A Discrete Kinetic Approximation of Entropy Solutions to Multidimensional Scalar Conservation Laws

Abstract We present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities. Under a suitable multidimensional generalization of the Whitham relaxation subcharacteristic condition, we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[3]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[4]  G. Papanicolaou,et al.  The fluid‐dynamical limit of a nonlinear model boltzmann equation , 1979 .

[5]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[6]  L. Tartar.  Some Existence Theorems for Semilinear Hyperbolic Systems in One Space Variable. , 1981 .

[7]  S. Levin.  Lectu re Notes in Biomathematics , 1983 .

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

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

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

[11]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[12]  Hal L. Smith.  Systems of ordinary differential equations which generate an order preserving flow. A survey of results , 1988 .

[13]  R. Illner,et al.  Discrete Velocity Models of the Boltzmann Equation: A Survey on the Mathematical ASPECTS of the Theory , 1988 .

[14]  Benoît Perthame,et al.  Global existence to the BGK model of Boltzmann equation , 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,et al.  Numerical passage from kinetic to fluid equations , 1991 .

[18]  Z. Xin.  The fluid-dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of Shocks , 1991 .

[19]  N. Bellomo,et al.  THE DISCRETE BOLTZMANN EQUATION: A REVIEW OF THE MATHEMATICAL ASPECTS OF THE INITIAL AND INITIAL-BOUNDARY VALUE PROBLEMS , 1991 .

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

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

[22]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[23]  B. S'evennec.  G'eom'etrie des syst`emes hyperboliques de lois de conservation , 1994 .

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

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

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

[27]  Sauro Succi,et al.  Recent Advances in Lattice Boltzmann Computing , 1995 .

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

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

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

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

[32]  J. Collet,et al.  Convergence of the relaxation approximation to a scalar nonlinear hyperbolic equation arising in chromatography , 1996 .

[33]  Shi Jin.  A Convex Entropy for a Hyperbolic System with Relaxation , 1996 .

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

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

[36]  R. Natalini,et al.  Weakly coupled systems of quasilinear hyperbolic equations , 1996, Differential and Integral Equations.

[37]  R. Natalini,et al.  L1Nonlinear Stability of Traveling Waves for a Hyperbolic System with Relaxation , 1996 .

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

[39]  R. Natalini,et al.  BV solutions and relaxation limit for a model in viscoelasticity , 1998, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

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

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

[42]  W. Yong.  A difference scheme for a stiff system of conservation laws , 1998, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[43]  P. Souganidis,et al.  Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates , 1998 .

[44]  ASYMPTOTIC LIMIT OF INITIAL BOUNDARY VALUE PROBLEMS FOR CONSERVATION LAWS WITH RELAXATIONAL EXTENSIONS , 1998 .

[45]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.