For the Maxwell equations in time-dependent media only finite difference schemes with time-dependent conductivity are known. In this paper we present a numerical scheme based on the Magnus expansion and operator splitting that can handle time-dependent permeability and permittivity too. We demonstrate our results with numerical tests.
@article{bwmeta1.element.doi-10_2478_s11533-011-0074-3,
author = {Istv\'an Farag\'o and \'Agnes Havasi and Robert Horv\'ath},
title = {Numerical solution of the Maxwell equations in time-varying media using Magnus expansion},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {137-149},
zbl = {1243.78049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0074-3}
}
István Faragó; Ágnes Havasi; Robert Horváth. Numerical solution of the Maxwell equations in time-varying media using Magnus expansion. Open Mathematics, Tome 10 (2012) pp. 137-149. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0074-3/
[1] Bagrinovskiĭ K.A., Godunov S.K., Difference schemes for multidimensional problems, Dokl. Akad. Nauk. SSSR, 1957, 115, 431–433 (in Russian) | Zbl 0087.12201
[2] Botchev M., Faragó I., Havasi Á., Testing weighted splitting schemes on a one-column transport-chemistry model, International Journal of Environment and Pollution, 2004, 22(1–2), 3–16 | Zbl 1151.65343
[3] Botchev M.A., Faragó I., Horváth R., Application of operator splitting to the Maxwell equations including a source term, Appl. Numer. Math., 2009, 59(3–4), 522–541 http://dx.doi.org/10.1016/j.apnum.2008.03.031 | Zbl 1159.78346
[4] Csomós P., Faragó I., Error analysis of the numerical solution of split differential equations, Math. Comput. Modelling, 2008, 48(7–8), 1090–1106 http://dx.doi.org/10.1016/j.mcm.2007.12.014 | Zbl 1187.65084
[5] Csomós P., Faragó I., Havasi Á., Weighted sequential splittings and their analysis, Comput. Math. Appl., 2005, 50(7), 1017–1031 http://dx.doi.org/10.1016/j.camwa.2005.08.004 | Zbl 1086.65053
[6] Faragó I., Havasi Á., Horváth R., On the order of operator splitting methods for non-autonomous systems (submitted) | Zbl 1243.65077
[7] Fante R., Transmission of electromagnetic waves into time-varying media, IEEE Trans. Antennas and Propagation, 1971, 19(3), 417–424 http://dx.doi.org/10.1109/TAP.1971.1139931
[8] Felsen L., Whitman G., Wave propagation in time-varying media, IEEE Trans. Antennas and Propagation, 1970, 18(2), 242–253 http://dx.doi.org/10.1109/TAP.1970.1139657
[9] Harfoush F.A., Taflove A., Scattering of electromagnetic waves by a material half-space with a time-varying conductivity, IEEE Trans. Antennas and Propagation, 1991, 39(7), 898–906 http://dx.doi.org/10.1109/8.86907
[10] Horváth R., Uniform treatment of numerical time-integrations of the Maxwell equations, In: Proceedings Scientific Computing in Electrical Engineering, Eindhoven, June 23–28, 2002, Math. Ind., 4, Springer, Berlin, 2003, 231–239 | Zbl 1108.78020
[11] Hundsdorfer W., Verwer J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Ser. Comput. Math., 33, Springer, Berlin, 2003 | Zbl 1030.65100
[12] Karlsfeld S., Oteo J.A., Recursive generation of higher-order terms in the Magnus expansion, Phys. Rev. A, 1989, 39(7), 3270–3273 http://dx.doi.org/10.1103/PhysRevA.39.3270
[13] Lee J.H., Kalluri D.K., Three-dimensional FDTD simulation of electromagnetic wave transformation in a dynamic inhomogeneous magnetized plasma, IEEE Trans. Antennas and Propagation, 1999, 47(7), 1146–1151 http://dx.doi.org/10.1109/8.785745
[14] Magnus W., On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 1954, 7(4), 649–673 http://dx.doi.org/10.1002/cpa.3160070404 | Zbl 0056.34102
[15] Marchuk G.I., Splitting Methods, Nauka, Moscow, 1988 (in Russian) | Zbl 0653.65065
[16] Moan P.C., Oteo J.A., Ros J., On the existence of the exponential solution of linear differential systems, J. Phys. A, 1999, 32(27), 5133–5139 http://dx.doi.org/10.1088/0305-4470/32/27/311 | Zbl 0945.34004
[17] Strang G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 1968, 5(3), 506–517 http://dx.doi.org/10.1137/0705041 | Zbl 0184.38503
[18] Taflove A., Hagness S.C., Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed., Artech House, Boston, 2005
[19] Taylor C.D., Lam D.-H., Shumpert T.H., Electromagnetic scattering in time varying, inhomogeneous media, Interaction Notes, 41, Mississippi State University, State College, Mississippi, 1968
[20] Vorgul I., On Maxwell’s equations in non-stationary media, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2008, 366(1871), 1781–1788 http://dx.doi.org/10.1098/rsta.2007.2186 | Zbl 1153.78309
[21] Wu R., Gao B.-Q., The analysis of 3 dB microstrip directional coupler in time-varying media by FDTD method, In: 2nd International Conference on Microwave and Millimeter Wave Technology, 2000, ICMMT, Beijing, 2000, 375–378
[22] Yee K.S., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas and Propagation, 1966, 14(3), 302–307 http://dx.doi.org/10.1109/TAP.1966.1138693 | Zbl 1155.78304
[23] Zhang Y., Gao B.-Q., Propagation of cylindrical waves in media of time-dependent permittivity, Chinese Phys. Lett., 2005, 22(2), 446–449 http://dx.doi.org/10.1088/0256-307X/22/2/049
[24] Zlatev Z., Dimov I., Computational and Numerical Challenges in Environmental Modelling, Stud. Comput. Math., 13, Elsevier, Amsterdam, 2006 | Zbl 1120.65103