volume | PIER Journals

Abstract

A novel four-step weakly conditionally stable hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) algorithm in three-dimensional (3-D) domains is presented in this paper, which is suitable for a finer discretization in one dimension. Based on the exponential evolution operator (EEO), the Maxwell's equations in a matrix form can be split into four sub-procedures. Accordingly, the time step is divided into four sub-steps. In addition, by taking second-order central finite-difference approximation for both the temporal and spatial derivatives, the formulation of the proposed four-step HIE-FDTD method is obtained. The proposed four-step HIE-FDTD algorithm is implemented, in which the implicit scheme was applied only in one direction with a fine grid, and the explicit scheme was applied in two other directions with coarser grids. Compared with the existing HIE-FDTD methods, the proposed method has a weaker Courant-Friedrichs-Lewy (CFL) stability condition (and), which means that the proposed method can improve computational efficiency by taking larger time step size. Since the CFLN stability condition of the proposed method is determined by the smaller grid size of the two coarse grid sizes, the proposed method is suitable for analyzing the electromagnetic objects with fine structures in one direction effectively. Besides, the numerical dispersion analysis is given, and the (Δt ≤ 2Δx/c and Δt ≤ 2Δz/c) comparisons of the numerical dispersion analysis among the proposed method, traditional FDTD method, ADI-FDTD method, and two existing HIE-FDTD methods are given. Finally, to testify the computational accuracy and efficiency, numerical experiments of the five FDTD methods are presented.

1. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag., Vol. 14, No. 3, 302-307, May 1966.
doi:10.1109/TAP.1966.1138693

2. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed., Artech House, 2000.

3. Namiki, T., "A new FDTD algorithm based on alternating-direction implicit method," IEEE Trans. Microw. Theory Techn., Vol. 47, No. 10, 2003-Oct. 2007, 1999.
doi:10.1109/22.795075

4. Zheng, F., Z. Chen, and J. Zhang, "Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method," IEEE Trans. Microw. Theory Techn., Vol. 48, No. 9, 1550-1558, Sep. 2000.
doi:10.1109/22.868993

5. Chen, J., Z. Wang, and Y. C. Chen, "Higher-order alternative direction implicit FDTD method," Electron. Lett., Vol. 38, No. 22, 1321-1322, Oct. 2002.
doi:10.1049/el:20020911

6. Fu, W. and E. L. Tan, "Stability and dispersion analysis for higher order 3-D ADI-FDTD method," IEEE Trans. Antennas Propag., Vol. 53, No. 11, 3691-3696, Nov. 2005.

7. Sun, G. and C. W. Trueman, "Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method," IEEE Trans. Microw. Theory Techn., Vol. 54, No. 5, 2275-2284, May 2006.
doi:10.1109/TMTT.2006.873639

8. Tan, E. L., "Efficient algorithms for Crank-Nicolson-based finite difference time-domain methods," IEEE Trans. Microw. Theory Techn., Vol. 56, No. 2, 408-413, Feb. 2008.
doi:10.1109/TMTT.2007.914641

9. Lee, J. and B. Fornberg, "A split step approaches for the 3-D Maxwell's equations," J. Comput. Appl., Vol. 158, 485-505, 2003.
doi:10.1016/S0377-0427(03)00484-9

10. Fu, W. and E. L. Tan, "Development of split-step FDTD method with higher-order spatial accuracy," Electron. Lett., Vol. 40, No. 20, 1252-1253, Sep. 2004.
doi:10.1049/el:20046040

11. Chu, Q. X. and Y. D. Kong, "Three new unconditionally-stable FDTD methods with high-order accuracy," IEEE Trans. Antennas Propag., Vol. 57, No. 9, 2675-2682, Sep. 2009.
doi:10.1109/TAP.2009.2027045

12. Kong, Y. D. and Q. X. Chu, "High-order split-step unconditionally-stable FDTD methods and numerical analysis," IEEE Trans. Antennas Propag., Vol. 59, No. 9, 3280-3289, Sep. 2011.
doi:10.1109/TAP.2011.2161543

