lax-wendroff convergence theorems for a class of overset...

Lax-Wendroff Convergence Theorems for a Class of Overset Mesh Methods Based on

Higher Order Flux Reconstruction

Dr. Robert Tramel Kord Technologies

13th Symposium on Overset Composite Grids and Solution Technology

October 23, 2016

Conservation Law Based Update Schemes • In this talk convergence to a weak solution under grid refinement is demonstrated for a class of overset grid schemes based on higher order Flux Reconstruction (FR) as well as MUSCL schemes • The method is based on enforcing the conservation laws on a cell-wise basis rather than the traditional approaches involving interpolation • Sources of conservation error are clearly identified • Consider the following conservation law

2

0)( =∂

∂+

∂∂

xqf

tq

2016 Overset Grid Syposium

Conservation Law Based Update Schemes

3

• In order to solve this equation numerically on a single grid, the domain is partitioned into discrete cells

• The variable q is averaged over cell j as follows

•The value of the cell averaged variable in cell Ω𝑗 is updated as follows

( ) ( )( ) , ,

111

jjjjnj

nj qrqlFqrqlFx

tqq −∆∆

−= +++

jq

∆+

∆−=

2,

2xxxxj jj

etc. ;t n ; ) ,(1 2

2

∆=∆

= ∫∆

+

∆−

nn

xx

xx

nj tdxtxqx

qj

j


MUSCL Reconstruction • Here ql, qr are the values of q reconstructed from the cell averages and evaluated at the cell interfaces using the MUSCL procedure and F is a numerical flux function (Roe, Lax–Friedrichs, etc.)

• Extension to higher order in multiple dimensions requires large

stencil sizes. This is impractical for overset grids. 4

njq

(Barth and Ohlberger, 2004)


Flux Reconstruction1,2

2016 Overset Grid Syposium 5

Discontinuous Solution

• The solution in each cell is approximated by K pieces of data • A K-1 order polynomial is used to fit the data. • and are defined by extrapolating the polynomials to the cell faces

1. Huynh, H. T. "A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods." AIAA paper 4079 (2007): 2007. 2. A Novel Approach for Shock Capturing in Unstructured High-Order Methods, by Abhishek Sheshadri, Stanford University

1+kql

1+kqr

1+kql 1+kqr

http://www.nas.nasa.gov/publications/ams/2016/07-07-16.html

Flux Reconstruction


Discontinuous Flux

• A K-1 order polynomial is used to approximate the flux function

Flux Reconstruction


( )11, ++ kk qrqlF

Discontinuous Flux

• and are used to define an upwind flux at the face 1+kql 1+kqr

Flux Reconstruction


Continuous Flux

• Correction polynomials are used to create a continuous Kth order flux approximation F

𝑑𝑞𝑗,𝑘𝑑𝑑 = 𝐹𝑥 𝑗,𝑘

• The continuous flux polynomial is used to time advance the variables

Conservation Law Based Update Schemes • What is the “best” way to extend these approaches to cases of multiple overlapping grids?

•Rather than interpolate variables or fluxes seek alternate formulation

•Use data from both grids to reconstruct interface states •Consider two grids with some small amount of overlap

1 2 3 4

np-3 np-2 np-1 np

9

OVERLAP REGION GRID 1 GRID 2 2016 Overset Grid Syposium

Conservation Law Based Update Schemes •Let the value of the cell averaged variable in cell j of grid 1 be denoted as , and the cell centroid value of cell j be denoted as etc

•Let the left and right interface states at face k of grid 1 be denoted as and respectively •Let the face centroid of face k of grid 1 be denoted as respectively

q j1

1kql

1kqr

1kql

1kxf

1kqr

1kxf

1jxc

1jxc

10 2016 Overset Grid Syposium

Conservation Law Based Update Schemes • Consider computing the cell interface states and as follows

• The states are computed by constructing a least squares monotonically limited approximation to the solution gradient in cell np from mesh 1 using the cells below and then extrapolating q to the cell faces using this gradient

np 1 np-1 ( )( )12 1112 1

111

1111

npnpnp

npnpnpnpnp

xcxfqqqlxcxfqqql

−•∇+=−•∇+= ++

11+npql

21ql

11+npql

21ql


np 1 2

Conservation Law Based Update Schemes • Consider computing the cell interface states and as follows

