[Paper Review] Computational Electromagnetism with Variational Integrators and Discrete Differential Forms

Ari Stern, Yiying Tong+2 more|arXiv (Cornell University)|Jul 30, 2007

Numerical methods for differential equations12 citations

TL;DR

This paper introduces a family of variational, multisymplectic numerical methods for Maxwell’s equations using discrete differential forms in spacetime. It proves that Yee’s FDTD scheme and related methods are multisymplectic and derive from a discrete Lagrangian variational principle, unifying them under a geometric framework and enabling new structure-preserving methods.

ABSTRACT

In this paper, we introduce a general family of variational, multisymplectic numerical methods for solving Maxwell’s equations, using discrete differential forms in spacetime. 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

Motivation & Objective

• To develop structure-preserving numerical methods for computational electromagnetism that respect the geometric and symplectic structure of Maxwell’s equations.
• To unify existing methods like Yee’s FDTD under a common variational and multisymplectic framework using discrete differential forms.
• To generalize these methods to construct new, geometrically accurate schemes for solving Maxwell’s equations in spacetime.

Proposed method

• The method employs discrete differential forms to discretize spacetime, preserving the geometric structure of Maxwell’s equations at the discrete level.
• It formulates a discrete Lagrangian variational principle over a spacetime mesh, ensuring the resulting schemes are multisymplectic.
• The approach uses finite-difference-like stencils on structured grids, but derived from variational principles rather than heuristic discretization.
• The discrete forms are defined on a spacetime grid with dual meshes, enabling consistent discretization of differential forms and their exterior derivatives.
• The method ensures conservation of symplectic structure across both space and time, leading to long-term stability and accuracy.
• It generalizes the Yee scheme by showing it arises naturally from a discrete variational principle, thereby explaining its stability and structure preservation.

Experimental results

Research questions

• RQ1 Can Yee’s FDTD method be derived from a discrete variational principle and shown to be multisymplectic?
• RQ2 What is the geometric structure underlying classical electromagnetic solvers like FDTD?
• RQ3 How can discrete differential forms be used to construct new multisymplectic schemes for Maxwell’s equations?
• RQ4 What are the implications of variational structure for long-term numerical stability in electromagnetism?
• RQ5 How can the spacetime formulation of Maxwell’s equations be discretized while preserving symplectic and geometric properties?

Key findings

• Yee’s FDTD scheme is shown to be multisymplectic and derivable from a discrete Lagrangian variational principle, explaining its long-term stability.
• The discrete Lagrangian formulation provides a unified framework for understanding and generalizing existing electromagnetic solvers.
• The use of discrete differential forms ensures consistent and structure-preserving discretization of differential forms and their exterior derivatives.
• The method naturally leads to new multisymplectic schemes that inherit the geometric properties of the continuous equations.
• The approach demonstrates that geometric structure preservation is achievable in spacetime discretizations, improving numerical accuracy over long simulations.