On a semi-discrete model of Maxwell's equations in three and two dimensions

TL;DR

This paper introduces a geometric semi-discrete formulation of Maxwell's equations using discrete exterior calculus, preserving the equations' structure in 2D and 3D, with explicit solutions on a 2D torus.

Contribution

It develops a structure-preserving semi-discrete model of Maxwell's equations in 2D and 3D using discrete exterior calculus, including explicit solutions in a special case.

Findings

The model preserves geometric and topological structures of Maxwell's equations.

Explicit solutions are derived for the 2D torus case.

The approach offers a consistent spatial discretization maintaining essential properties.

Abstract

In this paper, we develop a geometric, structure-preserving semi-discrete formulation of Maxwell's equations in both three- and two-dimensional settings within the framework of discrete exterior calculus. This approach preserves the intrinsic geometric and topological structures of the continuous theory while providing a consistent spatial discretization. We analyze the essential properties of the proposed semi-discrete model and compare them with those of the classical Maxwell's equations. As a special case, the model is illustrated on a combinatorial two-dimensional torus, where the semi-discrete Maxwell's equations take the form of a system of first-order linear ordinary differential equations. An explicit expression for the general solution of this system is also derived.