Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit

TL;DR

This paper provides a mathematical analysis demonstrating that an asymptotic preserving scheme for linear kinetic equations remains stable and accurate across different regimes, with conditions depending on the mean free path.

Contribution

The paper offers a rigorous proof of the uniform stability and accuracy of the scheme under a CFL condition that adapts to the mean free path, using simple energy estimates.

Findings

Scheme is uniformly stable and accurate across regimes.

CFL condition adapts to mean free path size.

Analysis uses simple energy estimates.

Abstract

We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31, pp. 334-368, 2008] for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles. This property is satisfied under an explicitly given CFL condition. This condition tends to a parabolic CFL condition for small mean free paths, and is close to a convection CFL condition for large mean free paths. Ou r analysis is based on very simple energy estimates.