• The states are computed by constructing a least squares monotonically limited approximation to the solution gradient in cell 1 from mesh 2 using the cells below and then extrapolating q to the cell faces using this gradient

( )( )212 121212 1

21

11

21

21

11

xcxfqqqrxcxfqqqr npnp

−•∇+=−•∇+= ++

11+npqr

21qr

11+npqr

21qr


Data Exchange for Flux Reconstruction


1. Galbraith, M. C., A Discontinuous Galerkin Overset Solver, Ph.D. thesis, University of Cincinnati, 2013.

2. Crabill, J., Jameson, A. and Sitaraman, J.: A High-Order Overset Method on Moving and Deforming Grids. AIAA 2016-3225, AIAA Aviation,

AIAA Modeling and Simulation Technologies Conference, 13-17 June 2016, Washington, DC.

11+npqr

21ql

1. Extrapolate from boundary cell solution polynomial 2. Interpolate from donor cell solution polynomial 3. Compute upwind flux

21qr

11+npql

11+npql

11+npqr

( )1 11 1, ++ npnp qrqlF

•A consistent and convergent scheme should be achieved as grid refinement is performed if the distance between the cells centers used in the reconstructions approach zero under grid refinement, i.e.

•Here TV is the total variation

•A similar argument holds for meshes with small gaps between the grids

•The basic argument remains unchanged in the case of general overlap

Conservation Law Based Update Schemes

( ) ( )etc.

)(

thatin0 as and

21

11

21

21

11

21

2 1

1 1

2 1

1 1

2 1

1 1

xqTVxfxfqxfxfqqrqr

xqrqrqlql

npnpnp

npnp

∆≤−∇≤−•∇≅−

→∆→→

+++

++


Generalized Lax-Wendroff Theorem

• Recall that a function q(x,t) is considered to be a weak solution to the conservation law

If for any compact, differentiable function φ(x,t) the following integrals are satisfied1

0)( =∂

∂+

∂∂

xqf

tq

( )[ ] ( ) ( )dxxqxdtdxqfq xt ∫∫∫∞

∞−

∞

∞−

∞

−=+ 0,0, 0

φφφ


1. LeVeque, Randall J., and Randall J. Leveque. Numerical methods for conservation laws. Vol. 132. Basel: Birkhäuser, 1992.

( ) ( ) ( )( )

, , ),( -),( 1 1 1 1

1 1

1,11,11jjjjnj

nj

njnj qrqlFqrqlFtxx

tqqtx −∆∆

−= +++ φφ

{ ( ) ( )

( ) ( )( )

( ) ( )( ) }

, , ),(

, , ),(

-

),( -

),(

2 2 2 1

2 12

01

1 1 1 1

1 11

0

,21,22

01

,11,11

0

xqrqlFqrqlF

tx

xqrqlFqrqlF

tx

tqq

txtqq

txxt

jjjjnj

nj

jjjjnj

n

np

j

nj

nj

njnj

nj

nj

njn

np

j

∆

−−

∆

−−

=∆

+∆

∆∆

++∞

=

∞

=

++∞

=−∞=

+∞

=

∞

=

+∞

=−∞=

∑∑

∑∑

∑∑∑∑

φ

φ

φφ


• In order to demonstrate that these integrals are satisfied for the proposed method, first multiply the cell average update equation by for mesh 1, etc.

