von Neumann Stability Analysis of Globally Constraint-Preserving DGTD
and PNPM Schemes for the Maxwell Equations using Multidimensional Riemann
Solvers
The time-dependent equations of computational electrodynamics (CED) are
evolved consistent with the divergence constraints. As a result, there has been
a recent effort to design finite volume time domain (FVTD) and discontinuous
Galerkin time domain (DGTD) schemes that satisfy the same constraints and,
nevertheless, draw on recent advances in higher order Godunov methods. This
paper catalogues the first step in the design of globally constraint-preserving
DGTD schemes. The algorithms presented here are based on a novel DG-like method
that is applied to a Yee-type staggering of the electromagnetic field variables
in the faces of the mesh. The other two novel building blocks of the method
include constraint-preserving reconstruction of the electromagnetic fields and
multidimensional Riemann solvers; both of which have been developed in recent
years by the first author. We carry out a von Neumann stability analysis of the
entire suite of DGTD schemes for CED at orders of accuracy ranging from second
to fourth. A von Neumann stability analysis gives us the maximal CFL numbers
that can be sustained by the DGTD schemes presented here at all orders. It also
enables us to understand the wave propagation characteristics of the schemes in
various directions on a Cartesian mesh. We find that the CFL of DGTD schemes
decreases with increasing order. To counteract that, we also present
constraint-preserving PNPM schemes for CED. We find that the third and fourth
order constraint-preserving DGTD and P1PM schemes have some extremely
attractive properties when it comes to low-dispersion, low-dissipation
propagation of electromagnetic waves in multidimensions. Numerical accuracy
tests are also provided to support the von Neumann stability analysis