Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Lagrange-remap schemes in conservative form F. De Vuyst(1) C. Fochesato(2) R. Loubère(3) P. Rouzier(2) L. Saas(2) R. Motte(2) J.M. Ghidaglia(1) (1) CMLA CNRS UMR 8536 ENS Cachan, France (2) CEA, DAM, DIF, F-91297 Arpajon, France, (3) IMT, CNRS UMR 5219, Univ. Toulouse, France

ECCOMAS 2012, Vienna, Mini-Symposium on Numerical Methods for Multimaterial Compressible Flows, September 11, 2012 (1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Outline 1

Objective

2

Basics of Lagrange-remap schemes

3

Comparison: Lagrange-remap vs pure eulerian schemes

4

Conservative reformulation (1D staggered case)

5

Future works

6

Concluding remarks

7

References

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Lagrange-remap (LR) schemes in conservation form: why ?

1

Prevent from some misunderstandings

2

Benefits for numerical analysis

3

Benefits for reliable code implementation ?

4

Benefits for parallel implementation ?

5

Get ideas for improvements or new derivations

6

Reformulate geometry distorsion by fluxes only

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Lagrange-remap (LR) schemes – Construction Define a piecewise C 1 Lagrange transformation operator L (x , t;t0 , x 0 ) : d x/dt = u(x), x(t = 0) = x 0 . Use of Reynolds’s transport theorem: d dt

Z Ωt

q(x , t)d x =

Z Ωt

{∂t q + ∇ · (qu)} d x

for any moving domain Ωt , quantity q. Finite volume approximation Move the mesh according to a Lagrange operator L over [t n , t n+1 ) Project the quantities onto the initial mesh. (1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Lagrange-remap: true differences with pure eulerian schemes

(1st order) LR schemes are actually 5-point schemes (large stencil) 2D/3D case: genuinely multidimensional scheme (corner effects) Different treatment between contact and nonlinear waves (Lagrange step) Specific treatment of multimaterial interface made possible.

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Conservative reformulation (staggered variables, 1D case)

Geometric conservation law (GCL): n+1/2,L

n+1/2,L

hjn+1,L = h + ∆t (uj +1/2 − uj −1/2 ). Mass conservation law: ρ jn+1,L hjn+1,L = hρ nj . (1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Mass conservation equation i.e. ρ nj +1,L =

ρ nj

1+

n+1/2,L ∆t h (∆u)j

,

n+1/2,L

(∆u)j

n+1/2,L

n+1/2,L

:= uj +1/2 − uj −1/2 .

Projection step: ρ nj +1 =

1 h

Z Ij

Iρ

n+1,L

1 (x)dx = h

Z Ijn+1,L

... − ... + ... .

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Mass equation (cont.) Mass balance rewriting : under some convenient CFL condition, we have n+1/2,L

n+1/2,L

,n+1 ,n+1 uj −1/2 uj +1/2 + ∆t ρ upw hρ nj +1 = hjn+1,L ρ nj +1,L − ∆t ρ jupw j +1/2 +1/2

upw ,n+1 n+1/2,L ,n+1 n+1/2,L uj +1/2 + ∆t ρ jupw uj −1/2 +1/2

= hρ nj − ∆t ρ j +1/2

in the form ρ nj +1 = ρ nj −

∆t ³

h

n+1/2,n+1

Φm,j +1/2

n+1/2,n+1

Φm,j +1/2

n+1/2,n+1

− Φm,j −1/2

´

,

upw ,n+1 n+1/2,L uj +1/2 .

= ρ j +1/2

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Flux accuracy

n+1/2,n+1

Let δx =

∆t

2

Φm,j +1/2

upw ,n+1 n+1/2,L uj +1/2 .

= ρ j +1/2

u(x , t + ∆t/2).