• Next sum over j and n for each mesh as follows

( )nj tx ,1φ



18

• The following summation by parts formula is useful for the proving convergence of the proposed scheme to a weak solution

( ) ( ) SPB 1

10010

1 ∑∑=

−+=

+ −−−=−N

jjjjNN

N

jjjj aabbababba



19

• For the LHS use the SPB formula on the n sum

),(-),(

),(-),(

),( - ),(

1

12 2

,211 1

,1

1

1

0,2 0

2 0,1 0

1

∆∆+

∆∆∆−

∆∆−

∑∑∑

∑∑∞

=

−−

−∞=

∞

=

∞

=−∞=

j

njnjnj

njnjnp

j

nj

n

jjj

np

jjj

ttxtx

qxt

txtxqxt

qtxxqtxx

φφφφ

φφ

• Note that these terms are discrete approximations to the following integrals

dxdtqqdxxqxdxxqx tt

+−−− ∫ ∫∫∫ ∫

++

∞−

∞∞

∞−

∞1

1np

21

11np

21

xf

xf0

xf

xf

)0,()0,()0,()0,( φφφφ


( ) ( )

( ) ( ) ),(, ),(,

),(),(,

),(),(, t

0

212

12

1

11

11

1

2

21

22 2

11

11 1

0

∑

∑ ∑∑∞

=++

−∞=

∞

=

−−∞

=

∆−

∆∆∆−

∆

−∆+

∆

−∆∆

n

nnnpnpnp

np

j j

njnjjj

njnjjj

n

xtxqrqlF

xtx

qrqlFxt

xtxtx

qrqlFxx

txtxqrqlFx

φφ

φφφφ


20

• For the RHS use the SPB formula on the j sum

End Terms from SBP Formula



21

• The end terms at the grid interface become

( ) ( )

( )

( ) ( )1 11 12 12 112

0

112

1212

12

1

0

212

12

1

11

11

1

, , where

),(),(),( ,

),( , ),(

,

++

∞

=

∞

=++

−=∆

∆

∆+

∆

−∆∆

=

∆−

∆∆∆−

∑

∑

npnp

n

nnpnnpn

n

nnnpnpnp

qrqlFqrqlFF

xtxF

xtxtx

qrqlFxt

xtxqrqlF

xtx

qrqlFxt

φφφ

φφ



22

• We can show that ΔF12 → 0 as Δx → 0 as follows

),()( ),( 1112 nnpn

nnp txxqTVCtxF φφ ∆≤∆

0 as 0 ⇒∆⇒ x

( ) ( ) etc. ,)( 211 121211 1212 1 1 1 xqTVxfxfqxfxfqqrqr npnpnp ∆≤−∇≤−•∇≅− +++

( ) ( )( ) ( )

) , max(

)( ),( ,

, ,

2 1

1 1

2 1

1 112

2 1

1 1

2 1

2 1

1 1

2 1

2 1

2 112

1 1

1 1

2 1

2 112

qrqrqlqlCF

qrqrqrqlqlqlFqrqlFF

qrqlFqrqlFF

npnp

npnp

npnp

−−≤∆

−+−+−=∆

−=∆

++

++

++

• Given that

it follows that

F is Lipschitz Continuous



23

• Note that these terms are discrete approximations to the following integrals

dxdtqfdxdtqf xxf

xf

x

np

φφ ∫∫∫∫∞∞

∞−

∞

++

21

11

)()(00

• If then terms like 0 as 0211 1 →∆→−+ xxfxfnp

dxdtqqdxxqxdxxqx tt

+−−− ∫ ∫∫∫ ∫

++

∞−

∞∞

∞−

∞1

1np

21

11np

21

xf

xf0

xf

xf

)0,()0,()0,()0,( φφφφ

dxdtqdxxqx t∫∫∫∞

∞−

∞∞

∞−

−− φφ0

)0,()0,( become

and the integral conservation law is satisfied under the same caveats invoked for the single grid case


Observations Based on Lax-Wendroff Theorem (1)

24

• If standard linear interpolation methods are used then no GENERAL bounds may be established a priori for the ΔF12 term

• We end up with a source term in the integral conservation law which MAY not vanish under grid refinement in the presence of discontinuous solutions.

• These source terms become ODEs along characteristics and may globally pollute the solution Corrupt convergence rates Generate spurious waves

( ) ( )dttxtF R ,10 12 ϕ∫∞∆≈


Observations Based on Lax-Wendroff Theorem (2)

25

•These terms can and will corrupt solutions when: • Severe mismatches in cell sizes exist • Strong discontinuities exist in the grid interface region • Slow moving discontinuities exist in the grid interface region • All three exist simultaneously!


