A mimetic discretization of the macroscopic Maxwell equations in Hamiltonian form

TL;DR

This paper introduces a novel mimetic spectral element discretization for Maxwell's equations in Hamiltonian form, which preserves key physical laws and separates topological and metric components for improved numerical fidelity.

Contribution

The paper presents a new split exterior calculus mimetic spectral element method that discretizes Maxwell's equations while conserving Gauss's laws and separating topological from metric quantities.

Findings

Exact conservation of Gauss's laws in discretization

Successful application to a 1D Maxwell's equations test case

Introduction of a new discrete Hodge star operator

Abstract

A mimetic spectral element discretization, utilizing a novel Galerkin projection Hodge star operator, of the macroscopic Maxwell equations in Hamiltonian form is presented. The idea of splitting purely topological and metric dependent quantities is natural in the Hamiltonian modeling framework as the Poisson bracket is metric free with the Hamiltonian containing all metric information. This idea may be incorporated into the mimetic spectral element method by directly discretizing the Poincar\'e duality structure. This "split exterior calculus mimetic spectral element method" yields spatially discretized Maxwell's equations which are Hamiltonian and exactly and strongly conserve Gauss's laws. Moreover, the new discrete Hodge star operator is itself of interest as a partition of the purely topological and metric dependent portions of the Hodge star operator. As a simple test case, the numerical results of applying this method to a one-dimensional version of Maxwell's equations are given.