欧拉和磁流体力学ADER格式在结构网格上的高效实现--与Runge-Kutta方法的比较
Efficient Implementation of ADER Schemes for Euler and Magnetohydrodynamical Flows on Structured Meshes -- Comparison with Runge-Kutta Methods
摘要：
ADER（Arbitrary Derivative in Space and Time）方法用于求解双曲守恒律的时间演化问题，最近引起了人们的兴趣。ADER时间更新可以在单个步骤中执行，这在许多应用中是期望的。然而，以前的论文集中在理论上，而忽视了实现细节。本文的目的是使ADER计划访问提供两个有用的配方的方法，以及它们的实施细节的三维结构网格。因此，我们提供了一个详细的制定ADER计划的守恒律与非刚性源项在节点以及模态空间以及有用的实施相关的细节。我们还提供了详细的有效利用ADER计划在获得守恒律的数值通量以及电场的发散自由磁流体力学。本文还提出了一种基于WENO的从MHD的面心磁场获得区域平均磁场的有效方法。这里列出的方案已在第一作者的RIEMANN代码中实现。ADER计划的速度几乎是两倍的速度，强稳定性保持Runge-Kutta时间步进计划的精度，我们测试的所有订单。
英文摘要：
ADER (Arbitrary DERivative in space and time) methods for the time-evolution of hyperbolic conservation laws have recently generated a fair bit of interest. The ADER time update can be carried out in a single step, which is desirable in many applications. However, prior papers have focused on the theory while downplaying implementation details. The purpose of the present paper is to make ADER schemes accessible by providing two useful formulations of the method as well as their implementation details on three-dimensional structured meshes. We therefore provide a detailed formulation of ADER schemes for conservation laws with non-stiff source terms in nodal as well as modal space along with useful implementation-related detail. We also provide details for the efficient use of ADER schemes in obtaining the numerical flux for conservation laws as well as electric fields for divergence-free magnetohydrodynamics. An efficient WENO-based strategy for obtaining zone-averaged magnetic fields from face-centered magnetic fields in MHD is also presented. The schemes catalogued here have been implemented in the first author's RIEMANN code. The speed of ADER schemes is shown to be almost twice as fast as that of strong stability preserving Runge-Kutta time stepping schemes for all the orders of accuracy that we tested.
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