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

收稿日期:2024-01-22 修回日期:2024-06-23 接受日期:2024-08-12 出版日期:2025-09-20 发布日期:2025-05-23
通讯作者: Thomas Hagstrom, thagstrom@smu.edu;Daniel Appel?, appelo@vt.edu;Yann-Meing Law, yann-meing.law@polymtl.ca E-mail:thagstrom@smu.edu;appelo@vt.edu;yann-meing.law@polymtl.ca
基金资助:
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.

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.

摘要/Abstract

摘要： 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.

关键词: Maxwell’s equations, High-order methods, Hermite methods

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

中图分类号:

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.

参考文献

1. Appelö, D., Chen, R., Hagstrom, T.: A hybrid Hermite-discontinuous Galerkin method for hyperbolic systems with applications to Maxwell’s equations. J. Comput. Phys. 257, 501–520 (2013)
2. Appelö, D., Hagstrom, T.: Solving PDEs with Hermite interpolation. In: Lecture Notes in Computational Science, pp. 31–49. Springer, Berlin (2015)
3. Appelö, D., Hagstrom, T., Vargas, A.: Hermite methods for the scalar wave equation. SIAM J. Sci. Comput. 40, 3902–3927 (2018)
4. Banks, J.W., Buckner, B., Henshaw, W., Jenkinson, M., Kildeshev, A., Kovacic, G., Prokopeva, L., Schwendeman, D.: A high-order accurate scheme for Maxwell’s equations with a Generalized Dispersive Material (GDM) model and material interfaces. J. Comput. Phys. 412, 109424 (2020)
5. Bokil, V., Gibson, N.: Convergence analysis of Yee schemes for Maxwell’s equations in Debye and Lorentz dispersive media. Int. J. Numer. Anal. Mod. 11, 657–687 (2014)
6. Byrne, G.D., Hindmarsh, A.C.: A polyalgorithm for the numerical solution of ordinary differential equations. ACM Trans. Math. Softw. 1, 71–96 (1975)
7. Goodrich, J., Hagstrom, T., Lorenz, J.: Hermite methods for hyperbolic initial-boundary value problems. Math. Comput. 75, 595–630 (2006)
8. Hairer, E., Norsett, S., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Springer, New York (1992)
9. Holland, R.: Finite-difference solution of Maxwell’s equations in generalized nonorthogonal coordinates. IEEE Trans. Nucl. Sci. 30, 4589–4591 (1983)
10. Jiang, Y., Sakkaplangkul, P., Bokil, V., Cheng, Y., Li, F.: Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell’s equations in linear Lorentz media. J. Comput. Phys. 394, 100–135 (2019)
11. Joly, P.: Variational methods for time-dependent wave propagation problems. In: Ainsworth, M., Davies, P., Duncan, D., Martin, P., Rynne, B. (eds.) Topics in Computational Wave Propagation, pp. 201–264. Springer, Berlin (2003)
12. Law, Y.-M., Appelö, D.: The Hermite-Taylor correction function method for Maxwell’s equations. Commun. Appl. Math. Comput. 7, 347–371 (2023). https://doi.org/10.1007/s42967-023-00287-5
13. Loya, A.A., Appelö, D., Henshaw, W.D.: High order accurate Hermite schemes on curvilinear grids with compatibility boundary conditions. J. Comput. Phys. 522, 113597 (2025)
14. Mai, W., Campbell, S.D., Whiting, E.B., Kang, L., Werner, P.L., Chen, Y., Werner, D.H.: Prismatic discontinuous Galerkin time domain method with an integrated generalized dispersion model for efficient optical metasurface analysis. Opt. Mater. Express 10, 2542–2559 (2020)
15. Palik, E.D.: Handbook of Optical Constants of Solids II. Academic Press, San Diego (1998)
16. Prokopeva, L., Borneman, J., Kildishev, A.: Optical dispersion models for time-domain modeling of metal-dielectric nanostructures. IEEE Trans. Magn. 47, 1150–1153 (2011)
17. Qiang, R., Bao, H., Campbell, S.D., Prokopeva, L.J., Kildishev, A.V., Werner, D.H.: Continuousdiscontinuous Galerkin time domain (CDGTD) method with generalized dispersive material (GDM) model for computational photonics. Opt. Express 26, 29005–29016 (2018)
18. Shi, C., Li, J., Shu, C-W.: Discontinuous Galerkin methods for Maxwell’s equations in Drude metamaterials on unstructured meshes. J. Comput. Appl. Math. 342, 147–163 (2018)
19. Vargas, A., Chan, J., Hagstrom, T., Warburton, T.: GPU acceleration of Hermite methods for simulation of wave propagation. In: Lecture Notes in Computational Science, pp. 357–368. Springer, Berlin (2017)
20. Vargas, A., Hagstrom, T., Chan, J., Warburton, T.: Leapfrog time-stepping for Hermite methods. J. Sci. Comput. 30, 289–314 (2019)
21. Yang, W., Huang, Y., Li, J.: Developing a time-domain finite element method for the Lorentz metamaterial model and applications. J. Sci. Comput. 68, 438–463 (2016)

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Energy-Conserving Hermite Methods for Maxwell’s Equations

Energy-Conserving Hermite Methods for Maxwell’s Equations

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