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

Abstract

We present a novel fully fourth order in time and space finite difference method (FDM) 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 to compare with second-order schemes.