Modeling of dispersive media within the Discontinuous Galerkin Finite Element Time-Domain method

Abstract

Discontinuous-Galerkin Time-Domain (DGTD) methods are highly efficient solution methods for solving the time-dependent Maxwell's equations [1]. DGTD methods are high-order accurate and highly parallel. Recently, a Discontinuous Galerkin Finite Element Time-Domain (DGFETD) method has been introduced [2]. Both the electric and magnetic fields are expanded via hierarchical Nedelec curl-conforming mixed-order basis functions [3]. Similar to the DGTD method, tangential field continuity is weakly constrained across shared boundaries. The use of curl-conforming basis functions that satisfy Nedelec's criterion avoids the concern of spurious solutions [4]. Hence, the formulation avoids the need for penalty methods [1]. Furthermore, the method does not require upwind flux terms for stability. The focus of this paper is the simulation of electromagnetic waves propagating in complex dispersive media via the DGFETD method. The difficulty of simulating dispersive media in the time domain is that the constitutive relations require a convolution. Performing a direct convolution is far too expensive, so more efficient schemes must be introduced. Dispersive materials have been modeled with FDTD and FETD methods [5] as well as within DGTD methods [6–8]. To preserve high-order accuracy in the DGTD algorithm, auxiliary differential equation (ADE) based-methods have been proposed to represent the constitutive relations. In this paper, a high-order DGFETD algorithm is proposed for simulating the fields in dispersive media based on an ADE formulation. The method is applied to a Drude and multiple pole Debye and Lorentz dispersive media models. The proposed algorithm is validated through the simulation of a number of canonical problems demonstrating that the algorithm preserves the high-order accuracy of the DGFETD method.