Variational Integrators for Maxwell's Equations with Sources

TL;DR

This paper develops variational spacetime integrators for Maxwell's equations with sources by combining geometric discretization techniques with structure-preserving time integrators, extending previous work to include charge and current sources.

Contribution

It introduces a novel framework that merges mixed finite elements and variational integrators to discretize Maxwell's equations with sources in a structure-preserving manner.

Findings

Successfully incorporates sources into variational spacetime integrators.

Extends previous source-free Maxwell's equations discretization.

Demonstrates improved geometric structure preservation.

Abstract

In recent years, two important techniques for geometric numerical discretization have been developed. In computational electromagnetics, spatial discretization has been improved by the use of mixed finite elements and discrete differential forms. Simultaneously, the dynamical systems and mechanics communities have developed structure-preserving time integrators, notably variational integrators that are constructed from a Lagrangian action principle. Here, we discuss how to combine these two frameworks to develop variational spacetime integrators for Maxwell's equations. Extending our previous work, which first introduced this variational perspective for Maxwell's equations without sources, we also show here how to incorporate free sources of charge and current.