An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation

Authors:
Francis Filbet;Shi Jin
Affiliations:
Université de Lyon, CNRS, UMR 5208, Institut Camille Jordan, Université Lyon I, Villeurbanne Cedex, France 69622;Department of Mathematics, University of Wisconsin, Madison, USA 53706
Venue:
Journal of Scientific Computing
Year:
2011

Citing 8
Cited 4

Numerical passage from kinetic to fluid equations

SIAM Journal on Numerical Analysis
Rarefied flow computations using nonlinear model Boltzmann equations

Journal of Computational Physics
Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations

SIAM Journal on Scientific Computing
High order numerical methods for the space non-homogeneous Boltzmann equation

Journal of Computational Physics
Accurate numerical methods for the collisional motion of (heated) granular flows

Journal of Computational Physics
Solving the Boltzmann Equation in N log2 N

SIAM Journal on Scientific Computing
Implicit—Explicit Schemes for BGK Kinetic Equations

Journal of Scientific Computing
Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics

Journal of Computational Physics

Microscopically implicit-macroscopically explicit schemes for the BGK equation

Journal of Computational Physics
Exponential Runge-Kutta Methods for Stiff Kinetic Equations

SIAM Journal on Numerical Analysis
A hybrid method for hydrodynamic-kinetic flow - Part II - Coupling of hydrodynamic and kinetic models

Journal of Computational Physics
Exponential Runge-Kutta for the inhomogeneous Boltzmann equations with high order of accuracy

Journal of Computational Physics

Quantified Score

Hi-index	0.02

Visualization

Abstract

In this paper, we study a time discrete scheme for the initial value problem of the ES-BGK kinetic equation. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We study an implicit-explicit (IMEX) time discretization in which the convection is explicit while the relaxation term is implicit to overcome the stiffness. We first show how the implicit relaxation can be solved explicitly, and then prove asymptotically that this time discretization drives the density distribution toward the local Maxwellian when the mean free time goes to zero while the numerical time step is held fixed. This naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver for the implicit relaxation term. Moreover, it can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We also show that it is consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. Several numerical examples, in both one and two space dimensions, are used to demonstrate the desired behavior of this scheme.