电阻性相对论MHD方程的非结构网格甚高阶$\PNM$格式
Very High Order $\PNM$ Schemes on Unstructured Meshes for the Resistive Relativistic MHD Equations
摘要：
本文提出了一种在空间和时间上均优于二阶精度的方法，用于在多维非结构网格上数值求解电阻性相对论磁流体动力学（RRMHD）方程。所考虑的非线性系统是纯双曲的，并包含一个源项，一个电场的演变，成为刚性的电阻率的低值。对于空间离散，我们建议使用高阶$\PNM$格式，如在\cite{Dumbser2008}中介绍的双曲守恒律和一个高精度的不分裂的时间离散实现使用元素局部时空间断Galerkin方法提出的\cite{DumbserEnauxToro}的一维平衡律刚性源项。通过Dedner等人的\cite{Dedneretal}的发散清除程序，解释了磁场的无发散特性。为了验证我们的高阶方法，我们首先解决了一些数值测试的情况下，精确的分析参考解决方案是已知的，我们还显示了数值收敛研究的刚性极限的RRMHD方程使用$\PNM$计划从第三到第五阶的空间和时间的准确性。我们还提出了一些应用程序与冲击波，如一个经典的激波管问题，不同的电导率值，以及相对论MHD转子问题和相对论等效的Orszag-Tang涡问题。我们已经证明，所提出的方法可以同样很好地处理电阻区和刚性极限的理想相对论磁流体。由于这些原因，它提供了一个强大的工具，相对论天体物理模拟涉及的外观磁重联。
英文摘要：
In this paper we propose the first better than second order accurate method in space and time for the numerical solution of the resistive relativistic magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space dimensions. The nonlinear system under consideration is purely hyperbolic and contains a source term, the one for the evolution of the electric field, that becomes stiff for low values of the resistivity. For the spatial discretization we propose to use high order $\PNM$ schemes as introduced in \cite{Dumbser2008} for hyperbolic conservation laws and a high order accurate unsplit time discretization is achieved using the element-local space-time discontinuous Galerkin approach proposed in \cite{DumbserEnauxToro} for one-dimensional balance laws with stiff source terms. The divergence free character of the magnetic field is accounted for through the divergence cleaning procedure of Dedner et al. \cite{Dedneretal}. To validate our high order method we first solve some numerical test cases for which exact analytical reference solutions are known and we also show numerical convergence studies in the stiff limit of the RRMHD equations using $\PNM$ schemes from third to fifth order of accuracy in space and time. We also present some applications with shock waves such as a classical shock tube problem with different values for the conductivity as well as a relativistic MHD rotor problem and the relativistic equivalent of the Orszag-Tang vortex problem. We have verified that the proposed method can handle equally well the resistive regime and the stiff limit of ideal relativistic MHD. For these reasons it provides a powerful tool for relativistic astrophysical simulations involving the appearance of magnetic reconnection.
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