A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

JOURNAL OF COMPUTATIONAL PHYSICS

卷:229

Article

Filbet, Francis¹  Jin, Shi² 

[1] Univ Lyon 1, CNRS, Inst Camille Jordan, UMR 5208, F-69622 Villeurbanne, France

[2] Univ Wisconsin, Dept Math, Madison, WI 53706 USA

关键词: Boltzmann equation;    Asymptotic-preserving scheme;    Stiff source terms;   

DOI  :  10.1016/j.jcp.2010.06.017

来源: Elsevier

【 摘 要 】

In this paper, we propose a general time-discrete framework to design asymptotic-preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabolic equations. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free