Admissibility and asymptotic-preserving scheme

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1 Admissibility and asymptotic-preserving scheme F. Blachère 1, R. Turpault 2 1 Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes, 2 Institut de Mathématiques de Bordeaux (IMB), Bordeaux-INP GdR EGRIN, 04/06/2015, Piriac-sur-Mer

2 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28 Outline 1 General context and examples 2 Development of a new asymptotic preserving FV scheme 3 Conclusion and perspectives

3 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28 Outline 1 General context and examples 2 Development of a new asymptotic preserving FV scheme 3 Conclusion and perspectives

4 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 2/28 Problematic Hyperbolic systems of conservation laws with source terms: t W+div(F(W)) = γ(w)(r(w) W) (1) A: set of admissible states, W A R N, F: physical flux, γ > 0: controls the stiffness, R : A A: smooth function with some compatibility conditions (cf. [Berthon, LeFloch, and Turpault, 2013]).

5 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 3/28 Problematic Hyperbolic systems of conservation laws with source terms: t W+div(F(W)) = γ(w)(r(w) W) (1) Under compatibility conditions on R, when γt, (1) degenerates into a diffusion equation: t w div ( D(w) w ) = 0 (2) w R, linked to W, D: positive and definite matrix, or positive function.

6 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 4/28 Examples: isentropic Euler with friction t ρ+ x ρu + y ρv = 0 t ρu + x (ρu 2 + p(ρ))+ y ρuv = κρu t ρv + x ρuv + y (ρv 2 + p(ρ)) = κρv Formalism of (1) A = {(ρ,ρu,ρv) T R 3 /ρ > 0}, with: p (ρ) > 0, κ > 0 ( W = (ρ,ρu,ρv) T ρu, ρu F(W) = 2 + p,ρuv R(W) = (ρ, 0, 0) T ρv, ρuv,ρv 2 + p γ(w) = κ Limit diffusion equation t ρ div ( ) 1 κ p(ρ) = 0 ) T

7 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 5/28 Examples: shallow water equations (linear friction) t h+ x hu + y hv = 0 t hu + x (hu 2 + gh 2 /2)+ y huv = κ h hu η t hv + x huv + y (hv 2 + gh 2 /2) = κ h hv η Formalism of (1) A = {(h, hu, hv) T R 3 /h > 0}, with: κ > 0 ( W = (h, hu, hv) T hu, hu F(W) = 2 + p, huv hv, huv, hv 2 + p R(W) = (h, 0, 0) T γ(w) = κ h η Limit diffusion equation ( ) h η t h div κ gh2 = 0 2 ) T

8 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 6/28 Examples: shallow water equations (Manning friction) Formalism of (1) t h+ x hu + y hv = 0 t hu + x (hu 2 + gh 2 /2)+ y huv = κ h hu hu η t hv + x huv + y (hv 2 + gh 2 /2) = κ h hu hv η A = {(h, hu, hv) T R 3 /h > 0} ( W = (h, hu, hv) T hu, hu F(W) = 2 + p, huv hv, huv, hv 2 + p R(W) = (h, 0, 0) T γ(w) = κ h hu η ) T Limit equation t h div ( gh η κ ) h h = 0 h

9 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 7/28 Aim of an AP scheme Model: t W+div(F(W)) = γ(w)(r(w) W) γt Diffusion equation: t w div ( D(w) w ) = 0 consistent: t, x 0 consistent? Numerical scheme γt Limit scheme

10 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 8/28 Example of a non AP scheme in 1D t W+ x (F(W)) = γ(w)(r(w) W) (1) W = (ρ,ρu) T F(W) = (ρu,ρu 2 + p) T γ(w) = κ R(W) = (ρ, 0) T x i 1/2 x i+1/2 x i 1 x i x i+1 W n+1 i Wi n t = 1 ( ) Fi+1/2 F x i 1/2 +γ(w n i )(R(Wi n ) Wi n ) Limit ρ n+1 i t ρ n i = 1 2 x 2 ( bi+1/2 x(ρ n i+1 ρ n i ) b i 1/2 x(ρ n i ρ n i 1) )

