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

Archana Arya, Kaushik Kalyanaraman

TL;DR

This work develops an arbitrarily high-order, implicit leapfrog time discretization (LF$_R$) for Maxwell's equations using a three-field formulation with a fictitious pressure $p$ to enforce exact divergence constraints. The scheme employs de Rham-compatible finite element spaces (Whitney forms) for spatial discretization, preserving the problem's geometric structure. The authors prove discrete energy conservation for the semidiscretization and derive rigorous error estimates for both time semidiscretization and full space-time discretization, showing convergence of order $(\Delta t)^R$ in time and $h^r$ in space, with additional cross-term rates. Numerical experiments in $\mathbb{R}^2$ validate energy conservation and the predicted convergence behavior for various polynomial orders. The framework offers a robust, high-order, structure-preserving approach for Maxwell's equations with potential impact on scattering problems and physics-informed machine learning applications.

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 $\mathbb{R}^2$.

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

TL;DR

This work develops an arbitrarily high-order, implicit leapfrog time discretization (LF) for Maxwell's equations using a three-field formulation with a fictitious pressure to enforce exact divergence constraints. The scheme employs de Rham-compatible finite element spaces (Whitney forms) for spatial discretization, preserving the problem's geometric structure. The authors prove discrete energy conservation for the semidiscretization and derive rigorous error estimates for both time semidiscretization and full space-time discretization, showing convergence of order in time and in space, with additional cross-term rates. Numerical experiments in validate energy conservation and the predicted convergence behavior for various polynomial orders. The framework offers a robust, high-order, structure-preserving approach for Maxwell's equations with potential impact on scattering problems and physics-informed machine learning applications.

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 .
Paper Structure (11 sections, 3 theorems, 98 equations, 4 figures)

This paper contains 11 sections, 3 theorems, 98 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Three-field Formulation of Maxwell's Equations
  3. Warmup Example: LF$_6$ Scheme
  4. LF$_6$ Scheme in a Sum Notation
  5. The LF$_R$ Scheme
  6. A Note on Preliminaries
  7. Organization
  8. Characterization of Implicit LF$_R$ Scheme
  9. Time Discretization Stability
  10. Error Estimate for Full Discretization
  11. Numerical Results and Summary

Key Result

Theorem 1

For the semidiscretization using the LF$_R$ scheme as given in Equations eqn:maxwell_p_lfReqn:maxwell_E_lfReqn:maxwell_H_lfReqn:maxwell_p0_lfReqn:maxwell_E0_lfReqn:maxwell_H0_lfR, and for any fixed time step $\Delta t$ > 0 sufficiently small, we have that:

Figures (4)

  • Figure 1: Linear finite elements with LF$_6$: Plots of solutions $(E_h^{n - 1/2}, H_h^n)$ at different times for Example 1 of Section \ref{['sec:numerics']} using LF$_6$ and linear Whitney forms for the FEEC spaces. The solutions for $p$ are not shown due to them being identically $0$. We also show projections of the analytical solutions to the linear Whitney forms spaces at these time steps for a visual comparison.
  • Figure 2: Quadratic finite elements with LF$_6$: Plots of solutions $(E_h^{n - 1/2}, H_h^n)$ at different times for Example 1 of Section \ref{['sec:numerics']} using LF$_6$ and quadratic Whitney forms in FEEC. The solutions for $p$ are again not shown due to them being identically $0$. We also show projections of the analytical solutions to the quadratic Whitney forms spaces at these time steps for a visual comparison.
  • Figure 3: Linear finite elements with LF$_6$: Plots of solutions $(p_h^{n + 1/2}, E_h^{n - 1/2}, H_h^n)$ at different times for Example 2 of Section \ref{['sec:numerics']} using the LF$_6$ and linear Whitney forms in FEEC. We also show projections of the analytical solutions to the linear Whitney forms spaces at these time steps for a visual comparison.
  • Figure 4: Quadratic finite elements with LF$_6$: Plots of solutions $(p_h^{n - 1/2}, E_h^{n - 1/2}, H_h^n)$ at different times for Example 2 of Section \ref{['sec:numerics']} using the LF$_6$ and quadratic Whitney forms in FEEC. We also show projections of the analytical solutions to the quadratic Whitney forms spaces at these time steps for a visual comparison.

Theorems & Definitions (6)

  • Theorem 1: Discrete Energy Estimate
  • proof
  • Theorem 2: Discrete Error Estimate
  • proof
  • Theorem 3: Full Error Estimate
  • proof