Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations
Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations
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DOI：
10.1007/s00211-021-01209-4
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发表时间：
2020-02
影响因子：
2.1
通讯作者：
Kailiang Wu;Chi-Wang Shu
中科院分区：
文献类型：
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作者：
Kailiang Wu;Chi-Wang Shu
我们提出并分析了一类鲁棒的、均匀高阶精确的不连续伽辽金（DG）方案，用于通用网格上的多维相对论磁流体动力学（RMHD）。该方案的一个显着特征是其物理约束保持（PCP）特性，即它们被证明可以保持对流体速度的亚光速约束以及密度、压力和内能的正性。这是首次针对多维 RMHD 实现可证明的 PCP 高阶方案。为 RMHD 开发 PCP 高阶方案是非常理想的，但仍然是一项具有挑战性的任务，特别是在多维情况下，由于约束中固有的强非线性和无磁散度条件的影响。受偏微分方程层面的一些关键观察的启发，我们通过使用最近提出的可对称 RMHD 方程的局部无散 DG 方案作为基本方案、强制 DG 解的 PCP 性质的限制技术以及用于时间离散化的强稳定性保持方法来构建可证明的 PCP 方案。我们通过使用新颖的“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发散误差的影响，并使用复杂的估计来分析可对称 RMHD 系统中附加源项的有益效果，严格证明了 PCP 属性。提供了几个二维数值例子来进一步证实 PCP 属性并证明所提出的 PCP 方案的准确性、有效性和鲁棒性。
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.