A Comparative Study of Limiting Strategies in Discontinuous Galerkin Schemes for the $M_1$ Model of Radiation Transport

TL;DR

This paper evaluates different limiting strategies in high-order Discontinuous Galerkin schemes for the $M_1$ radiation transport model, focusing on maintaining realizability and reducing oscillations in 2D simulations.

Contribution

It extends a realizability limiting strategy to two dimensions and compares its effectiveness with TVBM limiters in DG schemes for the $M_1$ model.

Findings

Combination of realizability and TVBM limiters yields robust, accurate solutions.

Realizability limiter alone is insufficient for stability and accuracy.

Numerical results demonstrate improved solution quality with combined limiting strategies.

Abstract

The $M_1$ minimum entropy moment system is a system of hyperbolic balance laws that approximates the radiation transport equation, and has many desirable properties. Among them are symmetric hyperbolicity, entropy decay, moment realizability, and correct behavior in the diffusion and free-streaming limits. However, numerical difficulties arise when approximating the solution of the $M_1$ model by high order numerical schemes; namely maintaining the realizability of the numerical solution and controlling spurious oscillations. In this paper, we extend a previously constructed one-dimensional realizability limiting strategy to 2D. In addition, we perform a numerical study of various combinations of the realizability limiter and the TVBM local slope limiter on a third order Discontinuous Galerkin (DG) scheme on both triangular and rectangular meshes. In several test cases, we demonstrate that in general, a combination of the realizability limiter and a TVBM limiter is necessary to obtain a robust and accurate numerical scheme. Our code is published so that all results can be reproduced by the reader.