Stability-enhanced AP IMEX-LDG schemes for linear kinetic transport equations under a diffusive scaling

Abstract

Transport equations arise in many applications such as rarefied gas dynamics, neutron transport, and radiative transfer. In this work, we consider some linear kinetic transport equations in a diffusive scaling and design high order asymptotic preserving (AP) methods within the discontinuous Galerkin method framework, with the main objective to achieve unconditional stability in the diffusive regime when the Knudsen number ε≪1, and to achieve high order accuracy when ε=O(1) and when ε≪1. Initial layers are also taken into account. The ingredients to accomplish our goal include: model reformulations based on the micro-macro decomposition and the limiting diffusive equation, local discontinuous Galerkin (LDG) methods in space, globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta methods in time, and strategies to handle non-well prepared initial data. Formal asymptotic analysis is carried out for the continuous model within the micro-macro decomposed framework to derive the initial layer as well as the interior problem with an asymptotically consistent initial condition as ε→0, and it is also conducted for numerical schemes to show the AP property and to understand the numerical initial treatments in the presence of initial layers. Fourier type stability analysis is performed, and it confirms the unconditional stability in the diffusive regime, and moreover it gives the stability condition in the kinetic regime when ε=O(1). In the reformulation step, a weighted diffusive term is added and subtracted to remove the parabolic stiffness and enhance the numerical stability in the diffusive regime. Such idea is not new, yet our numerical stability and asymptotic analysis provide new mathematical understanding towards the desired properties of the weight function. Finally, numerical examples are presented to demonstrate the accuracy, stability, and asymptotic preserving property of the proposed methods, as well as the effectiveness of the proposed strategies in the presence of the initial layer.