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.

我们提出并分析了一类用于一般网格上的多维相对论磁流体动力学 (RMHD) 的稳健、一致的高阶精确不连续伽辽金 (DG) 方案。这些方案的一个显着特点是它们的物理约束保持 (PCP) 特性，即它们被证明可以保持对流体速度的亚腔约束以及密度、压力和内能的正性。这是第一次为多维 RMHD 实现可证明的 PCP 高阶方案。为 RMHD 开发 PCP 高阶方案是非常可取的，但仍然是一项具有挑战性的任务，尤其是在多维情况下，由于约束中固有的强非线性和无磁发散条件的影响。受到 PDE 级别的一些重要观察的启发，我们通过使用最近提出的对称 RMHD 方程的局部无发散 DG 方案作为基本方案、一种强制执行 DG 解的 PCP 性质的限制技术以及时间的强稳定性保持方法来构建可证明的 PCP 方案离散化。我们通过使用一种新颖的“准线性化”方法来处理高度非线性的物理约束，使用技术分裂来抵消散度误差的影响，以及使用复杂的估计来分析对称化中附加源项的有益影响，从而严格证明了 PCP 属性。 RMHD 系统。提供了几个二维数值例子，以进一步确认 PCP 特性，并证明所提出的 PCP 方案的准确性、有效性和鲁棒性。