17 research outputs found
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
Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design
Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first-order reductions of the Einstein field equations, or a novel first-order hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENO-like schemes for PDEs that support a curl-preserving involution. (Some insights into discontinuous Galerkin (DG) schemes are also drawn, though that is not the prime focus of this paper.) This is done for two- and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. In two dimensions, a von Neumann analysis of structure-preserving WENO-like schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems. Numerical results are also presented to show that the edge-centered curl-preserving (ECCP) schemes meet their design accuracy. This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy. By its very design, this paper is, therefore, intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs