Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations

• Published in: Numerische Mathematik 2021-07, Vol.148 (3), p.699-741
• Main Authors: Wu, Kailiang, Shu, Chi-Wang
• Format: Article
• Language: English
• Subjects: Computational fluid dynamics; Divergence; Error analysis; Galerkin method; Internal energy; Magnetohydrodynamics; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Multidimensional methods; Nonlinearity; Numerical Analysis; Numerical and Computational Physics; Relativistic effects; Robustness (mathematics); Simulation; Theoretical
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-021-01209-4

We propose and analyze a class of robust, uniformly high-order accurate discontinuous Galerkin (DG) schemes for multidimensional relativistic magnetohydrodynamics (RMHD) on general meshes. A distinct feature of the schemes is their physical-constraint-preserving (PCP) property, i.e., they are proven to preserve the subluminal constraint on the fluid velocity and the positivity of density, pressure, and internal energy. This is the first time that provably PCP high-order schemes are achieved for multidimensional RMHD. Developing PCP high-order schemes for RMHD is highly desirable but remains a challenging task, especially in the multidimensional cases, due to the inherent strong nonlinearity in the constraints and the effect of the magnetic divergence-free condition. Inspired by some crucial observations at the PDE level, we construct the provably PCP schemes by using the locally divergence-free DG schemes of the recently proposed symmetrizable RMHD equations as the base schemes, a limiting technique to enforce the PCP property of the DG solutions, and the strong-stability-preserving methods for time discretization. We rigorously prove the PCP property by using a novel “quasi-linearization” approach to handle the highly nonlinear physical constraints, technical splitting to offset the influence of divergence error, and sophisticated estimates to analyze the beneficial effect of the additional source term in the symmetrizable RMHD system. Several two-dimensional numerical examples are provided to further confirm the PCP property and to demonstrate the accuracy, effectiveness and robustness of the proposed PCP schemes.