Abstract

We explore the feasibility of using a local spectral time-domain (LSTD) method to solve Maxwell’s equations that arise in optical and electromagnetic applications. The discrete singular convolution (DSC) algorithm is implemented in the LSTD method for spatial derivatives. Fourier analysis of the dispersive error of the DSC algorithm indicates that its grid density requirement for accurate simulations can be as low as approximately two grid points per wavelength. The analysis is further confirmed by numerical experiments. Our study reveals that the LSTD method has the potential to yield high resolution for solving large-scale electromagnetic problems.

© 2003 Optical Society of America

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  1. T. A. Driscoll and B. Fornberg, J. Comput. Phys. 140, 47 (1998).
    [CrossRef]
  2. Q. H. Liu, IEEE Trans. Geosci. Remote Sens. 37, 917 (1999).
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    [CrossRef]
  5. J. B. Cole, Comput. Phys. 11, 287 (1997).
    [CrossRef]
  6. E. Turkel, in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, ed. (Artech House, Boston, Mass., 1998), Chap. 2.
  7. H. M. Jurgens and D. W. Zingg, SIAM J. Sci. Comput. 22, 1675 (2000).
    [CrossRef]
  8. M. Krumpholz and J. Oliger, IEEE Trans. Microwave Theory Tech. 44, 555 (1996).
    [CrossRef]
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  10. G. W. Wei, J. Chem. Phys. 110, 8930 (1999).
  11. G. Bao, G. W. Wei, and A. H. Zhou, “Analysis of regularized Whittaker–Kotel’nikov–Shannon sampling expansion” submitted to SIAM J. Numer. Anal.
  12. G. W. Wei, Y. B. Zhao, and Y. Xiang, Int. J. Numer. Meth. Eng. 55, 913 (2002).
    [CrossRef]
  13. G. W. Wei, J. Phys. A 33, 4935 (2000).
    [CrossRef]
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    [CrossRef]
  15. D. W. Zingg and T. T. Chisholm, Appl. Numer. Math. 31, 227 (1999).
    [CrossRef]

2002

G. W. Wei, Y. B. Zhao, and Y. Xiang, Int. J. Numer. Meth. Eng. 55, 913 (2002).
[CrossRef]

S. Y. Yang, Y. C. Zhou, and G. W. Wei, Comput. Phys. Commun. 143, 113 (2002).
[CrossRef]

2000

G. W. Wei, J. Phys. A 33, 4935 (2000).
[CrossRef]

H. M. Jurgens and D. W. Zingg, SIAM J. Sci. Comput. 22, 1675 (2000).
[CrossRef]

1999

G. W. Wei, J. Chem. Phys. 110, 8930 (1999).

Q. H. Liu, IEEE Trans. Geosci. Remote Sens. 37, 917 (1999).
[CrossRef]

D. W. Zingg and T. T. Chisholm, Appl. Numer. Math. 31, 227 (1999).
[CrossRef]

1998

T. A. Driscoll and B. Fornberg, J. Comput. Phys. 140, 47 (1998).
[CrossRef]

1997

J. B. Cole, Comput. Phys. 11, 287 (1997).
[CrossRef]

1996

M. Krumpholz and J. Oliger, IEEE Trans. Microwave Theory Tech. 44, 555 (1996).
[CrossRef]

1994

Z. Haras and S. Ta’asan, J. Comput. Phys. 114, 265 (1994).
[CrossRef]

1966

K. S. Yee, IEEE Trans. Antennas Propag. AP-14, 302 (1966).

Bao, G.

G. Bao, G. W. Wei, and A. H. Zhou, “Analysis of regularized Whittaker–Kotel’nikov–Shannon sampling expansion” submitted to SIAM J. Numer. Anal.

Chisholm, T. T.

D. W. Zingg and T. T. Chisholm, Appl. Numer. Math. 31, 227 (1999).
[CrossRef]

Cole, J. B.

J. B. Cole, Comput. Phys. 11, 287 (1997).
[CrossRef]

Driscoll, T. A.

T. A. Driscoll and B. Fornberg, J. Comput. Phys. 140, 47 (1998).
[CrossRef]

Fornberg, B.

T. A. Driscoll and B. Fornberg, J. Comput. Phys. 140, 47 (1998).
[CrossRef]

Haras, Z.

Z. Haras and S. Ta’asan, J. Comput. Phys. 114, 265 (1994).
[CrossRef]

Harvey, J. F.

L. P. B. Katehi, J. F. Harvey, and E. Tentzeris, in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, ed. (Artech House, Boston, Mass., 1998), Chap. 3.

Jurgens, H. M.

H. M. Jurgens and D. W. Zingg, SIAM J. Sci. Comput. 22, 1675 (2000).
[CrossRef]

Katehi, L. P. B.

L. P. B. Katehi, J. F. Harvey, and E. Tentzeris, in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, ed. (Artech House, Boston, Mass., 1998), Chap. 3.

Krumpholz, M.

M. Krumpholz and J. Oliger, IEEE Trans. Microwave Theory Tech. 44, 555 (1996).
[CrossRef]

Liu, Q. H.

Q. H. Liu, IEEE Trans. Geosci. Remote Sens. 37, 917 (1999).
[CrossRef]

Oliger, J.

M. Krumpholz and J. Oliger, IEEE Trans. Microwave Theory Tech. 44, 555 (1996).
[CrossRef]

Ta’asan, S.

Z. Haras and S. Ta’asan, J. Comput. Phys. 114, 265 (1994).
[CrossRef]

Tentzeris, E.

