Volume 13, Issue 4
Development of an Explicit Symplectic Scheme That Optimizes the Dispersion-relation Equation of the Maxwell's Equations

Tony W. H. Sheu, L. Y. Liang & J. H. Li

Commun. Comput. Phys., 13 (2013), pp. 1107-1133.

Published online: 2013-08

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  • Abstract

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 Ampe`re’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. Theremainingspatialderivativetermsinthesemi-discretizedFaraday’sandAmpe`re’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 proposeafourth-orderaccuratespacecenteredscheme whichminimizes thedifference between the exact and numerical dispersion relationequations. Through the computational exercises, the proposeddual-preservingsolver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell’s solutions. 

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@Article{CiCP-13-1107, author = {Tony W. H. Sheu, L. Y. Liang and J. H. Li}, title = {Development of an Explicit Symplectic Scheme That Optimizes the Dispersion-relation Equation of the Maxwell's Equations}, journal = {Communications in Computational Physics}, year = {2013}, volume = {13}, number = {4}, pages = {1107--1133}, abstract = {

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 Ampe`re’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. Theremainingspatialderivativetermsinthesemi-discretizedFaraday’sandAmpe`re’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 proposeafourth-orderaccuratespacecenteredscheme whichminimizes thedifference between the exact and numerical dispersion relationequations. Through the computational exercises, the proposeddual-preservingsolver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell’s solutions. 

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.280711.230312a}, url = {http://global-sci.org/intro/article_detail/cicp/7266.html} }
TY - JOUR T1 - Development of an Explicit Symplectic Scheme That Optimizes the Dispersion-relation Equation of the Maxwell's Equations AU - Tony W. H. Sheu, L. Y. Liang & J. H. Li JO - Communications in Computational Physics VL - 4 SP - 1107 EP - 1133 PY - 2013 DA - 2013/08 SN - 13 DO - http://dor.org/10.4208/cicp.280711.230312a UR - https://global-sci.org/intro/cicp/7266.html KW - AB -

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 Ampe`re’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. Theremainingspatialderivativetermsinthesemi-discretizedFaraday’sandAmpe`re’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 proposeafourth-orderaccuratespacecenteredscheme whichminimizes thedifference between the exact and numerical dispersion relationequations. Through the computational exercises, the proposeddual-preservingsolver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell’s solutions. 

Tony W. H. Sheu, L. Y. Liang & J. H. Li. (1970). Development of an Explicit Symplectic Scheme That Optimizes the Dispersion-relation Equation of the Maxwell's Equations. Communications in Computational Physics. 13 (4). 1107-1133. doi:10.4208/cicp.280711.230312a
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