# Numerical solution of the Maxwell equations in time-varying media using Magnus expansion

István Faragó; Ágnes Havasi; Robert Horváth

Open Mathematics (2012)

- Volume: 10, Issue: 1, page 137-149
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topIstván Faragó, Ágnes Havasi, and Robert Horváth. "Numerical solution of the Maxwell equations in time-varying media using Magnus expansion." Open Mathematics 10.1 (2012): 137-149. <http://eudml.org/doc/269193>.

@article{IstvánFaragó2012,

abstract = {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.},

author = {István Faragó, Ágnes Havasi, Robert Horváth},

journal = {Open Mathematics},

keywords = {Maxwell equations; Numerical solution; Magnus expansion; Operator splitting; numerical solution; operator splitting},

language = {eng},

number = {1},

pages = {137-149},

title = {Numerical solution of the Maxwell equations in time-varying media using Magnus expansion},

url = {http://eudml.org/doc/269193},

volume = {10},

year = {2012},

}

TY - JOUR

AU - István Faragó

AU - Ágnes Havasi

AU - Robert Horváth

TI - Numerical solution of the Maxwell equations in time-varying media using Magnus expansion

JO - Open Mathematics

PY - 2012

VL - 10

IS - 1

SP - 137

EP - 149

AB - 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.

LA - eng

KW - Maxwell equations; Numerical solution; Magnus expansion; Operator splitting; numerical solution; operator splitting

UR - http://eudml.org/doc/269193

ER -

## References

top- [1] Bagrinovskiĭ K.A., Godunov S.K., Difference schemes for multidimensional problems, Dokl. Akad. Nauk. SSSR, 1957, 115, 431–433 (in Russian) Zbl0087.12201
- [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 Zbl1151.65343
- [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 Zbl1159.78346
- [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 Zbl1187.65084
- [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 Zbl1086.65053
- [6] Faragó I., Havasi Á., Horváth R., On the order of operator splitting methods for non-autonomous systems (submitted) Zbl1243.65077
- [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 Zbl1108.78020
- [11] Hundsdorfer W., Verwer J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Ser. Comput. Math., 33, Springer, Berlin, 2003 Zbl1030.65100
- [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 Zbl0056.34102
- [15] Marchuk G.I., Splitting Methods, Nauka, Moscow, 1988 (in Russian) Zbl0653.65065
- [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 Zbl0945.34004
- [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 Zbl0184.38503
- [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 Zbl1153.78309
- [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 Zbl1155.78304
- [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 Zbl1120.65103

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.