Von Neumann stability analysis of globally divergence-free RKDG schemes for the induction equation using multidimensional Riemann solvers
Von Neumann stability analysis of globally divergence-free RKDG schemes for the induction equation using multidimensional Riemann solvers
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DOI：
10.1016/j.jcp.2017.01.056
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发表时间：
2017-05-01
影响因子：
4.1
通讯作者：
Kaeppeli, Roger
中科院分区：
文献类型：
--
作者：
Balsara, Dinshaw S.;Kaeppeli, Roger
本文重点研究了全局无发散的类龙格-库塔不连续伽辽金格式的感应方程的数值解。感应方程在数值MHD和其他类似系统中起着重要作用。它确保磁场以无散度的方式演化；这里给出的数值格式也有相同的性质。本文提出的算法是基于一种新的类似dg的方法，因为它适用于网格面中的磁场分量。（也就是说，这不是守恒定律的传统DG算法。）该方法的另外两个新组成部分包括磁场的无散度重建和多维黎曼解算；这两种方法都是近年来由第一作者提出的。由于该方法是线性的，因此在二维上进行了von Neumann稳定性分析，以了解其稳定性特性。我们在本文中发展的冯诺伊曼稳定性分析依赖于从模态到节点DG公式的转录，以便为节点值开发离散进化方程。然后将它们与合适的龙格-库塔时间步进策略耦合，以便可以分析整个方案的稳定性，该方案在空间和时间上都是合适的高阶。我们表明，我们的方案允许CFL数与传统的RKDG方案相当。我们还分析了该方法的波传播特性，并表明随着精度的增加，波的传播变得更加各向同性，并且在更大范围的长波模式下没有耗散。这为投资于高阶方法提供了强有力的理由。我们还使用von Neumann稳定性分析来证明无散度重建和多维黎曼解是全局无散度rkdg方案的基本算法成分。给出了类rkdg格式的数值精度分析，并与PNPM格式的精度进行了比较。发现PNPM在允许更大的CFL数的同时恢复了RKDG-like方案的大部分准确性。(C) 2017爱思唯尔公司版权所有。
In this paper we focus on the numerical solution of the induction equation using Runge-Kutta Discontinuous Galerkin (RKDG)-like schemes that are globally divergence-free. The induction equation plays a role in numerical MHD and other systems like it. It ensures that the magnetic field evolves in a divergence-free fashion; and that same property is shared by the numerical schemes presented here. The algorithms presented here are based on a novel DG-like method as it applies to the magnetic field components in the faces of a mesh. (I.e., this is not a conventional DG algorithm for conservation laws.) The other two novel building blocks of the method include divergence-free reconstruction of the magnetic field and multidimensional Riemann solvers; both of which have been developed in recent years by the first author.Since the method is linear, a von Neumann stability analysis is carried out in two dimensions to understand its stability properties. The von Neumann stability analysis that we develop in this paper relies on transcribing from a modal to a nodal DG formulation in order to develop discrete evolutionary equations for the nodal values. These are then coupled to a suitable Runge-Kutta timestepping strategy so that one can analyze the stability of the entire scheme which is suitably high order in space and time.We show that our scheme permits CFL numbers that are comparable to those of traditional RKDG schemes. We also analyze the wave propagation characteristics of the method and show that with increasing order of accuracy the wave propagation becomes more isotropic and free of dissipation for a larger range of long wavelength modes. This makes a strong case for investing in higher order methods. We also use the von Neumann stability analysis to show that the divergence-free reconstruction and multidimensional Riemann solvers are essential algorithmic ingredients of a globally divergence-free RKDG-like scheme.Numerical accuracy analyses of the RKDG-like schemes are presented and compared with the accuracy of PNPM schemes. It is found that PNPM retrieve much of the accuracy of the RKDG-like schemes while permitting a larger CFL number. (C) 2017 Elsevier Inc. All rights reserved.