Making a Synthesis of FDTD and DGTD Schemes for Computational Electromagnetics

Abstract

A novel class of discontinuous Galerkin time-domain (DGTD) schemes, invented by the first author, are presented that are capable of globally preserving the constraints that are inherent in Maxwell's equations. The methods share the same Yee-type mesh structure as finite-difference time-domain (FDTD) schemes for computational electrodynamics. Since FDTD schemes also preserve global constraints, the novelty of this work consists of making a synthesis of FDTD and DGTD schemes. While previous DG methods were based on applying identities involving Gauss’ law in weak form to the volumetric elements of a mesh, the newer methods are based on applying identities involving Stokes’ law in weak form to the facial elements of the mesh. This fundamental paradigm shift is crucial for obtaining the globally constraint-preserving DGTD methods in this paper. The new DGTD methods meet their design accuracies. The more accurate schemes are indeed more accurate even at the lowest resolutions. Moreover, as the mesh is refined, the schemes reach their design accuracies much faster. These benefits are all attributable to the subcell resolving ability of the DGTD schemes presented here. The higher order methods offer the lowest time to solution, especially when very high accuracies are demanded. Excellent scalability is also demonstrated.