Solving Maxwell's Equations with Mimetic Methods

TL;DR

This paper introduces a mimetic finite-difference method for solving Maxwell's equations that ensures physical consistency and demonstrates its effectiveness through numerical examples involving wave interactions and absorbing boundaries.

Contribution

It develops a mimetic operator framework that preserves key physical properties in computational electromagnetics, advancing beyond classical finite-difference methods.

Findings

Accurate simulation of wave interactions with dielectric materials.

Effective implementation of absorbing boundary conditions.

Demonstrated physical consistency of the mimetic approach.

Abstract

We present a mimetic finite-difference approach for solving Maxwell's equations in one and two spatial dimensions. After introducing the governing equations and the classical Finite-Difference Time-Domain (FDTD) method, we describe mimetic operators that satisfy a discrete analogue of the extended Gauss divergence theorem and show how they lead to a compact, physically consistent formulation for computational electromagnetics. Two numerical examples are presented: a one-dimensional sinusoidal wave interacting with a lossy dielectric slab, and a two-dimensional Gaussian pulse with Uniaxial Perfectly Matched Layer (UPML) absorbing boundary conditions. All implementations use the Mimetic Operators Library Enhanced (MOLE).