Relaxation Scheme for a Lattice-Boltzmann-type Discrete Velocity Model and Numerical Navier-Stokes Limit

A discrete velocity model based on a lattice?Boltzmann approximation is considered in the low Mach number limit. A numerical scheme for this model working uniformly in the incompressible Navier?Stokes limit is constructed. The scheme is induced by the asymptotic analysis of the Navier?Stokes limit and works uniformly for all ranges of mean free paths. In the limit the scheme reduces to an explicit finite difference scheme for the incompressible Navier?Stokes equation, the Chorin projection method with MAC grid. Numerical results are presented and the uniform convergence of the scheme is established numerically.

[1]  C. D. Levermore,et al.  Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms , 1996 .

[2]  Giovanni Russo,et al.  Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation , 1997 .

[3]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[4]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[5]  A. Klar.  An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit , 1998 .

[6]  Brian Wetton.  Analysis of the Spatial Error for a Class of Finite Difference Methods for Viscous Incompressible Flow , 1997 .

[7]  F. Golse,et al.  Fluid dynamic limits of kinetic equations. I. Formal derivations , 1991 .

[8]  T. Abe.  Derivation of the Lattice Boltzmann Method by Means of the Discrete Ordinate Method for the Boltzmann Equation , 1997 .

[9]  Shiyi Chen,et al.  Stability Analysis of Lattice Boltzmann Methods , 1993, comp-gas/9306001.

[10]  E. Larsen,et al.  Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II , 1989 .

[11]  Shi Jin,et al.  Physical symmetry and lattice symmetry in the lattice Boltzmann method , 1997 .

[12]  Raffaele Esposito,et al.  Incompressible Navier-Stokes and Euler Limits of the Boltzmann Equation , 1989 .

[13]  Axel Klar,et al.  A Numerical Method for Kinetic Semiconductor Equations in the Drift-Diffusion Limit , 1999, SIAM J. Sci. Comput..

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

[15]  Matthaeus,et al.  Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[16]  Lorenzo Pareschi,et al.  Diffusive Relaxation Schemes for Multiscale Discrete-Velocity Kinetic Equations , 1998 .

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

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

[19]  Pierre Lallemand,et al.  Lattice Gas Hydrodynamics in Two and Three Dimensions , 1987, Complex Syst..