Optimal, globally constraint-preserving, DG(TD)2 schemes for computational electrodynamics based on two-derivative Runge-Kutta timestepping and multidimensional generalized Riemann problem solvers – A von Neumann stability analysis

Abstract

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.