Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws

We introduce a general framework for kinetic BGK models. We assume to be given a system of hyperbolic conservation laws with a family of Lax entropies, and we characterize the BGK models that lead to this system in the hydrodynamic limit, and that are compatible with the whole family of entropies. This is obtained by a new characterization of Maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies. We deduce a general procedure to construct such BGK models, and we show how classical examples enter the framework. We apply our theory to isentropic gas dynamics and full gas dynamics, and in both cases we obtain new BGK models satisfying all entropy inequalities.

[1]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[2]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[3]  D. Pullin,et al.  Direct simulation methods for compressible inviscid ideal-gas flow , 1980 .

[4]  Rolf D. Reitz,et al.  One-dimensional compressible gas dynamics calculations using the Boltzmann equation , 1981 .

[5]  Jean-Pierre Croisille,et al.  Kinetic symmetrizations and pressure laws for the Euler equations , 1982 .

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

[7]  Y. Giga,et al.  A kinetic construction of global solutions of first order quasilinear equations , 1983 .

[8]  Y. Brenier.  Résolution d'équations d'évolution quasilinéaires en dimension N d'espace à l'aide d'équations linéaires en dimension N + 1 , 1983 .

[9]  Yann Brenier,et al.  Averaged Multivalued Solutions for Scalar Conservation Laws , 1984 .

[10]  Tai-Ping Liu.  Hyperbolic conservation laws with relaxation , 1987 .

[11]  J. C. Mandal,et al.  Higher Order Accurate Kinetic Flux Vector Splitting Method for Euler Equations , 1989 .

[12]  Benoît Perthame,et al.  Global existence to the BGK model of Boltzmann equation , 1989 .

[13]  Jean-Pierre Croisille,et al.  On the approximation of K-diagonalizable hyperbolic systems by finite elements - Applications to the Euler equations and to gaseous mixtures , 1989 .

[14]  Convexity in Hyperbolic Problems. Application to a Discontinuous Galerkin Method for the Resolution of the Polydimensional Euler Equations , 1989 .

[15]  B. Perthame,et al.  A kinetic equation with kinetic entropy functions for scalar conservation laws , 1991 .

[16]  B. Perthame,et al.  Boltzmann type schemes for gas dynamics and the entropy property , 1990 .

[17]  B. Perthame.  Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions , 1992 .

[18]  M. Pulvirenti,et al.  Weighted L∞ bounds and uniqueness for the Boltzmann BGK model , 1993 .

[19]  Gui-Qiang G. Chen,et al.  Zero relaxation and dissipation limits for hyperbolic conservation laws , 1993 .

[20]  Benoît Perthame,et al.  Maximum principle on the entropy and second-order kinetic schemes , 1994 .

[21]  B. Perthame,et al.  Kinetic formulation of the isentropic gas dynamics andp-systems , 1994 .

[22]  C. D. Levermore,et al.  Hyperbolic conservation laws with stiff relaxation terms and entropy , 1994 .

[23]  B. Perthame,et al.  A kinetic formulation of multidimensional scalar conservation laws and related equations , 1994 .

[24]  Jean-Pierre Croisille,et al.  Numerical simulation of the MHD equations by a kinetic-type method , 1995 .

[25]  Gui-Qiang G. Chen,et al.  Entropy flux-splittings for hyperbolic conservation laws part I: General framework , 1995 .

[26]  B. Perthame,et al.  Kinetic formulation for chromatography and some other hyperbolic systems , 1995 .

[27]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .

[28]  R. Natalini.  Convergence to equilibrium for the relaxation approximations of conservation laws , 1996 .

[29]  R. Winther,et al.  A system of conservation laws with a relaxation term , 1996 .

[30]  R. Natalini,et al.  Convergence of relaxation schemes for conservation laws , 1996 .

[31]  Jean-Pierre Croisille,et al.  Resolution of the system of equations of ideal magnetohydrodynamics by a finite volume kinetic-type method , 1996 .

[32]  C. D. Levermore,et al.  Moment closure hierarchies for kinetic theories , 1996 .

[33]  A. Tzavaras,et al.  Contractive relaxation systems and the scalar multidimensional conservation law , 1997 .

[34]  Entropies and weak solutions of the compressible isentropic Euler equations , 1997 .

[35]  Jean-Marc Moschetta,et al.  A Robust Low Diffusive Kinetic Scheme for the Navier-Stokes/Euler Equations , 1997 .

[36]  Convergence results for some conservation laws with reflux boundary condition and a relaxation term arising in chemical engineering , 1998 .

[37]  R. Natalini.  A Discrete Kinetic Approximation of Entropy Solutions to Multidimensional Scalar Conservation Laws , 1998 .

[38]  Y. Brenier,et al.  A kinetic formulation for multi-branch entropy solutions of scalar conservation laws , 1998 .

[39]  B. Perthame,et al.  Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics , 1998 .

[40]  Yann Brenier,et al.  RELAXATION LIMITS FOR A CLASS OF BALANCE LAWS WITH KINETIC FORMULATION , 1998 .

[41]  Roberto Natalini,et al.  Recent Mathematical Results on Hyperbolic Relaxation Problems , 1999 .

[42]  Athanasios E. Tzavaras,et al.  Materials with Internal Variables and Relaxation to Conservation Laws , 1999 .

[43]  D. Serre.  Relaxations semi-linaire et cintique des systmes de lois de conservation , 2000 .

[44]  Roberto Natalini,et al.  Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws , 2000, SIAM J. Numer. Anal..