High Order Asymptotic Preserving DG-IMEX Schemes for Discrete-Velocity Kinetic Equations in a Diffusive Scaling

TL;DR

This paper introduces high order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in diffusive scaling, accurately capturing macroscopic limits like heat and porous media equations while maintaining stability and high accuracy.

Contribution

It develops a novel high order DG-IMEX scheme with micro-macro reformulation for kinetic equations, ensuring asymptotic preservation and high order accuracy in the diffusive limit.

Findings

Scheme achieves high order accuracy in numerical tests.

Method remains stable across different regimes.

Accurately captures asymptotic limits as epsilon approaches zero.

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 spatial discretization of arbitrary order of accuracy with suitable numerical fluxes; high order globally stiffly accurate implicit-explicit 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 epsilon -> 0 is an explicit, consistent and 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.