SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A2230-A2261, January 2020.
This paper presents entropy symmetrization and high-order accurate entropy stable schemes for the relativistic magnetohydrodynamic (RMHD) equations. It is shown that the conservative RMHD equations are not symmetrizable and do not admit a thermodynamic entropy pair. To address this issue, a symmetrizable RMHD system, equipped with a convex thermodynamic entropy pair, is first proposed by adding a source term into the equations, providing an analogue to the nonrelativistic Godunov--Powell system. Arbitrarily high-order accurate entropy stable finite difference schemes are developed on Cartesian meshes based on the symmetrizable RMHD system. The crucial ingredients of these schemes include (i) affordable explicit entropy conservative fluxes which are technically derived through carefully selected parameter variables, (ii) a special high-order discretization of the source term in the symmetrizable RMHD system, and (iii) suitable high-order dissipative operators based on essentially nonoscillatory reconstruction to ensure the entropy stability. Several numerical tests demonstrate the accuracy and robustness of the proposed entropy stable schemes.

中文翻译：

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相对论MHD方程的熵对称和高阶精确熵稳定数值格式

SIAM科学计算杂志，第42卷，第4期，第A2230-A2261页，2020年1月。
本文提出了相对论磁流体动力学（RMHD）方程的熵对称和高阶精确熵稳定方案。结果表明，保守的RMHD方程是不可对称的，并且不允许热力学熵对。为了解决这个问题，首先提出了一个带有凸热力学熵对的可对称RMHD系统，方法是将源项添加到方程中，为非相对论的Godunov-Powell系统提供一个类似物。基于对称RMHD系统，在笛卡尔网格上开发了任意高阶精确熵稳定有限差分格式。这些方案的关键要素包括（i）价格合理的显式熵保守通量，这些通量是通过精心选择的参数变量从技术上得出的，（ii）可对称RMHD系统中源项的特殊高阶离散化，以及（iii）基于本质上非振荡重建的合适高阶耗散算符，以确保熵的稳定性。几个数值测试证明了所提出的熵稳定方案的准确性和鲁棒性。