13. Shibayama, J., M. Muraki, J. Yamauchi, and H. Nakano, "Efficient implicit FDTD algorithm based on locally one-dimensional scheme," Electron. Lett., Vol. 41, No. 19, 1046-1047, Sep. 2005.
doi:10.1049/el:20052381

14. Ahmed, I., E. Chua, E. P. Li, and Z. Chen, "Development of three-dimensional unconditionally stable LOD-FDTD method," IEEE Trans. Antennas Propag., Vol. 56, No. 11, 3596-3600, Nov. 2008.
doi:10.1109/TAP.2008.2005544

15. Saxena, A. K. and K. V. Srivastava, "A three-dimensional unconditionally stable five-step LOD-FDTD method," IEEE Trans. Antennas Propag., Vol. 62, No. 3, 1321-1329, Mar. 2014.
doi:10.1109/TAP.2013.2293790

16. Chen, J. and J. Wang, "A novel WCS-FDTD method with weakly conditional stability," IEEE Trans. Electromagn. Compat., Vol. 49, No. 2, 419-426, May 2007.
doi:10.1109/TEMC.2007.897130

17. Wang, J. B., B. H. Zhou, C. Gao, B. Chen, and L. H. Shi, "An efficient one-step leapfrog WCS-FDTD method," IEEE Antennas Wireless Propag. Lett., Vol. 13, 1088-1091, 2014.
doi:10.1109/LAWP.2014.2329054

18. Huang, B. K., G. Wang, Y. S. Jiang, and W. B. Wang, "A hybrid implicit-explicit FDTD scheme with weakly conditional stability," Microw. Opt. Technol. Lett., Vol. 39, No. 2, 97-101, Oct. 2003.
doi:10.1002/mop.11138

19. Chen, J. and J. Wang, "A 3D hybrid implicit-explicit FDTD scheme with weakly conditional stability," Microw. Opt. Technol. Lett., Vol. 48, 2291-2294, Nov. 2006.

20. Chen, J. and J. Wang, "Comparison between HIE-FDTD method and ADI-FDTD mehtod," Microw. Opt. Technol. Lett., Vol. 49, No. 5, 1001-1005, May 2007.
doi:10.1002/mop.22340

21. Chen, J. and J. Wang, "Numerical simulation using HIE-FDTD method to estimate various antennas with fine scale structures," IEEE Trans. Antennas Propag., Vol. 55, No. 12, 3603-3612, Dec. 2007.
doi:10.1109/TAP.2007.910338

22. Chen, J. and J. Wang, "A three-dimensional semi-implicit FDTD scheme for calculation of shielding effectiveness of enclosure with thin slots," IEEE Trans. Electromagn. Compat., Vol. 49, No. 2, 354-360, Feb. 2007.
doi:10.1109/TEMC.2007.893329

23. Zhang, Q., B. Zhou, and J. B. Wang, "A novel hybrid implicit-explicit FDTD algorithm with more relaxed stability condition," IEEE Antennas Wireless Propag. Lett., Vol. 12, 1372-1375, 2013.
doi:10.1109/LAWP.2013.2283861

24. Zhang, Q. and B. H. Zhou, "A novel HIE-FDTD method with large time-step size," IEEE Antennas Propag. Magaz., Vol. 57, No. 2, 24-28, Apr. 2015.
doi:10.1109/MAP.2015.2420011

25. Wang, J. B., B. H. Zhou, L. H. Shi, C. Gao, and B. Chen, "A novel 3-D HIE-FDTD method with one-step leapfrog scheme," IEEE Trans. Microw. Theory Techn., Vol. 62, No. 6, 1275-1283, Jun. 2014.
doi:10.1109/TMTT.2014.2320692

26. Wang, J. B., J. L. Wang, B. H. Zhou, and C. Gao, "An efficient 3-D HIE-FDTD method with weaker stability condition," IEEE Trans. Antennas Propag., Vol. 64, No. 3, 998-1004, Mar. 2016.
doi:10.1109/TAP.2015.2513100

Dr. Yong-Dan Kong

Dr. Chu-Bin Zhang

Dr. Min Lai

Prof. Qing-Xin Chu