Asymptotic preserving IMEX-DG-S schemes for linear kinetic transport equations based on Schur complement

TL;DR

This paper introduces a new family of asymptotic preserving schemes for linear kinetic transport equations that avoid problem-dependent reformulations, ensuring stability and efficiency across regimes.

Contribution

The paper proposes IMEX-DG-S schemes that directly solve the micro-macro system using Schur complement, providing a stable, efficient, and reformulation-free approach for diffusive limits.

Findings

Schemes are formally shown to be asymptotic preserving.

Energy and Fourier stability analyses confirm uniform stability.

Numerical examples demonstrate the schemes' effectiveness.

Abstract

We consider a linear kinetic transport equation under a diffusive scaling, that converges to a diffusion equation as the Knudsen number $\varepsilon\rightarrow0$. In [3, 21], to achieve the asymptotic preserving (AP) property and unconditional stability in the diffusive regime with $\varepsilon\ll 1$, numerical schemes are developed based on an additional reformulation of the even-odd or micro-macro decomposed version of the equation. The key of the reformulation is to add a weighted diffusive term on both sides of one equation in the decomposed system. The choice of the weight function, however, is problem-dependent and ad-hoc, and it can affect the performance of numerical simulations. To avoid issues related to the choice of the weight function and still obtain the AP property and unconditional stability in the diffusive regime, we propose in this paper a new family of AP schemes, termed as IMEX-DG-S schemes, directly solving the micro-macro decomposed system without any further reformulation. The main ingredients of the IMEX-DG-S schemes include globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta (RK) temporal discretizations with a new IMEX strategy, discontinuous Galerkin (DG) spatial discretizations, discrete ordinate methods for the velocity space, and the application of the Schur complement to the algebraic form of the schemes to control the overall computational cost. The AP property of the schemes is shown formally. With an energy type stability analysis applied to the first order scheme, and Fourier type stability analysis applied to the first to third order schemes, we confirm the uniform stability of the methods with respect to $\varepsilon$ and the unconditional stability in the diffusive regime. A series of numerical examples are presented to demonstrate the performance of the new schemes.