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IEEE Transactions on Antennas and Propagation
Volume 46 Number 8, August 1998

Reducing the Phase Error for Finite-Difference Methods Without Increasing the Order

Page 1194.

Abstract:

The phase error in finite-difference (FD) methods is related to the spatial resolution and thus limits the maximum grid size for a desired accuracy. Greater accuracy is typically achieved by defining finer resolutions or implementing higher order methods. Both these techniques require more memory and longer computation times. In this paper, new modified methods are presented which are optimized to problems of electromagnetics. Simple methods are presented that reduce numerical phase error without additional processing time or memory requirements. Furthermore, these methods are applied to both the Helmholtz equation in the frequency domain and the finite-difference time-domain (FDTD) method. Both analytical and numerical results are presented to demonstrate the accuracy of these new methods.

References

1. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302-307, Apr. 1966.
2. B. Enquist and A. Majda, "Radiation boundary conditions for the numerical simulation of waves," Math Comp., vol. 31, pp. 629-651, 1977.
3. K. K. Mei and J. Fang, "Superabsorption: A method to improve absorbing boundary conditions," IEEE Trans. Antennas Propagat., vol. 40, pp. 1001-1010, Sept. 1992.
4. R. Higdon, "Absorbing boundary conditions for difference approximations to the multidimensional wave equation," Math Comp., vol. 47, pp. 437-459, Oct. 1986.
5. Z. P. Liao, H. L. Wong, B. P. Yang, and Y. F. Yuan, "A transmitting boundary for transient wave analysis," Scientia Sinica (Ser. A), vol. 27, pp. 1063-1073, Oct. 1984.
6. J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," Computational Phys., vol. 114, pp. 185-200, Oct. 1994.
7. W. C. Chew and W. H. Weedon, "A 3-D perfectly matched medium from modified Maxwell's equation with stretched coordinates," Microwave Opt. Technol. Lett., vol. 7, pp. 599-604, Sept. 1994.
8. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propagat., vol. 43, pp. 1460-1463, Dec. 1995.
9. R. Lee and A. C. Cangellaris, "A study of discretization error in the finite element approximation of wave solutions," IEEE Trans. Antennas Propagat., vol. 40, pp. 542-549, May 1992.
10. A. C. Cangellaris and R. Lee, "On the accuracy of numerical wave simulations based on finite methods," J. Electromagn. Waves Applicat., vol. 6, no. 12, pp. 1635-1653, 1992.
11. A. Taflove, Computational Electrodynamics the Finite-Difference Time-Domain Method.Norwood, MA: Artech House, 1995.
12. J. B. Cole, "A high accuracy FDTD algorithm to solve microwave propagation and scattering problems on a coarse grid," IEEE Trans. Microwave Theory Tech., vol. 43, pp. 2053-2058, Sept. 1995.
13. --, "A nearly exact second-order finite-difference time-domain wave propagation algorithm on a coarse grid," Comput. Phys., vol. 8, no. 6, pp. 730-734, 1994.
14. M. F. Hadi, M. Piket-May, and E. T. Thiele, "A modified FDTD (2,4) scheme for modeling electrically large structures with high phase accuracy," in 12th Annu. Rev. Progress Appl. Comput. Electromagn., Monterey, CA, Mar. 1996, vol. 2, pp. 1023-1030.
15. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics.Boca Raton, FL: CRC, 1993.