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

Prince Chidyagwai, Martin Frank+2 more|arXiv (Cornell University)|Jun 30, 2017

Computational Fluid Dynamics and Aerodynamics48 references3 citations

TL;DR

This paper extends a one-dimensional realizability limiter to 2D for discontinuous Galerkin schemes solving the $M_1$ radiation transport model. It demonstrates that combining the realizability limiter with a TVBM slope limiter is essential for maintaining solution realizability and suppressing spurious oscillations in third-order DG schemes on both triangular and rectangular meshes.

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.

Motivation & Objective

• To address numerical instabilities in high-order discontinuous Galerkin schemes for the $M_1$ radiation transport model, particularly loss of realizability and spurious oscillations.
• To extend a previously developed one-dimensional realizability limiting strategy to two-dimensional problems.
• To evaluate the performance of various combinations of realizability and TVBM slope limiters in maintaining solution accuracy and stability.
• To ensure the numerical scheme preserves key physical properties such as entropy decay and correct behavior in diffusion and free-streaming limits.
• To provide open-source code to enable reproducibility of all reported results.

Proposed method

• Adapting a one-dimensional realizability limiter to two-dimensional triangular and rectangular meshes using local reconstruction techniques.
• Implementing a third-order discontinuous Galerkin spatial discretization for the $M_1$ moment system.
• Applying a TVBM (Total Variation Bounded Minmod) slope limiter to control spurious oscillations in element-wise polynomial reconstructions.
• Combining the realizability limiter with the TVBM limiter to enforce both realizability and bounded variation in the solution.
• Using a predictor-corrector time integration scheme to advance the solution in time while preserving stability.
• Validating the scheme on benchmark test cases to assess accuracy, realizability, and oscillation control.

Experimental results

Research questions

• RQ1 How can a one-dimensional realizability limiter be effectively extended to two-dimensional triangular and rectangular meshes?
• RQ2 What is the relative contribution of the realizability limiter versus the TVBM slope limiter in maintaining solution stability and accuracy?
• RQ3 Can a combination of both limiters ensure robustness across diverse test cases, including those with strong gradients and discontinuities?
• RQ4 How do the different limiter combinations affect the preservation of physical properties such as entropy decay and realizability?
• RQ5 To what extent does the choice of mesh type (triangular vs. rectangular) influence the performance of the limiting strategy?

Key findings

• The extension of the one-dimensional realizability limiter to two dimensions successfully maintains moment realizability on both triangular and rectangular meshes.
• Using only the realizability limiter leads to persistent spurious oscillations in regions with steep gradients, indicating insufficient control over solution slopes.
• The TVBM slope limiter alone fails to preserve realizability in cases with high-order polynomial approximations and strong radiation fields.
• A combined strategy of realizability and TVBM limiters consistently produces stable, oscillation-free, and physically realizable solutions across all test cases.
• The proposed scheme maintains correct asymptotic behavior in both the diffusion and free-streaming limits, confirming the preservation of key physical properties.
• The open-source implementation enables full reproducibility of all numerical results, supporting transparency and further research.