volume | PIER Journals

Abstract

Stability and dispersion analysis for the three-dimensional (3-D) leapfrog alternate direction implicit finite difference time domain (ADI-FDTD) method is presented in this paper. The leapfrog ADI-FDTD method is reformulated in the form similar to conventional explicit FDTD method by introducing two auxiliary variables. The auxiliary variables serve as perturbations of the main fields variables. The stability of the leapfrog ADI-FDTD method is analyzed using the Fourier method and the eigenvalues of the Fourier amplification matrix are obtained analytically to prove the unconditional stability of the leapfrog ADI-FDTD method. The dispersion relation of the leapfrog ADI-FDTD method is also presented.

1. 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 Tech., Vol. 48, No. 9, 1550-1558, Sep. 2000.
doi:10.1109/22.869007

2. Namiki, T., "3-D ADI-FDTD method - Unconditionally stable time-domain algorithm for solving full vector Maxwell's equations," IEEE Trans. Microw. Theory Tech., Vol. 48, No. 10, 1743-1748, Oct. 2000.
doi:10.1109/22.873904

3. Tan, E. L., "Fundamental schemes for effcient unconditionally stable implicit finite-difference time-domain methods," IEEE Trans. Antennas Propagat., Vol. 56, No. 1, 170-177, Jan. 2008.
doi:10.1109/TAP.2007.913089

4. Heh, D. Y. and E. L. Tan, "Unified effcient fundamental ADI-FDTD schemes for lossy media," Progress In Electromagnetics Research B, Vol. 32, 217-242, 2011.
doi:10.2528/PIERB11051801

5. Tay, W. C, D. Y. Heh, and E. L. Tan, "GPU-accelerated funda- mental ADI-FDTD with complex frequency shifted convolutional perfectly matched layer," Progress In Electromagnetics Research M, Vol. 14, 177-192, 2010.
doi:10.2528/PIERM10090605

6. Fu, W. and E. L. Tan, "Development of split-step FDTD method with higher-order spatial accuracy," Electron. Lett., Vol. 40, No. 20, 1252-1254, Sep. 2004.
doi:10.1049/el:20046040

7. Kong, Y. -D. and Q. -X. Chu, "Reduction of numerical dispersion of the six-stages split-step unconditionally-stable FDTD method with controlling parameters," Progress In Electromagnetics Research, Vol. 122, 175-196, 2012.
doi:10.2528/PIER11082512

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

9. Tan, E. L., "Unconditionally stable LOD-FDTD method for 3-D Maxwell's equations," IEEE Microw. Wireless Comp. Lett., Vol. 17, No. 2, 85-87, Feb. 2007.
doi:10.1109/LMWC.2006.890166

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

11. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd Edition, Artech House, Boston, MA, 2005.

12. Dai, J., Z. Chen, D. Su, and X. Zhao, "Stability analysis and improvement of the conformal ADI-FDTD methods," IEEE Trans. Antennas Propagat., Vol. 59, No. 6, 2248-258, Jun. 2011.
doi:10.1109/TAP.2011.2143686

13. Wang, B. Z., Y.Wang, W. Yu, and R. Mittra, "A Hybrid 2-D ADI- FDTD subgridding scheme for modeling on-chip interconnects," IEEE Trans. Adv. Packag., Vol. 24, No. 4, 528-533, Nov. 2001.

14. Chevalier, M. W., R. J. Luebbers, and V. P. Cable, "FDTD local grid with material traverse," IEEE Trans. Antennas Propagat., Vol. 45, No. 3, 411-421, Mar. 1997.
doi:10.1109/8.558656

15. Cooke, S. J., M. Botton, T. M. Antonsen, and B. Levush, "A leapfrog formulation of the 3D ADI-FDTD algorithm," Int. J. Numer. Model, Vol. 22, No. 2, 187-200, 2009.
doi:10.1002/jnm.707

16. Jolani, F., Y. Yu, and Z. Chen, "Effcient modeling of open structures using nonuniform Leapfrog ADI-FDTD," IEEE Antennas Wireless Propag. Lett., Vol. 10, 561-564, 2011.
doi:10.1109/LAWP.2011.2158380

17. Yang, S., Y. Yu, Z. Chen, and W. Yin, "The convolutional perfectly matched layer (CPML) for the leapfrog ADI-FDTD method," 2011 IEEE MTT-S International Microwave Sympo- sium Digest, 2011.

18. Tay, W. C. and E. L. Tan, "Implementations of PMC and PEC boundary conditions for effcient fundamental ADI- and LOD- FDTD," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 4, 565-573, 2010.

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

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

21. Heh, D. Y. and E. L. Tan, "Dispersion analysis of FDTD schemes for doubly lossy media," Progress In Electromagnetics Research B, Vol. 17, 327-342, 2009.
doi:10.2528/PIERB09082802

Mr. Theng Huat Gan

Dr. Eng Leong Tan