Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms

TL;DR

This paper introduces a structure-preserving discretization framework for electromagnetism using variational integrators and discrete differential forms, leading to new multisymplectic numerical methods that improve energy conservation and flexibility in mesh design.

Contribution

It develops a unified variational framework for electromagnetism that generalizes existing schemes and introduces new asynchronous integrators for better numerical properties.

Findings

Yee's scheme is multisymplectic and derived from a variational principle.

Generalization of Yee scheme to unstructured 4D spacetime meshes.

Introduction of an asynchronous variational integrator with excellent conservation properties.

Abstract

In this paper, we develop a structure-preserving discretization of the Lagrangian framework for electromagnetism, combining techniques from variational integrators and discrete differential forms. This leads to a general family of variational, multisymplectic numerical methods for solving Maxwell's equations that automatically preserve key symmetries and invariants. In doing so, we demonstrate several new results, which apply both to some well-established numerical methods and to new methods introduced here. First, we show that Yee's finite-difference time-domain (FDTD) scheme, along with a number of related methods, are multisymplectic and derive from a discrete Lagrangian variational principle. Second, we generalize the Yee scheme to unstructured meshes, not just in space but in 4-dimensional spacetime. This relaxes the need to take uniform time steps, or even to have a preferred time coordinate at all. Finally, as an example of the type of methods that can be developed within this general framework, we introduce a new asynchronous variational integrator (AVI) for solving Maxwell's equations. These results are illustrated with some prototype simulations that show excellent energy and conservation behavior and lack of spurious modes, even for an irregular mesh with asynchronous time stepping.