Analysis of Asymptotic Preserving DG-IMEX Schemes for Linear Kinetic Transport Equations in a Diffusive Scaling

Abstract

In this paper, some theoretical aspects will be addressed for the asymptotic preserving DG-IMEX schemes recently proposed in [J. Jang, F. Li, J.-M. Qiu and T. Xiong, submitted, arxiv:1306.0227] for kinetic transport equations under a diffusive scaling. We will focus on the methods that are based on discontinuous Galerkin (DG) spatial discretizations with the $P^k$ polynomial space and a first order IMEX temporal discretization, and apply them to two linear models: the telegraph equation and the one-group transport equation in slab geometry. In particular, we will establish uniform numerical stability with respect to Knudsen number $\epsilon$ using energy methods, as well as error estimates for any given $\epsilon$. When $\epsilon \to 0$, a rigorous asymptotic analysis of the schemes is also obtained. Though the methods and the analysis are presented for one dimension in space, they can be generalized to higher dimensions directly.

arXiv comment20 pages