A Sixth Order Energy-Conserved Method for Three-Dimensional Time-Domain Maxwell’s Equations

In this paper, a sixth order energy-conserved method is proposed for solving the three-dimensional time-domain Maxwell’s equations. Based on the method of lines, the spatial derivatives of the Maxwell’s equations are approximated with the aid of Fourier pseudo-spectral methods. The resulting ordinary differential equations can be cast as a canonical Hamiltonian system. Then, a fully-discretized scheme is generated via utilizing a sixth order average vector field method to discretize the Hamiltonian system. The proposed scheme is unconditionally stable, non-dissipative and preserves the five discrete energy conservation laws, the momentum conservation law and the symplecticity. The rigorous error estimate is established based on the energy method, which show that the proposed method is of sixth order accuracy in time and spectral accuracy in space in the discrete L-norm. The error estimate is optimal, and especially the constant in the error estimate is proved to be only O(T ). Furthermore, the proposed scheme can preserve the discrete divergence exactly and its numerical dispersion relation is also investigated in detail. Finally, a fast solver is applied to solve the discrete linear system. Numerical results further verify our theoretical analysis. AMS subject classification: 65M12, 65M15, 65M70