The numerical interface coupling of nonlinear hyperbolic systems of conservation laws : II. The case of systems
ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 649-692.

We study the theoretical and numerical coupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling in the linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem. We discuss both approaches in the case of the coupling of two fluid models at a material contact discontinuity, the models being the usual gas dynamics equations with different equations of state. We also study the coupling of two-temperature plasma fluid models and illustrate the approach by numerical simulations.

DOI : 10.1051/m2an:2005029
Classification : 35L50, 35L65, 65M12, 65M30, 65-04, 76M12
Mots-clés : conservation laws, Riemann problem, boundary value problems, interface coupling, finite volume schemes 
@article{M2AN_2005__39_4_649_0,
     author = {Godlewski, Edwige and Thanh, Kim-Claire Le and Raviart, Pierre-Arnaud},
     title = {The numerical interface coupling of nonlinear hyperbolic systems of conservation laws : {II.} {The} case of systems},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {649--692},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {4},
     year = {2005},
     doi = {10.1051/m2an:2005029},
     mrnumber = {2165674},
     zbl = {1095.65084},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2005029/}
}
TY  - JOUR
AU  - Godlewski, Edwige
AU  - Thanh, Kim-Claire Le
AU  - Raviart, Pierre-Arnaud
TI  - The numerical interface coupling of nonlinear hyperbolic systems of conservation laws : II. The case of systems
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2005
SP  - 649
EP  - 692
VL  - 39
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2005029/
DO  - 10.1051/m2an:2005029
LA  - en
ID  - M2AN_2005__39_4_649_0
ER  - 
%0 Journal Article
%A Godlewski, Edwige
%A Thanh, Kim-Claire Le
%A Raviart, Pierre-Arnaud
%T The numerical interface coupling of nonlinear hyperbolic systems of conservation laws : II. The case of systems
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2005
%P 649-692
%V 39
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2005029/
%R 10.1051/m2an:2005029
%G en
%F M2AN_2005__39_4_649_0
Godlewski, Edwige; Thanh, Kim-Claire Le; Raviart, Pierre-Arnaud. The numerical interface coupling of nonlinear hyperbolic systems of conservation laws : II. The case of systems. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 649-692. doi : 10.1051/m2an:2005029. http://archive.numdam.org/articles/10.1051/m2an:2005029/

[1] R. Abgrall and S. Karni, Computations of compressible multifluids. J. Comput. Phys. 169 (2001) 594-623. | Zbl

[2] J.J. Adimurthi and G.D. Veerappa Gowda, Godunov-type methods for conservation laws with a flux function discontinuous in space. SIAM J. Numer. Anal. 42 (2004) 179-208. | Zbl

[3] E. Audusse and B. Perthame, Uniqueness for a scalar conservation law with discontinuous flux via adapted entropies, Inria research report No. 5261 (2004), France. | Zbl

[4] D. Bale, R. Leveque, S. Mitran and J. Rossmanith, A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24 (2002) 955-978. | Zbl

[5] T. Barberon, Modélisation mathématique et numérique de la cavitation dans les écoulements multiphasiques compressibles. Thesis, University of Toulon, France (2002).

[6] F. Coquel, E. Godlewski, P.-A. Raviart et al., Numerical coupling of models in the context of fluid flows, work in preparation.

[7] S. Cordier, Hyperbolicity of the hydrodynamic model of plasmas under the quasi-neutrality hypothesis. Math. Methods Appl. Sci. 18 (1995) 627-647. | Zbl

[8] B. Després, Lagrangian systems of conservation laws. Invariance properties of Lagrangian systems of conservation laws, approximate Riemann solvers and the entropy condition. Numer. Math. 89 (2001) 99-134. | Zbl

[9] S. Diehl, On scalar conservation laws with point source and discontinuous flux function. SIAM J. Numer. Anal. 26 (1995) 1425-1451. | Zbl

[10] F. Dubois and P. Le Floch, Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71 (1988) 93-122. | Zbl

[11] R. Fedkiw, T. Aslam, B. Merriman and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152 (1999) 457-492. | Zbl

[12] G. Gallice, Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates. Numer. Math. 94 (2003) 673-713. | Zbl

[13] M. Gisclon, Étude des conditions aux limites pour un système strictement hyperbolique via l'approximation parabolique. J. Math. Pures Appl. 75 (1996) 485-508. | Zbl

[14] M. Gisclon and D. Serre, Étude des conditions aux limites pour un système hyperbolique, via l'approximation parabolique. C. R. Acad. Sci. Paris, Série I 319 (1994) 377-382. | Zbl

[15] M. Gisclon and D. Serre, Conditions aux limites pour un système strictement hyperbolique fournies par le schéma de Godunov. RAIRO Modél. Math. Anal. Numér. 31 (1997) 359-380. | EuDML | Numdam | Zbl

[16] E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Appl. Math. Sci. 118, Springer, New York (1996). | MR | Zbl

[17] E. Godlewski and P.-A. Raviart, The numerical coupling of nonlinear hyperbolic systems of conservation laws: I. The scalar case. Numer. Math. 97 (2004) 81-130. | Zbl

[18] M. Göz and C.-D. Munz, Approximate Riemann solvers for fluid flow with material interfaces. Numerical methods for wave propagation (Manchester, 1995), Kluwer Acad. Publ., Dordrecht. Fluid Mech. Appl. 47 (1998) 211-235. | Zbl

[19] J.M. Greenberg, A.Y. Leroux, R. Baraille and A. Noussair, Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 (1997) 1980-2007. | Zbl

[20] A. Harten, P.D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35-61. | Zbl

[21] E. Isaacson and B. Temple, Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52 (1992) 1260-1278. | Zbl

[22] K. Karlsen, N. Risebro and J. Towers, Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J. Numer. Anal. 22 (2002) 623-664. | Zbl

[23] R. Klausen and N. Risebro, Stability of conservation laws with discontinuous coefficients. J. Differential Equations 157 (1999) 41-60. | Zbl

[24] C. Klingenberg and N.H. Risebro, Stability of a resonant system of conservation laws modeling polymer flow with gravitation, J. Differential Equations 170 (2001) 344-380. | Zbl

[25] S. Kokh, Aspects numériques et théoriques de la modélisation des écoulements diphasiques compressibles par des méthodes de capture d'interface. Thesis, University Paris 6, France (2001).

[26] K.-C. Le Thanh and P.-A. Raviart, Un modèle de plasma partiellement ionisé. Rapport CEA-R-6036, France (2003).

[27] W.K. Lyons, Conservation laws with sharp inhomogeneities. Quart. Appl. Math. 40 (1983) 385-393. | Zbl

[28] S. Mishra, Convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function. Ntnu Preprints on Conservation Laws 2003-077 (2003). | MR | Zbl

[29] C.-D. Munz, On Godunov-type schemes for Lagrangian gas dynamics. SIAM J. Numer. Anal. (1994), 17-42. | Zbl

[30] T. Pougeard Dulimbert, Extraction de faisceaux d'ions à partir de plasmas neutres: Modélisation et simulation numérique. Thesis, University Paris 6, France (2001).

[31] N. Seguin and J. Vovelle, Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci. 13 (2003) 221-257. | Zbl

[32] D. Serre, Systèmes de lois de conservation I and II. Diderot éditeur, Paris (1996). | MR

[33] J. Towers, A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39 (2001) 1197-1218. | Zbl

[34] Y.B. Zel'Dovich and Y.P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena, Vol. II. Academic Press (1967).

Cité par Sources :