr = 0order of multiwavelets (Legendre polynomials).

is solved with a hybrid scheme where fourth order operators are used for the region

of large gradients and second order operators are used for the regions next to the

boundaries. With constant proportion of high order points under mesh refinement,

the convergence is second order. The comparison with the solution with second order

operators throughout the computational domain, also included in Figure 8.4, shows

that the error of the the hybrid scheme is smaller.

Figure 8.5 (a) shows the spatial convergence employing three computational do-

mains separated by two interfaces. The middle domain is solved with second order

SBP and the left and right domains with fourth order SBP. The number of points in

the second order region remains constant (20), as the high order domains are refined.

Figure 8.5 (b) depicts the spatial convergence with three domains, all solved with

fourth order SBP. The proportion of points in each region remains the same so the

interface locations do not change when the grids are refined.


50 100 200

10−4

10−3

10−2

m

∥∥ρM − ρ∥∥2 ,2

SBP 2-4-2

∥∥ρM − ρ∥∥2 ,2

SBP 2

∥∥vM − v∥∥2 ,2

SBP 2-4-2

∥∥vM − v∥∥2 ,2

SBP 2

Figure 8.4: Comparison of 2,2 norm of errors, three solution regions SBP2-4-2 versus single region solved with SBP 2, t = 0.1. The proportion offourth order points remains constant during mesh refinement. Np = 8,Nr = 0 order of multiwavelets (Legendre polynomials).

50 80 140 26010

−6

10−5

10−4

10−3

10−2

m

∥∥ρP − ρ∥∥2 ,2

∥∥vP − v∥∥2 ,2

∥∥EP − E∥∥2 ,2

ord. 2 dec .

ord. 3 dec .

(a) SBP4-SBP2-SBP4, fixed number of SBP2points.

30 60 120 240

10−7

10−6

10−5

10−4

10−3

10−2

m

∥∥ρP − ρ∥∥2 ,2

∥∥vP − v∥∥2 ,2

∥∥EP − E∥∥2 ,2

ord. 3 dec .

ord. 4 dec .

(b) Three SBP4 schemes coupled by two inter-faces.

Figure 8.5: Spatial convergence with three regions and two interfaces.t = 0.05. Np = 8, Nr = 0 order of multiwavelets (Legendre polynomials).


8.4.2 Non-smooth Riemann problem

With the hybrid scheme as depicted schematically in Figure 8.2, we solve the problems

(8.5)-(8.7) with the boundary conditions in (8.4) and assuming ξ ∼ U [−0.05, 0.05].

Figure 8.6 shows the variances of density, velocity, energy and pressure at t = 0.05.

The error from the interface is not significant compared to the error due to the

stochastic truncation and spatial resolution. A relatively fine mesh and high order

MW expansion is required to capture the variance of the solution. Especially high

order MW coefficients exhibit sharp spatial variation. Thus, to attain a given level

of accuracy, more spatial grid points are required for the stochastic Galerkin problem

compared to the deterministic problem.

Figure 8.7 depicts the convergence of pressure statistics with increasing order of

MW on a fixed spatial grid of 400 points. In the analysis of regularity in Section 8.2,

we anticipated the solution to develop a larger number of weaker discontinuities as

the order of MW expansion increases. This behavior can be observed in Figure 8.7.

All (visible) discontinuities are located in the right domain where the shock-capturing

method is used.

8.5 Conclusions

In order to efficiently solve fluid flow problems, a feasible strategy is to locally adapt

the numerical method to the smoothness of the solution whenever these properties are

known or can be estimated. A two-phase Riemann problem with uncertain initial dis-

continuity location has been investigated with respect to the smoothness properties of

the MW coefficients of the solution. Whereas the corresponding deterministic prob-

lem has a discontinuous solution profile, the stochastic modes of the gPC expansion

of the true solution are smooth.

A symmetrization and combination of conservative and non-conservative formu-

lation leads to a generalized energy estimate for the stochastic Galerkin system, just

as for the case of the deterministic Euler equations. Under certain smoothness as-

sumptions, stability at the interfaces can be obtained for the symmetrized system.


Var(ρ)

0.4 0.5 0.6 0.70

0.02

0.04

0.06Exact

MW

Var(v)

0.4 0.5 0.6 0.70

0.05

0.1

0.15Exact

MW

Var(E)

0.4 0.5 0.6 0.70

0.05

0.1

0.15

0.2

0.25Exact

MW

Var(p)

0.4 0.5 0.6 0.70

0.01

0.02

0.03

0.04

0.05Exact

MW

Figure 8.6: Variances at t = 0.05, m = 400, fourth order SBP (left re-gion) single interface (dashed blue line) and HLL-MUSCL (right region),(Np, Nr) = (0, 5) (Haar wavelets).


Mean(p)

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8N

r=2

Nr=3

Nr=4

Nr=5

Exact

(a) Mean pressure.

Mean(p)

0.52 0.54 0.56 0.58 0.6 0.62 0.64

0.05

0.1

0.15

0.2

0.25

Nr=2

Nr=3

Nr=4

Nr=5

Exact

(b) Mean pressure in the proximity of the de-terministic shock.

