An efficient and energy decaying discontinuous Galerkin method for Maxwell's equations for the Cole-Cole dispersive medium

TL;DR

This paper develops an efficient discontinuous Galerkin method with energy decay properties for simulating electromagnetic wave propagation in Cole-Cole dispersive media, addressing nonlocal polarization effects with stability and optimal convergence.

Contribution

It introduces a new energy function, a stable approximate system using diffusive representation, and an optimal-order DG scheme with a fast algorithm for Maxwell's equations in dispersive media.

Findings

The proposed method achieves optimal convergence order.

Numerical results confirm the efficiency and stability of the scheme.

The energy decay behavior is effectively captured by the new energy function.

Abstract

In this work, we investigate the propagation of electromagnetic waves in the Cole-Cole dispersive medium by using the discontinuous Galerkin (DG) method to solve the coupled time-domain Maxwell's equations and polarization equation. We define a new and sharpened total energy function for the Cole-Cole model, which better describes the behaviors of the energy than what is available in the current literature. A major theme in the time-domain numerical modeling of this problem has been tackling the difficulty of handling the nonlocal term involved in the time-domain polarization equation. Based on the diffusive representation and the quadrature formula, we derive an approximate system, where the convolution kernel is replaced by a finite number of auxiliary variables that satisfy local-in-time ordinary differential equations. To ensure the resulted approximate system is stable, a nonlinear constrained optimization numerical scheme is established to determine the quadrature coefficients. By a special choice of the numerical fluxes and projections, we obtain {for the constant coefficient case } an optimal-order convergence result for the semi-discrete DG scheme. The temporal discretization is achieved by the standard two-step backward difference formula and a fast algorithm with linear complexity is constructed. Numerical examples are provided for demonstrating the efficiency of the proposed algorithm, validating the theoretical results and illustrating the behaviors of the energy.