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New Energy-Conserved Identitiesand Super-Convergence of the Symmetric Ec-S-Fdtd Scheme for Maxwell’s Equations in 2D

Published online by Cambridge University Press:  20 August 2015

Liping Gao  and
Dong Liang
Liping Gao*
Affiliation:
Department of Computational and Applied Mathematics, School of Sciences, China University of Petroleum, Qingdao, 266555, P.R. China
Dong Liang*
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J1P3, Canada
*
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

The symmetric energy-conserved splitting FDTD scheme developed in is a very new and efficient scheme for computing the Maxwell’s equations. It is based on splitting the whole Maxwell’s equations and matching the x-direction and y-direction electric fields associated to the magnetic field symmetrically. In this paper, we make further study on the scheme for the 2D Maxwell’s equations with the PEC boundary condition. Two new energy-conserved identities of the symmetric EC-S-FDTD scheme in the discrete H1-norm are derived. It is then proved that the scheme is uncondi-tionally stable in the discrete H1-norm. By the new energy-conserved identities, the super-convergence of the symmetric EC-S-FDTD scheme is further proved that it is of second order convergence in both time and space steps in the discrete H1-norm. Numerical experiments are carried out and confirm our theoretical results.

Keywords

65N0665N1265N15Symmetric EC-S-FDTDenergy-conservedunconditional stabilitysuper convergenceMaxwell’s equationssplitting
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 5 , May 2012 , pp. 1673 - 1696
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Chen, W., Li, X. and Liang, D., Symmetric energy-conserved splitting FDTD scheme for the Maxwell’s equations. Commun. Comput. Phys., 6 (2009), 804–825.Google Scholar
[2]Chen, W., Li, X. and Liang, D., Energy-Conserved Splitting FDTD Methods for Maxwell’s Equations. Numer. Math., 108 (2008), 445–485.Google Scholar
[3]Dendy, J. E. Jr., and Fairweather, G., Alternating-direction Galerkin methods for parabolic and hyperbolic problems on rectangular polygons. SIAM J. Numer. Anal., 12 (1975), 144–163.Google Scholar
[4]Douglas, J. Jr., and Rachford, H. H., On the numerical solution of heat conduction problems in two and three space variables. Trans. Amer. Math. Soc., 82 (1956), 421–439.CrossRefGoogle Scholar
[5]Douglas, J. Jr., and Dupont, T., Alternating-direction Galerkin methods on rectangules. In: Proc. Sympos. Numer. Solution of Partial Differential Equations II (ed. Hubbard, B.), Academic Press, New York, 1971, pp. 133–214.Google Scholar
[6]Gao, L., Zhang, B. and Liang, D., The splitting finite difference time domain methods for Maxwell’s equations in two dimensions. J. Comput. Appl. Math., 205 (2007), 207–230.Google Scholar
[7]Gao, L., Zhang, B. and Liang, D., Splitting finite difference methods on staggered grids for the three dimensional time dependent Maxwell’s equations. Commun. Comput. Phys., 4 (2008), 405–432.Google Scholar
[8]Garcia, S. G., Lee, T. W. and Hagness, S. C., On the accuracy of the ADI-FDTD method. IEEE Anten. Wireless Propagat. Letters, 1 (2002), 31–34.Google Scholar
[9]Gedney, S. D., Liu, G., Roden, J. A. and Zhu, A., Perfectly matched layer media with CFS for an unconditional stable ADI-FDTD method. IEEE Trans. Anten. Propagat., AP-49 (2001), 1554–1559.CrossRefGoogle Scholar
[10]Holland, R., Implicit three-dimensional finite differencing of Maxwell’s equations. IEEE Trans. Nucl. Sci. NS-31 (1984), 1322–1326.Google Scholar
[11]Leis, R., Initial Boundary Value Problems in Mathematical Physics, John Wiley, New York, 1986.CrossRefGoogle Scholar
[12]Monk, P. and Süli, E., A convergence analysis of Yee’s scheme on nonuniform grid. SIAM J. Numer. Anal., 31 (1994), 393–412.CrossRefGoogle Scholar
[13]Namiki, T., A new FDTD algorithm based on alternating-direction implicit method. IEEE Trans. Microwave Theory Tech., 47 (1999), 2003–2007.CrossRefGoogle Scholar
[14]Peaceman, D. W. and Rachford, H. H., The numerical solution of parabolic and elliptic difference equations. J. Soc. Ind. Appl. Math., 42 (1955), 28–41.Google Scholar
[15]Taflove, A. and Brodwin, M. E., Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell equations. IEEE Trans. Microwave Theory Tech., 23 (1975), 623–630.CrossRefGoogle Scholar
[16]Taflove, A. and Hagness, S., Computational Electrodynamics: The Finite-Difference TimeDomain Method (2nd Ed.), Artech House, Boston, 2000.Google Scholar
[17]Nicolaides, R. A. and Wang, D. Q., Convergence analysis of a covolume scheme for Maxwell’s equations in three dimensions. Math. Comput., 67 (1998), 947–963.Google Scholar
[18]Yee, K. S., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Anten. Propagat., AP-14 (1966), 302–307.Google Scholar
[19]Zhao, A. P., Analysis of the numerical dispersion of the 2-D alternating-direction implicit FDTD method, IEEE Trans. on Microwave Theory and Tech., 50 (2002), 1156–1164.CrossRefGoogle Scholar
[20]Zheng, F., Chen, Z. and Zhang, J., A finite-ditime-domain method without the Courant stability conditions. IEEE Microwave and Guide Letters, 9 (1999), 441–443.Google Scholar
[21]Zheng, F., Chen, Z. and Zhang, J., Toward the development of a three-dimensional unconditionally stable finite difference time-domain method. IEEE Trans. Microwave Theory Tech. MTT-48 (2000), 1550–1558.Google Scholar
[22]Zheng, F. and Chen, Z., Numerical dispersion analysis of the unconditionally stable 3D ADI-FDTD method. IEEE Trans. Microwave Theory Tech. MTT-49 (2001), 1006–1009.Google Scholar