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Convergence of an ADI splitting for Maxwell’s equations. (English) Zbl 1323.78018

The authors investigate the convergence of an alternating direction implicit method for Maxwell’s equations on a cuboid going back to F. H. Zheng, Z. Z. Chen and J. Z. Zhang [“Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method”, IEEE Trans. Microw. Theory Tech. 48, 1550–1558 (2000; doi:10.1109/22.869007)]. The main result of the paper is the proof of second-order convergence in time independent on the spatial discretization. The authors use an explicit formula of the global error for the semi-discretization derived in a framework of operator semigroup theory and regularity results of the Cauchy problem. So the convergence analysis respects the unboundedness of the involved differential operators. Numerical experiments confirm the second-order convergence in time independently on the mesh size and the impact of the smoothness of coefficients on the accuracy.

MSC:

78M20 Finite difference methods applied to problems in optics and electromagnetic theory
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q61 Maxwell equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
47D03 Groups and semigroups of linear operators
65J10 Numerical solutions to equations with linear operators
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[1] Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21(9), 823-864 (1998) · Zbl 0914.35094
[2] Chen, W., Li, X., Liang, D.: Energy-conserved splitting FDTD methods for Maxwell’s equations. Numer. Math. 108(3), 445-485 (2008) · Zbl 1185.78020
[3] Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151(3), 221-276 (2000) · Zbl 0968.35113
[4] Dautray, R., Lions, J.-L.: Mathematical analysis and numerical methods for science and technology. In: Spectral Theory and Applications. With the Collaboration of Michel Artola and Michel Cessenat, Vol. 3, 2nd edn. Springer, Berlin (2000a) · Zbl 0942.35001
[5] Dautray, R., Lions, J.-L.: Mathematical analysis and numerical methods for science and technology. In: Evolution Problems. I. With the Collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Vol. 5, 2nd edn. Springer, Berlin (2000b) · Zbl 0956.35003
[6] Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Volume 194 of Graduate Texts in Mathematics. Springer, Berlin (2000) · Zbl 0952.47036
[7] Faragó, I., Horváth, R., Schilders, W.H.A.: Investigation of numerical time-integrations of Maxwell’s equations using the staggered grid spatial discretization. Int. J. Numer. Model. Electron. Netw. Dev. Fields 18(2), 149-169 (2005) · Zbl 1108.78016
[8] Gao, L., Zhang, B., Liang, D.: The splitting finite-difference time-domain methods for Maxwell’s equations in two dimensions. J. Comput. Appl. Math. 205(1), 207-230 (2007) · Zbl 1122.78021
[9] Garcia, S.G., Lee, Tae-Woo, Hagness, S.C.: On the accuracy of the ADI-FDTD method. IEEE Antennas Wirel. Propag. Lett. 1(1), 31-34 (2002)
[10] Gonzáles García, S., Rubio Bretones, A., Gómez Martín, R., Hagness, S.C.: Accurate implementation of current sources in the ADI-FDTD scheme. IEEE Antennas Wirel. Propag. Lett. 3(1), 141-144 (2004)
[11] Gonzáles García, S., Godoy Rubio, R., Rubio Bretones, A., Gómez Martín, R.: On the dispersion relation of ADI-FDTD. IEEE Microw. Wirel. Compon. Lett. 16(6), 354-356 (2006)
[12] Hairer, E., Lubich, Ch., Wanner, G.: Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Volume 31 of Springer Series in Computational Mathematics. Springer, Berlin, Heidelberg (2006) · Zbl 1094.65125
[13] Hansen, E., Ostermann, A.: Dimension splitting for evolution equations. Numer. Math. 108, 557-570 (2008) · Zbl 1149.65084
[14] Lee, J., Fornberg, B.: A split step approach for the 3-D Maxwell’s equations. J. Comput. Appl. Math. 158(2), 485-505 (2003) · Zbl 1029.65094
[15] Lee, J., Fornberg, B.: Some unconditionally stable time stepping methods for the 3D Maxwell’s equations. J. Comput. Appl. Math. 166(2), 497-523 (2004) · Zbl 1034.78019
[16] Leis, R.: Initial-Boundary Value Problems in Mathematical Physics. B. G. Teubner, Stuttgart (1986) · Zbl 0599.35001
[17] Namiki, T.: A new FDTD algorithm based on alternating-direction implicit method. IEEE Trans. Microw. Theory Tech. 47(10), 2003-2007 (1999)
[18] Namiki, T.: 3-D ADI-FDTD method-unconditionally stable time-domain algorithm for solving full vector Maxwell’s equations. IEEE Trans. Microw. Theory Tech. 48(10), 1743-1748 (Oct 2000)
[19] Rauch, J.: Partial Differential Equations, Volume 128 of Graduate Texts in Mathematics. Springer, New York (1991) · Zbl 0742.35001
[20] Reed, M., Simon, B.: Methods of modern mathematical physics I. Functional Analysis, 2nd edn. Academic Press, Inc., Harcourt Brace Jovanovich, Publishers, New York (1980) · Zbl 0459.46001
[21] Sz-Nagy, B.: Spektraldarstellung Linearer Transformationen des Hilbertschen Raumes. Berichtigter Nachdruck, Volume 39 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin, New York (1967) (reprint of the 1942 original edition edition) · Zbl 0146.12602
[22] Taflove, A., Hagness, S.C.: Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edn. Artech House Publishers, Norwood (2005) · Zbl 0963.78001
[23] Tucsnak, M., Weiss, G.: Observation and control for operator semigroups. Birkhäuser, Basel (2009) · Zbl 1188.93002
[24] Verwer, J.G.: Component splitting for semi-discrete Maxwell equations. BIT 51(2), 427-445 (2011) · Zbl 1221.65247
[25] Verwer, J.G.: Composition methods, Maxwell’s equations, and source terms. SIAM J. Numer. Anal. 50(2), 439-457 (2012) · Zbl 1263.78015
[26] Verwer, J.G., Botchev, M.A.: Unconditionally stable integration of Maxwell’s equations. Linear Algebra Appl. 431(3-4), 300-317 (2009) · Zbl 1170.78004
[27] Yee, K.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14(3), 302-307 (1966) · Zbl 1155.78304
[28] Zheng, F., Chen, Z., Zhang, J.: Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method. IEEE Trans. Microw. Theory Tech. 48(9), 1550-1558 (2000)
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