Numerical Capture of Shock Solutions of Nonconservative Hyperbolic Systems via Kinetic Functions

This paper reviews recent contributions to the numerical approximation of solutions of nonconservative hyperbolic systems with singular viscous perturbations. Various PDE models for complex compressible materials enter the proposed framework. Due to lack of a conservative form in the limit systems, associated weak solutions are known to heavily depend on the underlying viscous regularization. This small scales sensitiveness drives the classical approximate Riemann solvers to grossly fail in the capture of shock solutions. Here, small scales sensitiveness is encoded thanks to the notion of kinetic functions so as to consider a set of generalized jump conditions. To enforce for validity these jump conditions at the discrete level, we describe a systematic and effective correction procedure. Numerical experiments assess the relevance of the proposed method.

[1]  Eitan Tadmor,et al.  The numerical viscosity of entropy stable schemes for systems of conservation laws. I , 1987 .

[2]  Frédéric Coquel,et al.  Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory , 1993 .

[3]  Frédéric Coquel,et al.  THE RIEMANN PROBLEM FOR THE MULTI-PRESSURE EULER SYSTEM , 2005 .

[4]  Robert L. Pego,et al.  Stable viscosity matrices for systems of conservation laws , 1985 .

[5]  Frédéric Coquel,et al.  EULER EQUATIONS WITH SEVERAL INDEPENDENT PRESSURE LAWS AND ENTROPY SATISFYING EXPLICIT PROJECTION SCHEMES , 2006 .

[6]  Christophe Chalons,et al.  Numerical Approximation of the Navier-Stokes Equations with Several Independent Specific Entropies , 2003 .

[7]  Frédéric Coquel,et al.  Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes , 2005, Numerische Mathematik.

[8]  C. Schmeiser,et al.  Travelling wave analysis and jump relations for Euler-Poisson model in the quasineutral limit , 1995 .

[9]  Pierre-Arnaud Raviart,et al.  A NONCONSERVATIVE HYPERBOLIC SYSTEM MODELING SPRAY DYNAMICS. PART I: SOLUTION OF THE RIEMANN PROBLEM , 1995 .

[10]  J. Colombeau,et al.  Elementary introduction to new generalized functions , 1985 .

[11]  A. Tzavaras,et al.  Representation of weak limits and definition of nonconservative products , 1999 .

[12]  F. Coquel,et al.  Numerical Methods for Weakly Ionized Gas , 1998 .

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

[14]  Travelling Wave Solutions of a Convective Diffusive System with First and Second Order Terms in Nonconservation Form , 1999 .

[15]  Bruno Després,et al.  Numerical resolution of a two-component compressible fluid model with interfaces , 2007 .

[16]  A. I. Vol'pert.  THE SPACES BV AND QUASILINEAR EQUATIONS , 1967 .

[17]  J. Colombeau,et al.  Multiplications of distributions in elasticity and hydrodynamics , 1988 .

[18]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[19]  P. Floch.  Shock Waves for Nonlinear Hyperbolic Systems in Nonconservative Form , 1989 .

[20]  T. O’Neil.  Geometric Measure Theory , 2002 .

[21]  P. Floch.  Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form , 1988 .

[22]  Tai-Ping Liu,et al.  Existence theory for nonlinear hyperbolic systems in nonconservative form , 1993 .

[23]  T. Hou,et al.  Why nonconservative schemes converge to wrong solutions: error analysis , 1994 .

[24]  Frédéric Coquel,et al.  Nonlinear projection methods for multi-entropies Navier-Stokes systems , 2007, Math. Comput..

[25]  Christophe Berthon.  Contribution a l'analyse numerique des equations de navier-stokes compressibles a deux entropies specifiques. Applications a la turbulence compressible , 1999 .

[26]  C. Chalons,et al.  Bilans d'entropie discrets dans l'approximation numerique des chocs non classiques. Application aux equations de Navier-Stokes multi-pression 2D et a quelques systemes visco-capillaires , 2002 .