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.
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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/m2an:2005029/
[1] Computations of compressible multifluids. J. Comput. Phys. 169 (2001) 594-623. | Zbl
and ,[2] Godunov-type methods for conservation laws with a flux function discontinuous in space. SIAM J. Numer. Anal. 42 (2004) 179-208. | Zbl
and ,[3] Uniqueness for a scalar conservation law with discontinuous flux via adapted entropies, Inria research report No. 5261 (2004), France. | Zbl
and ,[4] A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24 (2002) 955-978. | Zbl
, , and ,[5] Modélisation mathématique et numérique de la cavitation dans les écoulements multiphasiques compressibles. Thesis, University of Toulon, France (2002).
,[6] Numerical coupling of models in the context of fluid flows, work in preparation.
, , et al.,[7] Hyperbolicity of the hydrodynamic model of plasmas under the quasi-neutrality hypothesis. Math. Methods Appl. Sci. 18 (1995) 627-647. | Zbl
,[8] 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] On scalar conservation laws with point source and discontinuous flux function. SIAM J. Numer. Anal. 26 (1995) 1425-1451. | Zbl
,[10] Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71 (1988) 93-122. | Zbl
and ,[11] A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152 (1999) 457-492. | Zbl
, , and ,[12] 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] Étude des conditions aux limites pour un système strictement hyperbolique via l'approximation parabolique. J. Math. Pures Appl. 75 (1996) 485-508. | Zbl
,[14] 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
and ,[15] 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
and ,[16] Numerical approximation of hyperbolic systems of conservation laws. Appl. Math. Sci. 118, Springer, New York (1996). | MR | Zbl
and ,[17] The numerical coupling of nonlinear hyperbolic systems of conservation laws: I. The scalar case. Numer. Math. 97 (2004) 81-130. | Zbl
and ,[18] 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
and ,[19] Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 (1997) 1980-2007. | Zbl
, , and ,[20] On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35-61. | Zbl
, and ,[21] Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52 (1992) 1260-1278. | Zbl
and ,[22] Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J. Numer. Anal. 22 (2002) 623-664. | Zbl
, and ,[23] Stability of conservation laws with discontinuous coefficients. J. Differential Equations 157 (1999) 41-60. | Zbl
and ,[24] Stability of a resonant system of conservation laws modeling polymer flow with gravitation, J. Differential Equations 170 (2001) 344-380. | Zbl
and ,[25] 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] Un modèle de plasma partiellement ionisé. Rapport CEA-R-6036, France (2003).
and ,[27] Conservation laws with sharp inhomogeneities. Quart. Appl. Math. 40 (1983) 385-393. | Zbl
,[28] 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] On Godunov-type schemes for Lagrangian gas dynamics. SIAM J. Numer. Anal. (1994), 17-42. | Zbl
,[30] Extraction de faisceaux d'ions à partir de plasmas neutres: Modélisation et simulation numérique. Thesis, University Paris 6, France (2001).
,[31] 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
and ,[32] Systèmes de lois de conservation I and II. Diderot éditeur, Paris (1996). | MR
,[33] A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39 (2001) 1197-1218. | Zbl
,[34] Physics of shock waves and high-temperature hydrodynamic phenomena, Vol. II. Academic Press (1967).
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