11 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 9/28 State-of-the-art for AP schemes in 1D 1 control of numerical diffusion: telegraph equations: [Gosse and Toscani, 2002], M1 model: [Buet and Després, 2006], [Buet and Cordier, 2007], [Berthon, Charrier, and Dubroca, 2007],... Euler with gravity and friction: [Chalons, Coquel, Godlewski, Raviart, and Seguin, 2010], 2 ideas of hydrostatic reconstruction used in well-balanced scheme used to have AP properties: Euler with friction: [Bouchut, Ounaissa, and Perthame, 2007], 3 using convergence speed and finite differences: [Aregba-Driollet, Briani, and Natalini, 2012], 4 generalization of Gosse and Toscani: [Berthon and Turpault, 2011], [Berthon, LeFloch, and Turpault, 2013].

12 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 10/28 State-of-the-art for AP schemes in 2D Cartesian and admissible meshes = 1D unstructured meshes: 1 MPFA based scheme: [Buet, Després, and Franck, 2012], 2 using the diamond scheme (Coudière, Vila, and Villedieu) for the limit scheme: [Berthon, Moebs, Sarazin-Desbois, and Turpault, 2015], 3 SW with Manning-type friction: [Duran, Marche, Turpault, and Berthon, 2015].

13 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28 Outline 1 General context and examples 2 Development of a new asymptotic preserving FV scheme 3 Conclusion and perspectives

14 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 11/28 Notations x L2 n K,i1 x L1 M is an unstructured mesh constituted of polygonal cells K, x K is the center of the cell K, x L3 i 2 x K i 1 i 5 x L5 L is the neighbour of K by the interface i, i = K L, E K are the interfaces of K, i 3 i 4 e i is length of the interface i, K is the area of the cell K, x J2 x L4 n K,i is the normal vector to the interface i.

15 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 12/28 Aim of the development for any 2D unstructured meshes, for any system of conservation laws which could be written as (1), under a hyperbolic CFL: stability, preservation of A, preservation of the asymptotic behaviour, max K M i E K ( b K,i t x ) 1 2.

16 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28 Outline 1 General context and examples 2 Development of a new asymptotic preserving FV scheme Choice of a limit scheme Hyperbolic part Numerical results for the hyperbolic part Scheme for the complete system Results for the complete system 3 Conclusion and perspectives

17 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 13/28 Aim of an AP scheme Model: t W+div(F(W)) = γ(w)(r(w) W) γt Diffusion equation: t w div ( D(w) w ) = 0 consistent: t, x 0 consistent? Numerical scheme γt Limit scheme

18 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 14/28 Choice of the limit scheme FV scheme to discretize diffusion equations: t w div(d(w) w) = 0 (2) Choice: scheme developed by Droniou and Le Potier conservative and consistent, preserves A, nonlinear. (D i w K ) n K,i = ν J K,i (w)(w J w K ) J S K,i S K,i the set of points used for the reconstruction on edges i of cell K ν J K,i (w) 0

19 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 15/28 Presentation of the DLP scheme x A M K,i x L nk,i nl,i x K M L,i i x B = J S K,i ωk,i J X J M L,i = w MK,i = J S K,i ωk,i J w J J S L,i ωl,i J X J w ML,i = J S L,i ωl,i J w J M K,i

20 Presentation of the DLP scheme Two approximations i w K n K,i i w L n L,i = w M K,i w K KM K,i = w M L,i w L LM L,i Convex combination: γ K,i +γ L,i = 1, γ K,i 0, γ L,i 0 i w K n K,i = γ K,i (w) i w K n K,i +γ L,i (w) i u L n L,i = J S K,i ν J K,i (w)(w J w K ), with : ν J K,i (w) 0 Properties of the DLP scheme consistent with the diffusion equation on any mesh, satisfy the maximum principle. F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 16/28

