Stability of a Leap-Frog Discontinuous Galerkin Method for Time-Domain Maxwell's Equations in Anisotropic Materials

Abstract

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-Müller. 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.

Convergence of an explicit iterative leap-frog discontinuous Galerkin method for time-domain Maxwell’s equations in anisotropic materials

“…holds, where C is a generic positive constant independent of ∆t and the mesh size h. Proof: Follows the steps of the proof of Theorem 4.2 in [1].…”

“…At the outer cell boundaries we set Z + = Z − . The coupling between elements is introduced via the numerical flux as in [1].…”

“…The leap-frog DG method in anisotropic materials which is discussed in [4] leads to a locally implicit method for the case of SM-ABC. In [1] a fully explicit in time leap-frog DG method is investigated in the same framework. The error estimates derived therein show that the method is only of first order convergent in time when SM-ABC are considered.…”

“…In the present work we propose an iterative predictor-corrector method based on the explicit method investigated in [1], resulting a fully explicit method that is second order convergent in time for the SM-ABC case. In the Section 2 we prove that the explicit iterative method converges to a second order in time implicit method and we deduce the a priori error estimates for the fully discrete scheme.…”

“…The unknowns related to the electric field are approximated at integer time-stations t m and are denoted by Êm k = Êk (., t m ). The unknowns related to the magnetic field are approximated at half-integer timestations t m+1/2 = (m + 1 2 )∆t and are denoted by Ĥm+1/2 k = Ĥk (., t m+1/2 ). With the above setting , we can now formulate the iterative leap-frog DG method.…”