An Arbitrarily Higher-Order Time Implicit Scheme for Maxwell's Equations

Abstract

We propose an arbitrarily higher (even) order implicit leapfrog scheme for time discretization of a three-field formulation of Maxwell's equations. We use this in conjunction with an arbitrarily higher-order and compatible discretization using finite element spaces that form a de Rham complex. In doing so, we provide a generalization of an earlier work building from~\cite{ArKa2025} and~\cite{ArKa2026}. We prove stability, demonstrate energy conservation, and characterize the asymptotic convergence of the error for the time semidiscretization as well as for the full spatial and temporal discretization of this Maxwell's system. We also provide some numerical validation using computational examples in $\mathbb{R}^2$.