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The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. II: The case of systems. (English) Zbl 1095.65084
[For part I see E. Godlewski and P.-A. Raviart, Numer. Math. 97, No. 1, 81–130 (2004; Zbl 1063.65080).]
In the modelling of complex problems, different mathematical models are frequently used in different regions of interest. In many cases it leads to couple different systems of partial differential equations of different sizes at various artificial boundaries. This work is the one in a series of the authors’ papers devoted to the study of the coupling of nonlinear hyperbolic systems from both mathematical and numerical points of view.
Indeed, the authors study the coupling of one-dimensional nonlinear hyperbolic systems of conservation laws. A detailed analysis of the coupling is given for linear problems. The nonlinear case are approached by either linearization or by the solution of a Riemann problem. The authors study both approaches for (i) the coupling of two fluid models at a material contact discontinuity (gas dynamic equations in Lagrangian coordinates) and (ii) the coupling of two-temperature fluid models for a quasi-natural ionized plasma with different current density but the same equation of state. Some numerical implementations are also presented.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics, general
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
76M12 Finite volume methods applied to problems in fluid mechanics
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References:
[1] R. Abgrall and S. Karni , Computations of compressible multifluids . J. Comput. Phys. 169 ( 2001 ) 594 - 623 . Zbl 1033.76029 · Zbl 1033.76029 · doi:10.1006/jcph.2000.6685
[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 1081.65082 · Zbl 1081.65082 · doi:10.1137/S003614290139562X
[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 1071.35079 · Zbl 1071.35079 · doi:10.1017/S0308210500003863
[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 1034.65068 · Zbl 1034.65068 · doi:10.1137/S106482750139738X
[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 0828.35076 · Zbl 0828.35076 · doi:10.1002/mma.1670180805
[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 0990.65098 · Zbl 0990.65098 · doi:10.1007/PL00005465
[9] S. Diehl , On scalar conservation laws with point source and discontinuous flux function . SIAM J. Numer. Anal. 26 ( 1995 ) 1425 - 1451 . Zbl 0852.35094 · Zbl 0852.35094 · doi:10.1137/S0036141093242533
[10] F. Dubois and P. Le Floch , Boundary conditions for nonlinear hyperbolic systems of conservation laws . J. Differential Equations 71 ( 1988 ) 93 - 122 . Zbl 0649.35057 · Zbl 0649.35057 · doi:10.1016/0022-0396(88)90040-X
[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 0957.76052 · Zbl 0957.76052 · doi:10.1006/jcph.1999.6236
[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 1092.76044 · Zbl 1092.76044 · doi:10.1007/s00211-002-0430-0
[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 0869.35061 · Zbl 0869.35061
[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 0808.35075 · Zbl 0808.35075
[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 . Numdam | Zbl 0873.65087 · Zbl 0873.65087 · eudml:193841
[16] E. Godlewski and P.-A. Raviart , Numerical approximation of hyperbolic systems of conservation laws . Appl. Math. Sci. 118, Springer, New York ( 1996 ). MR 1410987 | Zbl 0860.65075 · Zbl 0860.65075
[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 1063.65080 · Zbl 1063.65080 · doi:10.1007/s00211-003-0438-5
[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 0962.76572 · Zbl 0962.76572
[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 0888.65100 · Zbl 0888.65100 · doi:10.1137/S0036142995286751
[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 0565.65051 · Zbl 0565.65051 · doi:10.1137/1025002
[21] E. Isaacson and B. Temple , Nonlinear resonance in systems of conservation laws . SIAM J. Appl. Math. 52 ( 1992 ) 1260 - 1278 . Zbl 0794.35100 · Zbl 0794.35100 · doi:10.1137/0152073
[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 1014.65073 · Zbl 1014.65073 · doi:10.1093/imanum/22.4.623
[23] R. Klausen and N. Risebro , Stability of conservation laws with discontinuous coefficients . J. Differential Equations 157 ( 1999 ) 41 - 60 . Zbl 0935.35097 · Zbl 0935.35097 · doi:10.1006/jdeq.1998.3624
[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 0977.35083 · Zbl 0977.35083 · doi:10.1006/jdeq.2000.3826
[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 0516.35053 · Zbl 0516.35053
[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 2177880 | Zbl 1096.35085 · Zbl 1096.35085 · doi:10.1137/030602745
[29] C.-D. Munz , On Godunov-type schemes for Lagrangian gas dynamics . SIAM J. Numer. Anal. ( 1994 ), 17 - 42 . Zbl 0796.76057 · Zbl 0796.76057 · doi:10.1137/0731002
[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 1078.35011 · Zbl 1078.35011 · doi:10.1142/S0218202503002477
[32] D. Serre , Systèmes de lois de conservation I and II . Diderot éditeur, Paris ( 1996 ). MR 1459988
[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 1055.65104 · Zbl 1055.65104 · doi:10.1137/S0036142900374974
[34] Y.B. Zel’dovich and Y.P. Raizer , Physics of shock waves and high-temperature hydrodynamic phenomena , Vol. II. Academic Press ( 1967 ).
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