Development of an Explicit Symplectic Scheme that Optimizes the Dispersion-Relation Equation of the Maxwell's Equations (Communications in Computational Physics, 2013)
In this paper an explicit finite-difference time-domain scheme for solvingthe Maxwell's equations in non-staggered grids is presented. The proposed schemefor solving the Faraday's and Ampère's equations in a theoretical manner is aimed topreserve discrete zero-divergence for the electric and magnetic fields. The inherent local conservation laws in Maxwell's equations are also preserved discretely all the timeusing the explicit second-order accurate symplectic partitioned Runge-Kutta scheme.The remaining spatial derivative terms in the semi-discretized Faraday's and Ampère'sequations are then discretized to provide an accurate mathematical dispersion relationequation that governs the numerical angular frequency and the wavenumbers in twospace dimensions. To achieve the goal of getting the best dispersive characteristics, wepropose a fourth-order accurate space centered scheme which minimizes the differencebetween the exact and numerical dispersion relation equations. Through the computational exercises, the proposed dual-preserving solver is computationally demonstratedto be efficient for use to predict the long-term accurate Maxwell's solutions.