21 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28 Outline 1 General context and examples 2 Development of a new asymptotic preserving FV scheme Choice of a limit scheme Hyperbolic part Numerical results for the hyperbolic part Scheme for the complete system Results for the complete system 3 Conclusion and perspectives

22 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 17/28 Aim of an AP scheme Model: t W+div(F(W)) = γ(w)(r(w) W) γt Diffusion equation: t w div ( D(w) w ) = 0 consistent: t, x 0 consistent? Numerical scheme γt Limit scheme

23 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 18/28 Scheme for the hyperbolic part Theorem W n+1 K = W n K t K i E K F i (W K,W L,...) n K,i (3) We assume that the conservative flux F i has the following properties: 1 Consistency: if W n K W then F i n K,i = F(W) n K,i, 2 Admissibility: a νk,i J 0, F i n K,i = νk,i J F KJ η KJ J S K,i b e i νk,i J η KJ = 0. i E K J S K,i Then the scheme (3) is stable, and preserves A under the classical following CFL condition: ( ) t b KJ 1. (4) δj K max K M J E K

24 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 19/28 Example of fluxes 1 TP flux: 2 HLL-DLP flux: F i (W K,W L ) n K,i = F(W K)+F(W L ) 2 F i (W K,W L,W J ) n K,i = J S K,i ν J K,i ( F(WK )+F(W J ) 2 n K,i b KL (W L W K ) ) η KJ b KJ (W J W K ) But... 1 Fully respect the theorem, but not consistent in the diffusion limit 2 Does not respect the second property of admissibility, but consistent in the limit

25 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 20/28 Procedure to preserve A W n HLL-DLP flux W n+1 W n+1 PAD Yes TP flux No 1 W n+1 is computed with the HLL-DLP flux and with the CFL (4), 2 Physical Admissiblility Detection (PAD): if W n+1 A then the time iterations can continue, else: property 2b is enforced by using the TP flux on all not-admissible cells, t and W n+1 are re-computed with the TP flux.

26 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28 Outline 1 General context and examples 2 Development of a new asymptotic preserving FV scheme Choice of a limit scheme Hyperbolic part Numerical results for the hyperbolic part Scheme for the complete system Results for the complete system 3 Conclusion and perspectives

27 Wind tunnel with step F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 21/28

28 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28 Outline 1 General context and examples 2 Development of a new asymptotic preserving FV scheme Choice of a limit scheme Hyperbolic part Numerical results for the hyperbolic part Scheme for the complete system Results for the complete system 3 Conclusion and perspectives

29 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 22/28 Scheme for the complete model Complete system t W+div(F(W)) = γ(w)(r(w) W) (1) W n+1 K = W n K t K i E K e i F K,i n K,i, (5) Construction of F K,i F K,i n K,i = J S K,i ν J K,i F KJ η KJ F KJ η KJ =α KJ F KJ η KJ (α KJ α KK )F(W n K ) η KJ (1 α KJ )b KJ (R(W n K ) Wn K ),

30 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 23/28 Scheme for the complete model Is the scheme with the source term AP? generally not... Equivalent formulation Rewrite (1) into: t W+div(F(W)) = γ(w)(r(w) W), (1) = γ(w)(r(w) W)+(γ γ)w, t W+div(F(W)) = (γ(w)+γ)( R(W) W). (6) with: γ(w)+γ > 0 R(W) = γr(w)+γw γ+γ

31 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 24/28 Limit scheme Introduction of γ t W+div(F(W)) = (γ(w)+γ(w))( R(W) W) Rescaling γ γ ε t t ε h n+1 K = h n K + i E K t K e i = h n+1 K J S K,i ν J,h K,i = h n K + i E K t K e i b 2 KJ 2(γ K + γ J K,i )δh KJ ν J h η K,i J S K,i ( hj h K ), κ K (gh 2 J /2 gh2 K /2).

