An Asymptotic Preserving Scheme for the ES-BGK model

TL;DR

This paper introduces an asymptotic-preserving IMEX scheme for the ES-BGK kinetic model that efficiently handles stiff collision terms and accurately captures both Euler and Navier-Stokes limits without nonlinear solvers.

Contribution

It proposes a novel IMEX discretization that is explicit for convection and implicit for relaxation, ensuring asymptotic preservation and computational efficiency.

Findings

The scheme drives the distribution toward Maxwellian as mean free time approaches zero.

It captures the Euler limit even with coarse time steps.

It is consistent with Navier-Stokes equations when viscosity and heat conduction are resolved.

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.