Energy-Conserving Hermite Methods for Maxwell’s Equations

doi:10.1007/s42967-024-00469-9

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (3): 1146-1173.doi: 10.1007/s42967-024-00469-9

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Energy-Conserving Hermite Methods for Maxwell’s Equations

Daniel Appel?¹, Thomas Hagstrom², Yann-Meing Law³

1 Department of Mathematics, Virginia Tech, Blacksburg 24060, VA, USA;
2 Department of Mathematics, Southern Methodist University, Dallas 75275, TX, USA;
3 Department of Mathematics and Industrial Engineering, Polytechnique Montréal, Montréal QC H3C 3A7, Quebec, Canada

Received:2024-01-22 Revised:2024-06-23 Accepted:2024-08-12 Online:2025-09-20 Published:2025-05-23
Supported by:
This work was funded in part by the National Science Foundation Grants DMS-2012296, DMS-2309687, and DMS-2210286. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. We also thank the anonymous referee whose suggestions improved our presentation of the results.

Abstract

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 the stability for time steps limited only by domain-of-dependence requirements along with error estimates in a special semi-norm associated with the interpolation process. Numerical experiments are presented which demonstrate that Hermite methods of very high order enable the efficient simulation of the electromagnetic wave propagation over thousands of wavelengths.

Key words: Maxwell’s equations, High-order methods, Hermite methods

CLC Number:

65M70

Daniel Appel?, Thomas Hagstrom, Yann-Meing Law. Energy-Conserving Hermite Methods for Maxwell’s Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 1146-1173.

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References

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Energy-Conserving Hermite Methods for Maxwell’s Equations

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