Discontinuous Galerkin (DG) methods have become mainstays in the accurate solution of hyperbolic systems, which suggests that they should also be important for computational electrodynamics (CED). Typically DG schemes are coupled with Runge-Kutta timestepping, resulting in RKDG schemes, which are also sometimes called DGTD schemes in the CED community. However, Maxwell's equations, which are solved in CED codes, have global mimetic constraints. In Balsara and Käppeli [von Neumann Stability Analysis of Globally Constraint-Preserving DGTD and PNPM Schemes for the Maxwell Equations using Multidimensional Riemann Solvers, Journal of Computational Physics, 376 (2019) 1108-1137] the authors presented globally constraint-preserving DGTD schemes for CED. The resulting schemes had excellent low dissipation and low dispersion properties. Their one deficiency was that the maximal permissible CFL of DGTD schemes decreased with increasing order of accuracy. The goal of this paper is to show how this deficiency is overcome. Because CED entails the propagation of electromagnetic waves, we would also like to obtain DG schemes for CED that minimize dissipation and dispersion errors even more than the prior generation of DGTD schemes.

Two recent advances make this possible. The first advance, which has been reported elsewhere, is the development of a multidimensional Generalized Riemann Problem (GRP) solver. The second advance relates to the use of Two Derivative Runge Kutta (TDRK) timestepping. This timestepping uses not just the solution of the multidimensional Riemann problem, it also uses the solution of the multidimensional GRP. When these two advances are melded together, we arrive at DG(TD)² schemes for CED, where the first “TD” stands for time-derivative and the second “TD” stands for the TDRK timestepping. The first goal of this paper is to show how DG(TD)² schemes for CED can be formulated with the help of the multidimensional GRP and TDRK timestepping. The second goal of this paper is to utilize the free parameters in TDRK timestepping to arrive at DG(TD)² schemes for CED that offer a uniformly large CFL with increasing order of accuracy while minimizing the dissipation and dispersion errors to exceptionally low values. The third goal of this paper is to document a von Neumann stability analysis of DG(TD)² schemes so that their dissipation and dispersion properties can be quantified and studied in detail.

At second order we find a DG(TD)² scheme with CFL of 0.25 and improved dissipation and dispersion properties; for a second order scheme. At third order we present a novel DG(TD)² scheme with CFL of 0.2571 and improved dissipation and dispersion properties; for a third order scheme. At fourth order we find a DG(TD)² scheme with CFL of 0.2322 and improved dissipation and dispersion properties. As an extra benefit, the resulting DG(TD)² schemes for CED require fewer synchronization steps on parallel supercomputers than comparable DGTD schemes for CED. We also document some test problems to show that the methods achieve their design accuracy.

不连续Galerkin（DG）方法已成为双曲系统精确解决方案的主要手段，这表明它们对于计算电动力学（CED）也应该很重要。通常，DG方案与Runge-Kutta时间步长结合使用，从而产生RKDG方案，在CED社区中有时也称为DGTD方案。但是，在CED代码中求解的麦克斯韦方程组具有全局模拟约束。在Balsara和Käppeli[冯·诺依曼中，采用多维Riemann解算器的麦克斯韦方程组的全局约束保持DGTD和PNPM格式的稳定性分析，Journal of Computational Physics，376（2019）1108-1137]作者介绍了CED的全局约束保留DGTD方案。所得方案具有优异的低耗散性和低分散性。他们的缺点之一是DGTD方案的最大允许CFL随着精度的提高而降低。本文的目的是说明如何克服这一缺陷。因为CED需要电磁波的传播，所以我们还希望获得比CDG的DG方案更能使耗散和色散误差最小化的技术，甚至比上一代DGTD方案还要多。

最近的两项进步使这成为可能。在其他地方已有报道的第一个进展是多维广义Riemann问题（GRP）求解器的开发。第二个进步与两个导数Runge Kutta（TDRK）时间步的使用有关。这个时间步伐不仅使用多维黎曼问题的解决方案，还使用多维GRP的解决方案。当这两项技术融合在一起时，我们得出了CED的DG（TD）²方案，其中第一个“ TD”代表时间导数，第二个“ TD”代表TDRK时间步长。本文的首要目标是展示如何借助多维GRP和TDRK时间步长来制定CED的DG（TD）²方案。的本文的第二个目标是利用TDRK时间步长中的自由参数得出CED的DG（TD）²方案，该方案可提供均匀大的CFL，并具有递增的精度，同时将耗散和色散误差最小化到极低的值。本文的第三个目标是记录DG（TD）²方案的冯·诺伊曼稳定性分析，以便可以对它们的耗散和色散特性进行量化和详细研究。

在二阶时，我们发现DG（TD）²方案的CFL为0.25，并改善了耗散和色散特性。用于二阶方案。在三阶中，我们提出了一种新颖的DG（TD）²方案，其CFL为0.2571，并改善了耗散和色散特性。三阶方案。在四阶时，我们发现DG（TD）²方案的CFL为0.2322，并改善了耗散和色散特性。作为一个额外的好处，与用于CED的同类DGTD方案相比，用于CED的DG（TD）²方案在并行超级计算机上所需的同步步骤更少。我们还记录了一些测试问题，以表明该方法达到了设计精度。