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A Novel Structure-Preserving Scheme for Three-Dimensional Maxwell’s Equations

A Novel Structure-Preserving Scheme for Three-Dimensional Maxwell’s Equations

Year:    2024

Author:    Chaolong Jiang, Wenjun Cai, Yushun Wang, Haochen Li

CSIAM Transactions on Applied Mathematics, Vol. 5 (2024), Iss. 4 : pp. 788–834

Abstract

In this paper, a novel structure-preserving scheme is proposed for solving the three-dimensional Maxwell’s equations. The proposed scheme can preserve all of the desired structures of the Maxwell’s equations numerically, including five energy conservation laws, two divergence-free fields, three momentum conservation laws and a symplectic conservation law. Firstly, the spatial derivatives of the Maxwell’s equations are approximated with Fourier pseudo-spectral methods. The resulting ordinary differential equations are cast into a canonical Hamiltonian system. Then, the fully discrete structure-preserving scheme is derived by integrating the Hamiltonian system using a sixth order average vector field method. Subsequently, an optimal error estimate is established based on the energy method, which demonstrates that the proposed scheme is of sixth order accuracy in time and spectral accuracy in space in the discrete $L^2$-norm. The constant in the error estimate is proved to be only $\mathcal{O}(T),$ where $T > 0$ is the time period. Furthermore, its numerical dispersion relation is analyzed in detail, and a customized fast solver is presented to efficiently solve the resulting discrete linear equations. Finally, numerical results are presented to validate our theoretical analysis.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/csiam-am.SO-2023-0047

CSIAM Transactions on Applied Mathematics, Vol. 5 (2024), Iss. 4 : pp. 788–834

Published online:    2024-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    47

Keywords:    Maxwell’s equations AVF method structure-preserving scheme dispersion relation divergence preservation.

Author Details

Chaolong Jiang

Wenjun Cai

Yushun Wang

Haochen Li