Var(p)

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05N

r=2

Nr=3

Nr=4

Nr=5

Exact

(c) Variance of pressure.

Var(p)

0.54 0.56 0.58 0.6 0.62 0.64

0

1

2

3

4

5

6

7

x 10−3

Nr=2

Nr=3

Nr=4

Nr=5

Exact

(d) Variance of pressure in the proximity of thedeterministic shock.

Figure 8.7: Convergence of the mean and variance of pressure with theorder of MW chaos, different orders of piecewise constant MW. t = 0.05,m = 400. Fourth order SBP (left domain) and HLL-MUSCL (right do-main).


The derived penalty matrices are transformed back to the conservative variable for-

mulation that is used in the numerical experiments.

The numerical results show that the convergence rate for the smooth problem

(smoothness enforced by the method of manufactured solutions) is second order when

fourth order and second order operators are combined and the proportion of second

order points remains constant during mesh refinement. However, the error is smaller

in this case compared to the case of a single domain solved with second order opera-

tors.

The two-phase non-smooth Riemann problem is reasonably well resolved with the

hybrid scheme combining high order SBP operators in the smooth regions with the

HLL solver and MUSCL reconstruction in the spatial region containing discontinu-

ities. A relatively large number of multiwavelets are needed to accurately represent

the stochastic solution. This in turn requires a fine spatial mesh for accurate resolu-

tion.

The framework presented here can be extended to time-dependent interfaces that

are adapted to the evolving regions of non-smooth solutions. A moving mesh based

on interfaces and SBP-operators has already been designed for deterministic problems

in [23] and this technique could be used for stochastic Galerkin systems. Depending

on the problem, different MW bases can be used in the different spatial regions for

efficient representation of the local uncertainty.

APPENDIX A. GENERATION OF MULTIWAVELETS 161

Appendix A

Generation of multiwavelets

Algorithm 1 Generation of multiwavelets (mother-wavelets (2.8))

Start with the set of functions f 1k

Npk=0, defined by

f 1k (ξ) =

ξk, ξ ∈ [−1, 0],

−ξk, ξ ∈ [0, 1],

0, otherwise.

STEP 1: Orthogonalize w.r.t. the monomials 1, ..., ξNp (Gram-Schmidt) to obtain

f 2k

Npk=0.

STEP 2:

for i← 0 to Np − 1 do

Make sure 〈f i+1i ξNp+i〉 6= 0 (otherwise reorder).

for j = i+ 1 to N0 do

w =〈f i+2j ξNp+i〉〈f i+2i ξNp+i〉

f i+3j ← f i+2

j − wf i+2i

end for

end for

STEP 3: Orthogonalize f i+2i

Npi=0 using G-S.

for i← Np to 0 do

ψWi (ξ)← Apply Gram-Schmidt to f i+2i .

end for

Output ψWi (ξ)Npi=0.

Appendix B

Proof of constant eigenvectors of

low-order MW triple product

matrices

B.1 Proof of constant eigenvectors of A

Proposition 9. The matrix A defined by (7.3) for Haar wavelets ψjPj=0 has constant

eigenvectors for all P + 1 = 2Nr , Nr ∈ N.

Sketch of proof. We will use induction on the order P of wavelet chaos to show that

the matrix A has constant eigenvectors for all orders P . In order to do this, we

will need certain features of the structure of A. To facilitate the notation, denote

P = P + 1. We can express A2P in terms of the matrix AP . Two properties of the

triple product 〈ψiψjψk〉 will be used to prove that A indeed has the matrix structure

presented.

Property 1: Let i ∈ 0, ..., P − 1, j = k ∈ P, ..., 2P − 1 and let j′ and j′′ be the

progenies of j. Then

〈ψiψ2j 〉 = 〈ψiψ2

j′〉 = 〈ψiψ2j′′〉.

162

APPENDIX B. EIGENVECTORS OF MW MATRICES 163

Property 2: Consider the indices i ∈ P, ..., 2P −1, j = k ∈ 2P, ..., 4P −1. Then

〈ψiψ2j 〉 =

P 1/2 if j first progeny of i

−P 1/2 if j second progeny of i

0 otherwise

.

As induction hypothesis, we assume that given AP for some P = 2Nr , Nr ∈ N,

the next order of triple product matrix A2P can be written

A2P =

[AP QPMP

MPQTP

Λ

]

where QP is the matrix of constant eigenvectors of AP satisfying ‖QP‖22 = P , MP =

diag(wP , ..., w2P−1) and Λ is diagonal and contains the eigenvalues of AP . Then, we

have that

[AP QPMP

MPQTP

Λ

][QP

±P 1/2I

]=

[QPΛ± P 1/2

QMP

PMP ± P 1/2Λ

]=

[QP

±P 1/2I

](Λ±P 1/2

M),

so the eigenvalues and eigenvectors ofA2P are given by Λ±P 1/2MP and [QP ,±P

1/2I]T ,

respectively. For the next order of expansion, 4P , we have

A4P =

[AP QPMP

MPQTP

Λ

] [QP ⊗ [1, 1]

P1/2I ⊗ [1,−1]

]M2P

M2P

[QP ⊗ [1, 1]

P1/2I ⊗ [1,−1]

]TΛ⊗ I2 + P

1/2MP ⊗

[1 0

0 −1

]

(B.1)

To see that this is indeed the structure of A4P , note that any non-zero matrix entry

not already present in A2P , can be deduced using properties 1 and 2, and scaling the

rows/columns by multiplication by the diagonal matrix M2P . The structure of A4P

follows from the construction of the Haar wavelet basis, but we do not give a proof

here.


