High order asymptotic preserving discontinuous Galerkin methods for gray radiative transfer equations

TL;DR

This paper develops high order asymptotic preserving discontinuous Galerkin methods for nonlinear gray radiative transfer equations, enabling larger time steps and efficient solutions while maintaining accuracy in different regimes.

Contribution

It introduces a novel high order AP DG scheme for nonlinear GRTEs using micro-macro decomposition and a Picard iteration, improving stability and efficiency.

Findings

Scheme is asymptotic preserving and accurate.

Allows hyperbolic time stepping without photon mean free path restrictions.

Numerical tests confirm high order accuracy and efficiency.

Abstract

In this paper, we will develop a class of high order asymptotic preserving (AP) discontinuous Galerkin (DG) methods for nonlinear time-dependent gray radiative transfer equations (GRTEs). Inspired by the work \cite{Peng2020stability}, in which stability enhanced high order AP DG methods are proposed for linear transport equations, we propose to pernalize the nonlinear GRTEs under the micro-macro decomposition framework by adding a weighted linear diffusive term. In the diffusive limit, a hyperbolic, namely $\Delta t=\mathcal{O}(h)$ where $\Delta t$ and $h$ are the time step and mesh size respectively, instead of parabolic $\Delta t=\mathcal{O}(h^2)$ time step restriction is obtained, which is also free from the photon mean free path. The main new ingredient is that we further employ a Picard iteration with a predictor-corrector procedure, to decouple the resulting global nonlinear system to a linear system with local nonlinear algebraic equations from an outer iterative loop. Our scheme is shown to be asymptotic preserving and asymptotically accurate. Numerical tests for one and two spatial dimensional problems are performed to demonstrate that our scheme is of high order, effective and efficient.