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 discontinuous Galerkin implicit-explicit (DG-IMEX) schemes recently proposed in [J. Jang, F. Li, J.-M. Qiu, and T. Xiong, High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling, http://arxiv.org/abs/1306.0227, 2013, submitted] 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 implicit-explicit (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 $\varepsilon$ using energy methods, as well as error estimates for any given $\varepsilon$. When $\varepsilon\rightarrow 0$, 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.