High Order Hierarchical Asymptotic Preserving Nodal Discontinuous Galerkin IMEX Schemes For The BGK Equation

TL;DR

This paper introduces a high-order hierarchical asymptotic preserving nodal discontinuous Galerkin IMEX scheme for the BGK equation, seamlessly integrating kinetic and hydrodynamic regimes with significant computational savings.

Contribution

It develops a hierarchy scheme within the NDG-IMEX framework that unifies kinetic, Euler, and Navier-Stokes regimes without hybridization, improving efficiency and accuracy.

Findings

Scheme accurately captures all three regimes.

Numerical results show significant computational savings.

Method demonstrates good performance across regimes.

Abstract

A class of high order asymptotic preserving (AP) schemes has been developed for the BGK equation in Xiong et. al. (2015) [37], which is based on the micro-macro formulation of the equation. The nodal discontinuous Galerkin (NDG) method with Lagrangian basis functions for spatial discretization and globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta (RK) scheme as time discretization are introduced with asymptotic preserving properties. However, it is only necessary to solve the kinetic equation when the hydrodynamic description breaks down. Motivated by the recent work in Filbet and Rey (2015) [23], it is more naturally to construct a hierarchy scheme under the NDG-IMEX framework without hybridization, as the formal analysis in [37] shows that when $\epsilon$ is small, the NDG-IMEX scheme becomes a local discontinuous Galerkin (LDG) scheme for the compressible Navier-Stokes equations, and when $\epsilon=0$ it is a discontinuous Galerkin (DG) scheme for the compressible Euler equations. Moveover, we propose to combine the kinetic regime with the hydrodynamic regime including both the compressible Euler and Navier-Stokes equations. Numerical experiments demonstrate very decent performance of the new approach. In our numerics, all three regimes are clearly divided, leading to great savings in terms of the computational cost.