32 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 25/28 Theorem W n+1 K = WK n t e i F K,i n K,i (5) K i E K = ( WK n t ) δj K [ F KJ F KK ] η KJ J E K ω KJ The scheme (5) is consistent with the system of conservation laws, under the same assumptions of the previous theorem. Moreover, it preserves the set of admissible states A under the CFL condition: max K M J E K ( b KJ t δ K J ) 1. (4)

33 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28 Outline 1 General context and examples 2 Development of a new asymptotic preserving FV scheme Choice of a limit scheme Hyperbolic part Numerical results for the hyperbolic part Scheme for the complete system Results for the complete system 3 Conclusion and perspectives

34 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 26/28 Comparison in the diffusion limit D Density DLP TP HLL-DLP-NoAP HLL-DLP-AP Position

35 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 27/28 Dam break 6 Without friction Height With friction Position

36 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28 Outline 1 General context and examples 2 Development of a new asymptotic preserving FV scheme 3 Conclusion and perspectives

37 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 28/28 Conclusion and perspectives Conclusion generic theory for various hyperbolic problems with asymptotic behaviours, first order scheme that preserve A and the asymptotic limit. Perspectives extend the limit scheme to take care of diffusion systems and nonlinear diffusion equation, high-order schemes.

38 Thanks for your attention.

39 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 29/28 References I D. Aregba-Driollet, M. Briani, and R. Natalini. Time asymptotic high order schemes for dissipative BGK hyperbolic systems. arxiv: v1, URL C. Berthon and R. Turpault. Asymptotic preserving HLL schemes. Numer. Methods Partial Differential Equations, 27(6): , ISSN X. doi: /num URL C. Berthon, P. Charrier, and B. Dubroca. An HLLC scheme to solve the M 1 model of radiative transfer in two space dimensions. J. Sci. Comput., 31 (3): , ISSN doi: /s URL

40 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 30/28 References II C. Berthon, P. G. LeFloch, and R. Turpault. Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations. Math. Comp., 82(282): , ISSN doi: /S URL C. Berthon, G. Moebs, C. Sarazin-Desbois, and R. Turpault. An asymptotic-preserving scheme for systems of conservation laws with source terms on 2D unstructured meshes. to appear, F. Bouchut, H. Ounaissa, and B. Perthame. Upwinding of the source term at interfaces for euler equations with high friction. Comput. Math. Appl., 53: , 2007.

41 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 31/28 References III C. Buet and S. Cordier. An asymptotic preserving scheme for hydrodynamics radiative transfer models: numerics for radiative transfer. Numer. Math., 108(2): , ISSN X. doi: /s x. URL C. Buet and B. Després. Asymptotic preserving and positive schemes for radiation hydrodynamics. J. Comput. Phys., 215(2): , ISSN doi: /j.jcp URL C. Buet, B. Després, and E. Franck. Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes. Numer. Math., 122(2): , ISSN X. doi: /s URL

42 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 32/28 References IV C. Chalons, F. Coquel, E. Godlewski, P.-A. Raviart, and N. Seguin. Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction. Math. Models Methods Appl. Sci., 20(11): , ISSN doi: /S X. URL Y. Coudière, J.-P. Vila, and P. Villedieu. Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem. M2AN Math. Model. Numer. Anal., 33(3): , ISSN X. doi: /m2an: URL J. Droniou and C. Le Potier. Construction and convergence study of schemes preserving the elliptic local maximum principle. SIAM J. Numer. Anal., 49(2): , ISSN doi: / URL

43 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 33/28 References V A. Duran, F. Marche, R. Turpault, and C. Berthon. Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes. J. Comput. Phys., 287: , ISSN doi: /j.jcp URL L. Gosse and G. Toscani. Asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. C. R., Math., Acad. Sci. Paris, 334: , 2002.

All-regime Lagrangian-Remap numerical schemes for the gas dynamics equations. Applications to the large friction and low Mach coefficients

All-regime Lagrangian-Remap numerical schemes for the gas dynamics equations. Applications to the large friction and low Mach coefficients Christophe Chalons LMV, Université de Versailles Saint-Quentin-en-Yvelines