High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling

Abstract

In this paper, we develop a family of high order asymptotic preserving schemes for some discrete-velocity kinetic equations under a diffusive scaling, that in the asymptotic limit lead to macroscopic models such as the heat equation, the porous media equation, the advection–diffusion equation, and the viscous Burgers' equation. Our approach is based on the micro–macro reformulation of the kinetic equation which involves a natural decomposition of the equation to the equilibrium and non-equilibrium parts. To achieve high order accuracy and uniform stability as well as to capture the correct asymptotic limit, two new ingredients are employed in the proposed methods: discontinuous Galerkin (DG) spatial discretization of arbitrary order of accuracy with suitable numerical fluxes; high order globally stiffly accurate implicit–explicit (IMEX) Runge–Kutta scheme in time equipped with a properly chosen implicit–explicit strategy. Formal asymptotic analysis shows that the proposed scheme in the limit of ε→0 is a consistent high order discretization for the limiting equation. Numerical results are presented to demonstrate the stability and high order accuracy of the proposed schemes together with their performance in the limit. Our methods are also tested for the continuous-velocity one-group transport equation in slab geometry and for several examples with spatially varying parameters.