A Comparative Study of Limiting Strategies in Discontinuous Galerkin Schemes for the $M_1$ Model of Radiation Transport
This work addresses numerical stability and accuracy for high-order simulations of radiation transport, but is incremental as it extends existing limiters to 2D and performs comparative tests.
The paper extends a 1D realizability limiting strategy to 2D for the M1 model of radiation transport and compares combinations of realizability and TVBM limiters on a third-order DG scheme. Results show that combining both limiters is necessary for robustness and accuracy.
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.