L. P. B. Katehi, J. F. Harvey, and E. Tentzeris, in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, ed. (Artech House, Boston, Mass., 1998), Chap. 3.

Turkel, E.

E. Turkel, in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, ed. (Artech House, Boston, Mass., 1998), Chap. 2.

Wei, G. W.

G. W. Wei, Y. B. Zhao, and Y. Xiang, Int. J. Numer. Meth. Eng. 55, 913 (2002).
[CrossRef]

S. Y. Yang, Y. C. Zhou, and G. W. Wei, Comput. Phys. Commun. 143, 113 (2002).
[CrossRef]

G. W. Wei, J. Phys. A 33, 4935 (2000).
[CrossRef]

G. W. Wei, J. Chem. Phys. 110, 8930 (1999).

G. Bao, G. W. Wei, and A. H. Zhou, “Analysis of regularized Whittaker–Kotel’nikov–Shannon sampling expansion” submitted to SIAM J. Numer. Anal.

Xiang, Y.

G. W. Wei, Y. B. Zhao, and Y. Xiang, Int. J. Numer. Meth. Eng. 55, 913 (2002).
[CrossRef]

Yang, S. Y.

S. Y. Yang, Y. C. Zhou, and G. W. Wei, Comput. Phys. Commun. 143, 113 (2002).
[CrossRef]

Yee, K. S.

K. S. Yee, IEEE Trans. Antennas Propag. AP-14, 302 (1966).

Zhao, Y. B.

G. W. Wei, Y. B. Zhao, and Y. Xiang, Int. J. Numer. Meth. Eng. 55, 913 (2002).
[CrossRef]

Zhou, A. H.

G. Bao, G. W. Wei, and A. H. Zhou, “Analysis of regularized Whittaker–Kotel’nikov–Shannon sampling expansion” submitted to SIAM J. Numer. Anal.

Zhou, Y. C.

S. Y. Yang, Y. C. Zhou, and G. W. Wei, Comput. Phys. Commun. 143, 113 (2002).
[CrossRef]

Zingg, D. W.

H. M. Jurgens and D. W. Zingg, SIAM J. Sci. Comput. 22, 1675 (2000).
[CrossRef]

D. W. Zingg and T. T. Chisholm, Appl. Numer. Math. 31, 227 (1999).
[CrossRef]

Appl. Numer. Math.

D. W. Zingg and T. T. Chisholm, Appl. Numer. Math. 31, 227 (1999).
[CrossRef]

Comput. Phys.

J. B. Cole, Comput. Phys. 11, 287 (1997).
[CrossRef]

Comput. Phys. Commun.

S. Y. Yang, Y. C. Zhou, and G. W. Wei, Comput. Phys. Commun. 143, 113 (2002).
[CrossRef]

IEEE Trans. Antennas Propag.

K. S. Yee, IEEE Trans. Antennas Propag. AP-14, 302 (1966).

IEEE Trans. Geosci. Remote Sens.

Q. H. Liu, IEEE Trans. Geosci. Remote Sens. 37, 917 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

M. Krumpholz and J. Oliger, IEEE Trans. Microwave Theory Tech. 44, 555 (1996).
[CrossRef]

Int. J. Numer. Meth. Eng.

G. W. Wei, Y. B. Zhao, and Y. Xiang, Int. J. Numer. Meth. Eng. 55, 913 (2002).
[CrossRef]

J. Chem. Phys.

G. W. Wei, J. Chem. Phys. 110, 8930 (1999).

J. Comput. Phys.

T. A. Driscoll and B. Fornberg, J. Comput. Phys. 140, 47 (1998).
[CrossRef]

Z. Haras and S. Ta’asan, J. Comput. Phys. 114, 265 (1994).
[CrossRef]

J. Phys. A

G. W. Wei, J. Phys. A 33, 4935 (2000).
[CrossRef]

SIAM J. Sci. Comput.

H. M. Jurgens and D. W. Zingg, SIAM J. Sci. Comput. 22, 1675 (2000).
[CrossRef]

Other

G. Bao, G. W. Wei, and A. H. Zhou, “Analysis of regularized Whittaker–Kotel’nikov–Shannon sampling expansion” submitted to SIAM J. Numer. Anal.

L. P. B. Katehi, J. F. Harvey, and E. Tentzeris, in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, ed. (Artech House, Boston, Mass., 1998), Chap. 3.

E. Turkel, in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, ed. (Artech House, Boston, Mass., 1998), Chap. 2.

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Figures (5)

Fig. 1
Fig. 1

Plots of the dispersion errors.

Fig. 2
Fig. 2

Geometry of the dielectric square problem.

Fig. 3
Fig. 3

Contour plots of an electric field and errors in the electric field. (a) Reference solution generated by use of the DSC algorithm with h=0.0025, (b) FD2 error (h=0.005), (c) FD4 error (h=0.01), (d) DSC error (h=0.01). Because the errors are small, the gray scales for the errors have much shorter ranges than that of the reference solution.

Fig. 4
Fig. 4

Absolute errors in the electric field across the diagonal of a dielectric square.

Fig. 5
Fig. 5

Plot of dispersive errors with M=200.

Tables (2)

Tables Icon

Table 1 Numerical Errors of the LSTD Method with M=200 and n2=2002a

Tables Icon

Table 2 Large-Scale Electromagnetic Studya

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

μDzt=Byx-Bxy, ϵBxt=-Dzy,    ϵByt=Dzx,
fnxk=-MMδα,σnx-xkfxk,    n=0,1,,
δh,σx-xk=sinx-xkπ/hx-xkπ/hexp-x-xk22σ2,
δh,σx-xk=sinx-xkπ/hx-xkπ/h.

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