A Fully Fourth Order Accurate Energy Stable Finite Difference Method for Maxwell's Equations in Metamaterials

TL;DR

This paper introduces a new fully fourth order accurate finite difference method for Maxwell's equations in metamaterials, ensuring energy stability and high precision in simulations.

Contribution

The paper develops a novel fourth order in time and space finite difference scheme for Maxwell's equations in metamaterials, with energy preservation properties.

Findings

Achieves fourth order convergence in numerical simulations

Demonstrates energy stability of the scheme

Outperforms second order schemes in accuracy

Abstract

We present a novel fully fourth order in time and space {finite difference method for the time domain Maxwell's equations} in metamaterials. We consider a Drude metamaterial model for the material response to incident electromagnetic fields. We consider the second order formulation of the system of partial differential equations that govern the evolution in time of electric and magnetic fields along with the evolution of the polarization and magnetization current densities. Our discretization employs fourth order staggering in space of different field components and the modified equation approach to obtain fourth order accuracy in time. Using the energy method, we derive energy relations for the continuous models, and design numerical schemes that preserve a discrete analogue of the energy relation. Numerical simulations are provided in one and two dimensional settings to illustrate fourth order convergence as well as compare with second order schemes.