∆t ρ (x + δx , t + ∆t)u(x + δx , t + )= 2 µ ¶ ∆t = ρ (x + δx , t + ∆t/2) + ∂t ρ + o(∆t) u(x + δx , t + ∆t/2)

2

= (ρ u)(x , t + ∆t/2) + δx ∂x (ρ u)(x , t + ∆t/2)) + = (ρ u)(x , t + ∆t/2) +

∆t

2

(u ∂t ρ )(x , t + ∆t/2) + o(∆t)

∆t

u (∂t ρ + ∂x (ρ u))(x , t + ∆t/2) + o(∆t) 2 = (ρ u)(x , t + ∆t/2) + o(∆t).

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Full result : mass conservation

ρ nj +1 = ρ nj −

∆t ³

h

n,n+1 n,n+1 Φm − Φm ,j +1/2 ,j −1/2

where

´

n+1/2,L

n,n+1 1,L Φm = ρ nj ++1/2 uj +1/2 ,j +1/2

and 1, L ρ nj ++1/2 =

1

Z

1,L δI n+1,L δVjn++1/2 j +1/2

I (ρ n+1,L )(x)dx .

(1) (2)

(3)

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Momentum conservation

ρ nj +1 + ρ nj ++11

2

1 ujn++1/2 =

ρ nj + ρ nj+1

2

ujn+1/2 −

∆t ³

h

,n+1 ,n+1 Φnmu − Φnmu ,j +1 ,j

´

(4)

with n+1/2,L

,n+1 Φnmu = u nj +1,L ,j

n+1/2,L

1, L 1, L (ρ nj −+1/2 uj −1/2 + ρ nj ++1/2 uj +1/2 )

2

n+1/2,L

+ (p + q)j

.

(5)

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Energy conservation

e n+1 = ρ n E en ρ nj +1 E j j − j

∆t ³

h

n,n+1 n,n+1 ΦmE − ΦmE ,j +1/2 ,j −1/2

´

(6)

with n+1/2,L

,n+1 1, L ΦnmE = e nj ++1/2 uj +1/2 ,j +1/2

n+1/2,L

n+1/2,L

n+1,L n+1,L µ 1 1 2 n+1,L (ρ j −1/2 uj −1/2 + ρ j +1/2 uj +1/2 ) (u )j + 2 2 2 n+1/2,L

n+1/2,L

n+1,L n+1,L ¶ 1 2 n+1,L (ρ j +1/2 uj +1/2 + ρ j +3/2 uj +3/2 ) n+1/2,L + (u )j +1 + Φpu ,j +1/2(7) 2 2

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Related works

B. Desprès, F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, Journal of Scientific Computing, 16 (2002), pp. 479-524. S. Del Pino, H. Jourdren, Arbitrary high-order schemes for the linear advection and wave equations: application to hydrodynamics and aeroacoustics, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 441-446. R. Loubère, J.-Ph. Braeunig, J.-M. Ghidaglia, A totally Eulerian finite volume solver for multi-material fluid flows: Enhanced Natural Interface Positioning (ENIP), European Journal of Mechanics B/Fluids, 31, 1-11 (2012).

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Future works

The multimaterial case The multidimensional case Generalizing the concept: toward Lagrange-Flux Conservative form implementation on GPU schemes

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

Concluding remarks

Lagrange-remap schemes may be written in conservative form (both centered or staggered version) Therefore LR may be implemented in a standard FV code Safe implementation for conservation Lagrange step allows for a different treatment of linear vs NL waves (multimaterial) Advantageous for GPU implementation

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

(1) C

Objective Basics of Lagrange-remap schemes Comparison: Lagrange-remap vs pure eulerian schemes Conservative reform

References

F. De Vuyst, C. Fochesato, R. Loubère, P. Rouzier, L. Saas, R. Motte and J.-M. Ghidaglia, Lagrange-remap schemes in conservation form, Comptes Rendus Mécanique, submitted (2012). Thank you for your attention

(1) [2ex] F. De Vuyst(1) , C. Fochesato(2) , R. Loubère(3) , P. Rouzier(2)Lagrange-remap , L. Saas(2) , R. schemes Motte(2)in , J.M. conservative Ghidagliaform

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