Development of an Explicit Symplectic Scheme that Optimizes the Dispersion-Relation Equation of the Maxwell's Equations
Tony W. H. Sheu 1*, L. Y. Liang 2, J. H. Li 21 Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan; Taida Institute of Mathematical Science (TIMS) & Center for Quantum Science and Engineering (CQSE), National Taiwan University, Taiwan.
2 Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan.
Received 28 July 2011; Accepted (in revised version) 23 March 2012
Available online 21 September 2012
In this paper an explicit finite-difference time-domain scheme for solving the Maxwell's equations in non-staggered grids is presented. The proposed scheme for solving the Faraday's and Ampere's equations in a theoretical manner is aimed to preserve discrete zero-divergence for the electric and magnetic fields. The inherent local conservation laws in Maxwell's equations are also preserved discretely all the time using the explicit second-order accurate symplectic partitioned Runge-Kutta scheme. The remaining spatial derivative terms in the semi-discretized Faraday's and Ampere's equations are then discretized to provide an accurate mathematical dispersion relation equation that governs the numerical angular frequency and the wavenumbers in two space dimensions. To achieve the goal of getting the best dispersive characteristics, we propose a fourth-order accurate space centered scheme which minimizes the difference between the exact and numerical dispersion relation equations. Through the computational exercises, the proposed dual-preserving solver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell's solutions.AMS subject classifications: 52B10, 65D18, 68U05, 68U07
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Key words: Maxwell's equations, non-staggered grids, zero-divergence, fourth-order, dual-preserving solver, dispersion relation equations.
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