GQL-Based Bound-Preserving and Locally Divergence-Free Central Discontinuous Galerkin Schemes for Relativistic Magnetohydrodynamics
Shengrong Ding, Kailiang Wu
TL;DR
This work addresses the challenge of simulating relativistic magnetohydrodynamics while strictly preserving physical bounds and the magnetic divergence-free constraint. It develops high-order central discontinuous Galerkin schemes that are provably bound-preserving and locally divergence-free by discretizing a symmetrizable modified RMHD system and employing geometric quasilinearization (GQL) to manage nonlinear admissibility constraints. A novel 2D cell-average decomposition (CAD) on overlapping meshes, combined with carefully discretized symmetrization source terms, yields a locally coupled BP mechanism with milder CFL conditions and improved efficiency. Comprehensive 1D and 2D RMHD tests across multiple EOSs demonstrate the schemes’ accuracy, robustness for ultra-relativistic and strongly magnetized flows, and practical relevance for astrophysical and laboratory plasmas.
Abstract
This paper develops novel and robust central discontinuous Galerkin (CDG) schemes of arbitrarily high-order accuracy for special relativistic magnetohydrodynamics (RMHD) with a general equation of state (EOS). These schemes are provably bound-preserving (BP), i.e., consistently preserve the upper bound for subluminal fluid velocity and the positivity of density and pressure, while also (locally) maintaining the divergence-free (DF) constraint for the magnetic field. For 1D RMHD, the standard CDG method is exactly DF, and its BP property is proven under a condition achievable by BP limiter. For 2D RMHD, we design provably BP and locally DF CDG schemes based on the suitable discretization of a modified RMHD system. A key novelty in our schemes is the discretization of additional source terms in the modified RMHD equations, so as to precisely counteract the influence of divergence errors on the BP property across overlapping meshes. We provide rigorous proofs of the BP property for our CDG schemes and first establish the theoretical connection between BP and discrete DF properties on overlapping meshes for RMHD. Owing to the absence of explicit expressions for primitive variables in terms of conserved variables, the constraints of physical bounds are strongly nonlinear, making the BP proofs highly nontrivial. We overcome these challenges through technical estimates within the geometric quasilinearization (GQL) framework, which converts the nonlinear constraints into linear ones. Furthermore, we introduce a new 2D cell average decomposition on overlapping meshes, which relaxes the theoretical BP CFL constraint and reduces the number of internal nodes, thereby enhancing the efficiency of the 2D BP CDG method. We implement the proposed CDG schemes for extensive RMHD problems with various EOSs, demonstrating their robustness and effectiveness in challenging scenarios.