A Novel Structure-Preserving Scheme for Three-Dimensional Maxwell’s Equations (CSIAM Transactions on Applied Mathematics, 2024)
In this paper, a novel structure-preserving scheme is proposed for solvingthe three-dimensional Maxwell’s equations. The proposed scheme can preserve allof the desired structures of the Maxwell’s equations numerically, including five energyconservation laws, two divergence-free fields, three momentum conservation laws anda symplectic conservation law. Firstly, the spatial derivatives of the Maxwell’s equations are approximated with Fourier pseudo-spectral methods. The resulting ordinarydifferential equations are cast into a canonical Hamiltonian system. Then, the fullydiscrete structure-preserving scheme is derived by integrating the Hamiltonian system using a sixth order average vector field method. Subsequently, an optimal errorestimate 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 thediscrete $L^2$-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 analyzedin detail, and a customized fast solver is presented to efficiently solve the resultingdiscrete linear equations. Finally, numerical results are presented to validate our theoretical analysis.
Keywords: Maxwell’s equations; AVF method; structure-preserving scheme; dispersion relation; divergence preservation.