•Shock capturing schemes needed to control TV(q)

• Follow Zang and Shu.1

• Let 𝑞 𝐾𝑁 be the solution points in cell Ω𝑁 •Use a discontinuity detector θ based on cell average values to control oscillations as follows: • Here we use Nichols discontinuity dector2 • One can also use the TVB filters of Engquist, et al3

FR Shock Capturing Considerations


𝑞 𝐾𝑁 → 𝜃 𝑞 𝐾𝑁 − 𝑞� + 𝑞�

1. Zhang, Xiangxiong, and Chi-Wang Shu. "On maximum-principle-satisfying high order schemes for scalar conservation laws." Journal of Computational Physics 229.9 (2010): 3091-3120.

2. Tramel, Robert W., Robert H. Nichols, and Pieter G. Buning. "Addition of improved shock-capturing schemes to OVERFLOW 2.1." AIAA Paper 3988 (2009): 2009. 3. Engquist, Björn, Per Lötstedt, and Björn Sjögreen. "Nonlinear filters for efficient shock computation." Mathematics of Computation 52.186 (1989): 509-537.

𝑞 𝐾𝑁 → 𝜃 𝑞 𝐾𝑁 − 𝑞� 𝐾𝑁 − 𝑞� 𝐾𝑁

Sample Cases


• Shock Wave Formation in Burgers Equation • Shock Propagation in Burgers Equation • K=5 • Cells overlap 1/8 ΔX • Shock location lies in the overset region • Use global Lax-Friedrichs flux • Use Gauss points and weights and Radau

correction polynomials • “Exact” solution for shock formation is a 4000 point

solution using a 5th order WENO-RBF method

𝑞 𝑥, 0 = sin (𝑥)

𝑞 𝑥, 0 = H (− 𝑥 − 𝑥𝑜 )

No Shock Treatment


Internal Cell Variables Cell Averages

No Shock Treatment


Comparison With Exact Solution

Dissipation Switch



Discontinuity Detector


Cell Number

θ

TVB Filter



No Shock Treatment



Dissipation Switch



TVB Filter



Discontinuity Detector


Cell Number

θ

Conclusions

39

• Within the context of this framework strict lack of conservation becomes merely another source of numerical error to be removed by grid refinement

• Convergence to a weak solution is guaranteed as grid independence is achieved

• MUSCL/WENO/RBF schemes can be mixed with Flux Reconstruction schemes

• Framework seamlessly blends block matching/overset grid grids


ACKNOWLEDGEMENTS

40

• Thanks to • Dr. Randy Leveque (U Wash) • Dr. John Benek (AFRL/RBAC)

for discussions and careful reading of earlier versions of this presentation


That’s All Folks

41

• Questions • Constructive Criticism


That’s All Folks

42

• Questions • Constructive Criticism


Lax-Wendroff Convergence Theorems for a Class of Overset Mesh Methods Based on Higher Order Flux ReconstructionConservation Law Based Update SchemesConservation Law Based Update SchemesMUSCL ReconstructionFlux Reconstruction1,2Flux ReconstructionFlux ReconstructionFlux ReconstructionConservation Law Based Update SchemesConservation Law Based Update SchemesConservation Law Based Update SchemesConservation Law Based Update SchemesData Exchange for Flux ReconstructionData Exchange for Flux ReconstructionConservation Law Based Update SchemesGeneralized Lax-Wendroff TheoremGeneralized Lax-Wendroff TheoremGeneralized Lax-Wendroff TheoremGeneralized Lax-Wendroff TheoremGeneralized Lax-Wendroff TheoremGeneralized Lax-Wendroff TheoremGeneralized Lax-Wendroff TheoremGeneralized Lax-Wendroff TheoremObservations Based on Lax-Wendroff Theorem (1)Observations Based on Lax-Wendroff Theorem (2)FR Shock Capturing ConsiderationsSample CasesNo Shock TreatmentNo Shock TreatmentDissipation SwitchDissipation SwitchDiscontinuity DetectorTVB FilterTVB FilterNo Shock TreatmentDissipation SwitchTVB FilterDiscontinuity DetectorConclusionsACKNOWLEDGEMENTSThat’s All FolksThat’s All Folks

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