One can verify that A4P given by (B.1) has the eigenvectors and eigenvalues

Q4P =

[QP ⊗ [1, 1]

P1/2IP ⊗ [1,−1]

]

±(2P )1/2I2P

,

Λ4P = Λ⊗ I2 + P1/2MP ⊗

[1 0

0 −1

]± (2P )1/2M2P ,

so the eigenvectors are constant (but the eigenvalues are variable in the coefficients

(wi)j through MP and M2P ). The base cases P = 1, P = 2, can easily be verified, so

by induction AP has constant eigenvectors for all P = 2Nr , Nr ∈ N.

B.2 Eigenvalue decompositions of A

B.2.1 Piecewise constant multiwavelets (Haar wavelets)

Nr = 2

Q =1

2

1 1 1 1

1 1 −1 −1√2 −

√2 0 0

0 0√

2 −√

2

, Λ = diag

u0 + u1 +√

2u2

u0 + u1 −√

2u2

u0 − u1 +√

2u3

u0 − u1 −√

2u3


Nr = 3

Q =

1 1 1 1 1 1 1 1

1 1 1 1 −1 −1 −1 −1

√2√

2 −√

2 −√

2 0 0 0 0

0 0 0 0√

2√

2 −√

2 −√

2

2 −2 0 0 0 0 0 0

0 0 2 −2 0 0 0 0

0 0 0 0 2 −2 0 0

0 0 0 0 0 0 2 −2

Λ = diag

u0 + u1 +√

2u2 + 2u4

u0 + u1 +√

2u2 − 2u4

u0 + u1 −√

2u2 + 2u5

u0 + u1 −√

2u2 − 2u5

u0 − u1 +√

2u3 + 2u6

u0 − u1 +√

2u3 − 2u6

u0 − u1 −√

2u3 + 2u7

u0 − u1 −√

2u3 − 2u7


B.2.2 Piecewise linear multiwavelets

Nr = 1

Q =

12

12

12

12

−√

3+14

√3−14

√3+14

−√

3−14

−12

12

−12

12

−√

3−14

−√

3+14

√3−14

−√

3+14

, Λ = diag

u0 −√

3+12u1 − u2 −

√3−12u3

u0 +√

3−12u1 + u2 −

√3+12u3

u0 +√

3+12u1 − u2 +

√3−12u3

u0 −√

3−12u1 + u2 −

√3+12u3

Nr = 2

Q =

1√8

1√8

1√8

1√8

1√8

1√8

1√8

1√8

√14+3

√3

8−√

14+3√

38

−√

14−3√

38

√14−3

√3

8

√3+1

8√

2−√

3+18√

2−√

3−18√

2

√3−1

8√

2

−√

3+14√

2−√

3+14√

2−√

3−14√

2−√

3−14√

2

√3−1

4√

2

√3−1

4√

2

√3+1

4√

2

√3+1

4√

2

√3+1

8√

2−√

3+18√

2

√3−1

8√

2−√

3−18√

2−√

14−5√

38

√14−5

√3

8

√14+5

√3

8−√

14+5√

38

0 − 12

12

0 0 12

− 12

0

0 −√

3−14

√3+14

0 0 −√

3+14

√3−14

0

− 12

0 0 12

12

0 0 − 12

√3−14

0 0 −√

3+14

√3+14

0 0 −√

3−14


Λ = diag

u0 +√

14+3√

38

u1 −√

3+12u2 +

√3+14u3 −

√2u6 +

√3−1√

2u7

u0 −√

14+3√

38

u1 −√

3+12u2 −

√3+14u3 −

√2u4 −

√3−1√

2u5

u0 −√

14−3√

38

u1 −√

3−12u2 +

√3−14u3 +

√2u4 +

√3+1√

2u5

u0 +√

14−3√

38

u1 −√

3−12u2 −

√3−14u3 +

√2u6 −

√3+1√

2u7

u0 +√

3+14u1 +

√3−12u2 −

√14−5

√3

8u3 +

√2u6 +

√3+1√

2u7

u0 −√

3+14u1 +

√3−12u2 +

√14−5

√3

8u3 +

√2u4 −

√3+1√

2u5

u0 −√

3−14u1 +

√3+12u2 +

√14+5

√3

8u3 −

√2u4 +

√3−1√

2u5

u0 +√

3−14u1 +

√3+12u2 −

√14+5

√3

8u3 −

√2u6 −

√3−1√

2u7

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