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IEEE Transactions on Antennas and Propagation
Volume 48 Number 2, February 2000

Table of Contents for this issue

Complete paper in PDF format

Numerical Stability of NonorthogonalFDTD Methods

Stephen D. Gedney, Senior Member, IEEE and J. Alan Roden Member, IEEE

Page 231.

Abstract:

In this paper, a sufficient test for the numerical stability of generalized grid finite-difference time-domain (FDTD) schemes is presented. It is shown that the projection operators of such schemes must be symmetric positive definite. Without this property, such schemes can exhibit late-time instabilities. The origin and the characteristics of these late-time instabilities are also uncovered. Based on this study, nonorthogonal grid FDTD schemes (NFDTD) and the generalized Yee (GY) methods are proposed that are numerically stable in the late time for quadrilateral prism elements, allowing these methods to be extended to problems requiring very long-time simulations. The study of numerical stability that is presented is very general and can be applied to most solutions of Maxwell's equations based on explicit time-domain schemes.

References

  1. A. Taflove, Computational Electrodynamics: The finite-difference time-domain method, Boston, MA: Artech House, 1995.
  2. T. G. Jurgens, A. Taflove, K. R. Umashankar and T. G. Moore, "Finite-difference time-domain modeling of curved surfaces", IEEE Trans. Antennas Propagat., vol. 40, pp.  357-366,  Apr.  1992.
  3. J.-F. Lee, R. Palendech and R. Mittra, "Modeling three-dimensional discontinuities in waveguides using nonorothogonal FDTD algorithm", IEEE Trans. Microwave Theory Tech., vol. 40, pp.  346-352, Feb.  1992.
  4. N. Madsen, "Divergence preserving discrete surface integral methods for Maxwell's equations using nonorthogonal unstructured grids", J. Computat. Phys., vol. 119, pp.  34-45, 1995.
  5. S. Gedney, F. Lansing and D. Rascoe, "A full-wave analysis of passive monolithic integrated circuit devices using a generalized Yee-algorithm", IEEE Trans. Microwave Theory Tech., vol. 44, pp.  1393-1400, Aug.  1996 .
  6. J. A. Roden, "Broadband electromagnetic analysis of complex structures with the finite-difference time-domain technique in general curvilinear coordinates", Ph.D., Dept. Elect. Eng., Univ. Kentucky, Lexington, KY, 1997.
  7. I. J. Craddock, C. J. Railton and J. P. McGeehan, "Derivation and application of a passive equivalent circuit for the finite-difference time-domain algorithm", IEEE Microwave Guided Wave Lett., vol. 6, pp.  40-42, Jan.  1996.
  8. N. Madsen, "Divergence preserving discrete surface integral methods for Maxwell's equations using nonorthogonal unstructured grids", Tech. Rep. UCRL-JC-109787, LLNL, Feb. 1992.
  9. S. Gedney and F. Lansing, "Explicit time-domain solutions of Maxwell's equations using nonorthogonal and unstructured grids,"in Finite Difference Time Domain Methods for Electrodynamic Analyzes, A. Taflove, Ed. New York: Artech House, 1995.
  10. J. F. Lee, R. Lee and A. Cangellaris, "Time-domain finite-element methods", IEEE Trans. Antennas Propagat., vol. 45, pp.  430-442, Mar.  1997 .
  11. B. Gustafsson, H.-O. Kreiss and J. Oliger, Time-Dependent Problems and Difference Methods, New York: Wiley, 1995.
  12. I. Bardi, O. Biro, K. Preis and B. R. Vrisk K. R., "Nodal and edge element analysis of inhomoheneous loaded 3-D cavities", IEEE Trans. Magn., vol. 28, pp.  1141-1145, Feb.  1992.
  13. S. Gedney and U. Navdariwala, "An unconditionally stable implicit finite-element time-domain solution of the vector wave equation", IEEE Microwave Guided Wave Lett., vol. 5, pp.  332-334, Oct.  1995.
  14. Zeland Software, Freemont, CA,
  15. S. D. Gedney, "An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices", IEEE Trans. Antennas Propagat., vol. 44, pp.  1630-1639, Dec.  1996.