Stability of a leap-frog discontinuous Galerkin method for time-domain Maxwell's equations in anisotropic materials

TL;DR

This paper analyzes the stability of a leap-frog discontinuous Galerkin method for solving time-domain Maxwell's equations in anisotropic materials, providing stability conditions and numerical validation.

Contribution

It introduces a stability condition for an explicit leap-frog DG method applied to anisotropic Maxwell's equations, considering boundary conditions and polynomial degrees.

Findings

Derived stability bounds depending on mesh size, flux choice, and polynomial degree.

Validated stability conditions with numerical experiments.

Highlighted the impact of anisotropic permittivity on numerical stability.

Abstract

In this work we discuss the numerical discretization of the time-dependent Maxwell's equations using a fully explicit leap-frog type discontinuous Galerkin method. We present a sufficient condition for the stability, for cases of typical boundary conditions, either perfect electric, perfect magnetic or first order Silver-M\"uller. The bounds of the stability region point out the influence of not only the mesh size but also the dependence on the choice of the numerical flux and the degree of the polynomials used in the construction of the finite element space, making possible to balance accuracy and computational efficiency. In the model we consider heterogeneous anisotropic permittivity tensors which arise naturally in many applications of interest. Numerical results supporting the analysis are provided.