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Energy-Conserving Hermite Methods for Maxwell's Equations

Daniel Appelo, Thomas Hagstrom, Yann-Meing Law-Kam-Cio

TL;DR

This paper develops energy-conserving Hermite methods for Maxwell's equations in dielectric and dispersive media, formulating a conservative discretization on cuboidal grids with vertex-based tensor-product DoF and Hermite-Birkhoff interpolation. It proves HB-seminorm stability under a CFL-like condition and derives HB-seminorm error estimates, while showing practical $L^2$ convergence compatibility with observed rates; the framework is extended to dispersive Lorentz/Sellmeier models via auxiliary fields and, optionally, dissipative corrections. Numerical experiments in 2D periodic domains demonstrate high-order accuracy and remarkable efficiency: high-order schemes propagate waves over thousands of wavelengths with modest DOF per wavelength, and results span dielectric, resonant Lorentz, and Sellmeier models. The work also discusses mapped-coordinate implementations and boundary/interface strategies, highlighting open theoretical questions on boundary stability and $L^2$ convergence in complex geometries.

Abstract

Energy-conserving Hermite methods for solving Maxwell's equations in dielectric and dispersive media are described and analyzed. In three space dimensions methods of order $2m$ to $2m+2$ require $(m+1)^3$ degrees-of-freedom per node for each field variable and can be explicitly marched in time with steps independent of $m$. We prove stability for time steps limited only by domain-of-dependence requirements along with error estimates in a special seminorm associated with the interpolation process. Numerical experiments are presented which demonstrate that Hermite methods of very high order enable the efficient simulation of electromagnetic wave propagation over thousands of wavelengths.

Energy-Conserving Hermite Methods for Maxwell's Equations

TL;DR

This paper develops energy-conserving Hermite methods for Maxwell's equations in dielectric and dispersive media, formulating a conservative discretization on cuboidal grids with vertex-based tensor-product DoF and Hermite-Birkhoff interpolation. It proves HB-seminorm stability under a CFL-like condition and derives HB-seminorm error estimates, while showing practical convergence compatibility with observed rates; the framework is extended to dispersive Lorentz/Sellmeier models via auxiliary fields and, optionally, dissipative corrections. Numerical experiments in 2D periodic domains demonstrate high-order accuracy and remarkable efficiency: high-order schemes propagate waves over thousands of wavelengths with modest DOF per wavelength, and results span dielectric, resonant Lorentz, and Sellmeier models. The work also discusses mapped-coordinate implementations and boundary/interface strategies, highlighting open theoretical questions on boundary stability and convergence in complex geometries.

Abstract

Energy-conserving Hermite methods for solving Maxwell's equations in dielectric and dispersive media are described and analyzed. In three space dimensions methods of order to require degrees-of-freedom per node for each field variable and can be explicitly marched in time with steps independent of . We prove stability for time steps limited only by domain-of-dependence requirements along with error estimates in a special seminorm associated with the interpolation process. Numerical experiments are presented which demonstrate that Hermite methods of very high order enable the efficient simulation of electromagnetic wave propagation over thousands of wavelengths.
Paper Structure (13 sections, 5 theorems, 81 equations, 8 figures, 4 tables)

This paper contains 13 sections, 5 theorems, 81 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Conservative Hermite Discretization of the Dispersive Maxwell System
  3. Dissipative Corrections
  4. Implementation in Mapped Coordinates and Compatiblity Conditions
  5. Stability and Convergence
  6. Extensions to the Dispersive System
  7. Numerical Experiments
  8. Dielectric Medium
  9. Dispersive Medium
  10. Resonant Case
  11. High-Frequency Case
  12. Sellmeier Model
  13. Conclusions and Open Issues

Key Result

Lemma 1

For the dielectric system, $M=\Gamma_V=\Gamma_W=0$, if $q=3m+2$ and (CFL) holds then the quantities $P^{\pm,h}$, $E_N^h$, $Q^{\pm,h}$ and $H_N^h$ computed from the approximations, $\tilde{E}^h$, $\tilde{H}^h$, to the symmetrized variables satisfy the evolution formulas:

Figures (8)

  • Figure 1: Convergence for $m=3-6$, $k=40$, $T=100$ in a dielectric medium. Errors are computed by comparing the Hermite interpolant of the numerical solution to the exact solution for each time step.
  • Figure 2: Comparison of accuracy for $m=3-6$, $k=40$, $T=100$ in the dielectric medium. Errors are computed by comparing the Hermite interpolant of the numerical solution to the exact solution for each time step.
  • Figure 3: Convergence for $m=3-6$, $k=40$, $T=100$ for the near-resonant mode in a Lorentz medium. Errors are computed by comparing the Hermite interpolant of the numerical solution to the exact solution for each time step.
  • Figure 4: Comparison of accuracy for $m=3-6$, $k=40$, $T=100$ in the Lorentz medium for the near-resonant mode. Errors are computed by comparing the Hermite interpolant of the numerical solution to the exact solution for each time step.
  • Figure 5: Convergence for $m=2-3$, $k=40$, $T=100$ for the high-frequency mode in a Lorentz medium. Errors are computed by comparing the Hermite interpolant of the numerical solution to the exact solution for each time step.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Corollary 1
  • Remark 1
  • proof