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

Francis Filbet (1) , Shi Jin (2)

1 ICJ - Institut Camille Jordan
2 DM - Department Mathematics

In this paper, we propose a general time-discrete framework to design asymptotic-preserving schemes for the initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations is 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 implemented 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 the 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 with 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

Domaines: Analyse numérique [math.NA]

Francis Filbet, Shi Jin. A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. Journal of Computational Physics, 2010, 229 (20), pp.7625-7648. ⟨10.1016/j.jcp.2010.06.017⟩. ⟨hal-00659668⟩