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Development of an Explicit Symplectic Scheme that Optimizes the Dispersion-Relation Equation of the Maxwell’s Equations

Published online by Cambridge University Press:  03 June 2015

Tony W. H. Sheu ,
L. Y. Liang  and
J. H. Li
Tony W. H. Sheu*
Affiliation:
Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan Taida Institute of Mathematical Science (TIMS), National Taiwan University, Taiwan Center for Quantum Science and Engineering (CQSE), National Taiwan University, Taiwan
L. Y. Liang
Affiliation:
Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan
J. H. Li
Affiliation:
Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan
*
Corresponding author.Email:[email protected]
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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 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.

Keywords

52B1065D1868U0568U07Maxwell’s equationsnon-staggered gridszero-divergencefourth-orderdualpreserving solverdispersion relation equations
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 4 , April 2013 , pp. 1107 - 1133
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Cai, J. X., Wang, Y. S., Wang, B., and Jiang, B., New multisymplectic self-adjoint scheme and its composition scheme for the time-domain Maxwell’s equations, Journal of Mathematical Physics, 47 (2006), pp. 1–18.Google Scholar
[2]Yee, K. S., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Transactions on Antenna Propagation, AP4 (1966), pp. 302307.Google Scholar
[3]Munz, C.-D., Ommes, P., Schneider, R., Sonnendriicker, E., and Voβ, U., Divergence correction techinques for Maxwell solvers based on a hyperbolic model, Journal of Computational Physics, 161 (2000), pp. 484–511.Google Scholar
[4]Cockburn, B., Li, F., and Shu, C.-W., nauocally divergence-free discontinuous Galerkin methods for the Maxwell equations, Journal of Computational Physics, 194 (2004), pp. 588–610.Google Scholar
[5]Bridges, T. J. and Reich, S., Multi-symplectic integration numerical scheme for Hamiltonian PDEs that conserves symplecticity, Phys. Lett. A, 284 (2001), pp. 184–193.Google Scholar
[6]Born, M. and Wolf, E., Principles of Optics, Pergamon Press, Oxford, (1964).Google Scholar
[7]Kong, L. H., Hong, J. L., and Zhang, J. J., Splitting multisymplectic integrators for Maxwell’s equations, Journal of Computational Physics, 229 (2010), pp. 4259–4278.CrossRefGoogle Scholar
[8]Anderson, N. and Arthurs, A. M., Helicity and variational principles for Maxwell’s equations, Int. J. Electron, 54 (1983), pp. 861–864.Google Scholar
[9]Brackbill, J. U. and Barnes, D. C., The effect of nonzero product of magnetic gradient and B on the numerical solution of the magnetohydrodynamic equations, Journal of Computational Physics, 35 (1980), pp. 426–430.Google Scholar
[10]Landau, L. D., Lifshitz, E. M. and Pitaevskii, L. P., Electrodynamics of Continuous Media, Pergamon Press, Oxford, (1984).Google Scholar
[11]Jiang, B., Wu, J., and Povinelli, L. A., The origin of spurious solutions in computational electromagnetics, NASA-TM-10692, E-9633, ICOMP-95–8,1995.Google Scholar
[12]Assous, F., Degond, P., Heintze, E., Raviart, P. A., and Serger, J., On a finite-element method for solving the three-dimensional Maxwell equations, Journal of Computational Physics, 109 (1993), pp. 222–237.Google Scholar
[13]Sheu, T. W. H., Hung, Y. W., Tsai, M. H., and Li, J. H., On the development of a triple-preserving Maxwell’s equations solver in non-staggered grids, Int. J. Numer. Meth. in Fluids, 63 (2010), pp. 1328–1346.Google Scholar
[14]Greif, C., Li, D., Schotzau, D., and Wei, X., A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics, Computer Methods in Applied Mechanics and Engineering, 199 (2010), pp. 2840–2855.Google Scholar
[15]Pinho, P., Domingues, M. O., Ferrira, P. J. S. G., Gomes, S. M., Gomide, A., Pereira, J. R., Inter-polating wavelets and adaptive finite difference schemes for solving Maxwell’s equations: The effects of gridding, IEEE Trans. on Magnetics, 43 (3) (2007), pp. 1013–1022.Google Scholar
[16]Liu, Y., Fourier analysis of numerical algorithms for the Maxwell equations, J. Computational Physics, 124, (1996), pp. 396–416.Google Scholar
[17]Munz, C.-D., Ommes, P., Schneider, R., A three-dimensional finite-volume solver for the Maxwell equations with divergence cleaning on unstructured meshes, Computer Physics Communications, 130, (2000), pp. 83–117.Google Scholar
[18]Sanz-Serna, J. M., Symplectic Runge-Kutta and related methods: recent results, Physica D, (1992), pp. 293–302.Google Scholar
[19]Monovasills, T. H., Kalogiratou, Z., and Simos, T. E., Symplectic partitioned Runge-Kutta methods with minimal phase-lag, Computer Physics Communications, 181 (2010), pp. 12511254.Google Scholar
[20]Jiang, L. L., Mao, J. F., and Wu, X. L., Symplectic finite-difference time-domain method for Maxwell equations, IEEE Transactions on Magnetics, 42(8) (2006), pp. 1991–1995.Google Scholar
[21]Taflove, A. and Umashankar, K. R., The finite-difference time-domain method for numerical modeling of electromagnetic wave interactions, Electromag., 10(1) (1990), pp. 105–126.Google Scholar
[22]Wang, S. and Teixeira, F. L., Dispersion relation-preserving FDTD algorithms for large-scale three-dimensional problems, IEEE Transactions on Antennas and Propagation, 51(8) (2003), pp. 1818–1828.Google Scholar
[23]Kim, J. K. and Lee, D. J., Optimized compact finite difference schemes with maximun resolu-tion, AIAA J., 34(5) (1996), pp. 887–893.Google Scholar
[24] C. Tam, K. W. and Webb, J. C., Dispersion-relation-preserving finite difference schemes for computational acoustics, Journal of Computational Physics, 107 (1993), pp. 262–281.Google Scholar
[25]Lele, S. K., Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics, 103 (1992), pp. 16–42.Google Scholar
[26]Bogry, C. and Bailly, C., A family of low dispersive and low dissipative explicit schemes for flow and noise computations, Journal of Computational Physics, 194 (2004), pp. 194–214.Google Scholar
[27]Gao, L., Zhang, B., and Liang, D., The splitting finite-difference time-domain methods for Maxwell’s equations in two dimensions, J. Comput. Applied Math., 205 (2007), pp. 207–230.Google Scholar
[28]Sacks, Z. S., Kinsland, D. M., Lee, R., and Lee, J. F., A perfectly matched anisotropic absorber for use as an absorbing boundary condition, IEEE T. Antenn. Propag., 43 (1995), pp. 16401643.Google Scholar
[29]Gedney, S. D., An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices, IEEE T. Antenn. Propag., 44 (1996), pp. 1630–1639.Google Scholar
[30]Born, M. and Wolf, E., Principles of Optics, Cambridge University Press, Cambridge, (1999) pp.759–774.Google Scholar
[31]Berenger, J. P., A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, 114 (1994), pp. 185–200.Google Scholar
[32]Mekis, A., Chen, J. C., Kurland, I., Fan, S., Villeneuve, P. R., and Joannopoulos, J. D., High transmission through sharp bends in photonic crystal waveguides, Physical Review Letters, 77 (1996), pp. 3787–3790.CrossRefGoogle ScholarPubMed