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

TL;DR

This paper introduces a general framework for designing asymptotic preserving schemes for kinetic equations with stiff sources, enabling stable and accurate simulations across different regimes without nonlinear solvers.

Contribution

The authors propose a novel penalization approach using a BGK-type relaxation term that simplifies the solution process and ensures asymptotic preservation in kinetic equations.

Findings

Scheme is uniformly stable across small Knudsen numbers

Captures Euler limit without resolving small scales

Consistent with Navier-Stokes equations when resolved

Abstract

In this paper, we propose a general framework to design asymptotic preserving schemes for the Boltzmann kinetic 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 natually 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 parabilic equations.