The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: II. The case of systems

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.

[1]  Jérôme Jaffré,et al.  Godunov-Type Methods for Conservation Laws with a Flux Function Discontinuous in Space , 2004, SIAM J. Numer. Anal..

[2]  Claus-Dieter Munz,et al.  Approximate Riemann Solvers for Fluid Flow with Material Interfaces , 1998 .

[3]  Stefan Diehl,et al.  On scalar conservation laws with point source and discontinuous flux function , 1995 .

[4]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[5]  P. Floch,et al.  Boundary conditions for nonlinear hyperbolic systems of conservation laws , 1988 .

[6]  Gérard Gallice,et al.  Positive and Entropy Stable Godunov-type Schemes for Gas Dynamics and MHD Equations in Lagrangian or Eulerian Coordinates , 2003, Numerische Mathematik.

[7]  Nils Henrik Risebro,et al.  STABILITY OF CONSERVATION LAWS WITH DISCONTINUOUS COEFFICIENTS , 1999 .

[8]  B. Perthame,et al.  Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies , 2005, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[9]  John D. Towers,et al.  Upwind difference approximations for degenerate parabolic convection–diffusion equations with a discontinuous coefficient , 2002 .

[10]  Siam Staff,et al.  Godunov-Type Methods for Conservation Laws with a Flux Function Discontinuous in Space , 2004 .

[11]  Edwige Godlewski,et al.  The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. The scalar case , 2004, Numerische Mathematik.

[12]  Bruno Després,et al.  Lagrangian systems of conservation laws , 2001, Numerische Mathematik.

[13]  Rémi Abgrall,et al.  Computations of compressible multifluids , 2001 .

[14]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[15]  Modélisation mathématique et numérique de la cavitation dans les écoulements multiphasiques compressibles , 2002 .

[16]  Christian Klingenberg,et al.  Stability of a Resonant System of Conservation Laws Modeling Polymer Flow with Gravitation , 2001 .

[17]  J. Greenberg,et al.  Analysis and Approximation of Conservation Laws with Source Terms , 1997 .

[18]  E. Isaacson,et al.  Nonlinear resonance in systems of conservation laws , 1992 .

[19]  Thierry Pougeard-Dulimbert.  Extraction de faisceaux d'ions à partir de plasmas neutres : modélisation et simulation numérique , 2001 .

[20]  Nicolas Seguin,et al.  ANALYSIS AND APPROXIMATION OF A SCALAR CONSERVATION LAW WITH A FLUX FUNCTION WITH DISCONTINUOUS COEFFICIENTS , 2003 .

[21]  Siddhartha Mishra.  Convergence of Upwind Finite Difference Schemes for a Scalar Conservation Law with Indefinite Discontinuities in the Flux Function , 2005, SIAM J. Numer. Anal..

[22]  Randall J. LeVeque,et al.  A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions , 2002, SIAM J. Sci. Comput..

[23]  Denis Serre,et al.  Conditions aux limites pour un système strictement hyperbolique fournies, par le schéma de Godunov , 1997 .

[24]  M. Gisclon,et al.  Etude des conditions aux limites pour des systèmes strictement hyperboliques, via l'approximation parabolique , 1994 .

[25]  W. Lyons,et al.  Conservation laws with sharp inhomogeneities , 1983 .

[26]  Claus-Dieter Munz,et al.  On Godunov-type schemes for Lagrangian gas dynamics , 1994 .

[27]  A. Visintin,et al.  Theoretical and numerical results on the two-phase stefan problem , 1989 .

[28]  John D. Towers.  A Difference Scheme for Conservation Laws with a Discontinuous Flux: The Nonconvex Case , 2001, SIAM J. Numer. Anal..

[29]  S. P. Gill,et al.  Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena , 2002 .

[30]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .