時域有限差分法是許多數值電磁分析工具之ㄧ。這個方法廣泛的應用在模擬天線、電子封裝和波導等等。然而，此方法並不適用於大型結構的物體。FDTD模擬大型物體時會需要使用較多的記憶體。

最近，時域偽譜法被引入用來求解Maxwell’s equation。此方法採用複利葉轉換來計算空間中的微分。根據Nyquist取樣定理，每波長只需要取樣2個網格，因此有效的模擬大型物體。本論文描述PSTD結合FDTD的方法應用在不同的方向。FDTD使用的方向適合細小的結構以及PSTD使用的方向適合大型的結構。

The finite-difference time-domain (FDTD) method is one of the most popular numerical electromagnetic analysis tools. This method has been applied to a wide variety of problems such as antennas, electronic packaging, waveguides, etc. However, it is not suitable for large scale structures. The enormous memory requirement prohibits the use of FDTD to a large electrical size.

Recently, the pseudospectral time-domain (PSTD) method has been introduced for solution of Maxwell’s equation. This method adopts the Fourier transform algorithm to perform the spatial derivatives. According to Nyquist sampling theorem, it requires only 2 cells per wavelength, so that it is possible to efficiently model larger scale problems. This thesis describes a combination of PSTD and FDTD method applied in different directions. The FDTD be applied to directions along fine structures and the PSTD be applied in direction along large structures.

目錄……..…………………………………………………………............................IV
圖表目錄…..……………………………………………………….......................... .V
第一章 序論……..……………………………………………………………………1
1.1 概述…..……………………………………………………………………...1
1.2 論文大綱…………………………………………………………………….4
第二章 FDTD演算法……...…………………………………………………………5
2.1 FDTD公式推導……………..………...………………….………………….5
2.2 Courant穩定準則….……. …………………………………………………..9
第三章 PSTD演算法…..……. …………………………………….……………….10
3.1 PSTD公式推導……….…………………………………………………….10
3.2 數值色散分析.…..…….…………………………………………………...12
3.3 穩定準則分析…………...…….….…………………….………………….16
3.4 Berenger的PML吸收邊界條件………………………...………………….20
3.5 全場/散射場公式……...…………..……………………………………….24
3.6 PSTD的數值模擬………………………….……………...………………..27
第四章 PSTD結合FDTD……………………….…………………………………..31
4.1 PSTD結合FDTD之公式推導………….……..…………………………..31
4.2 PSTD結合FDTD之數值色散與穩定性分析.…….……..………………..34
4.3 FDTD結合FDTD之數值模擬……….……………….……..….………….37
第五章 結論………………………………..………………………………………..49
參考文獻……………………………………………………………………………..50

[1] K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations," IEEE Trans. Antennas Propagat, vol. 14, pp. 300-307, May. 1966.
[2] A. Taflove, "Computational Electrodynamics : The Finite-Difference Time-Domain Method, 1995.
[3] T. Namiki, "A new FDTD algorithm based on alternating-direction implicit method," IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2003-2007, Oct. 1999.
[4] E. A. Navarro, N. T. Sangary and J. Litva, "Some considerations on the accuracy of the nonuniform FDTD method and its application to waveguide analysis when combined with the perfectly matched layer technique," IEEE Trans. Microw. Theory Tech., vol. 44, pp. 1115-1124, July. 1996.
[5] M. Okoniewski, E. Okoniewska and M. Stuchly, "Three-dimensional subgridding algorithm for FDTD," IEEE Trans. Antennas Propag., vol. 45, pp. 422-429, Mar. 1997.
[6] J. W. Cooley and J. W. Tukey, "An Algorithm for the Machine Calculation of Complex Fourier Series," Math. Comput., vol. 19, pp. 297-301, 1965.
[7] N. Nguyen and Q. H. Liu, "Regular Fourier matrices and nonuniform fast Fourier transforms," SIAM J. SCI. Comput., vol. 21, pp. 283-293, 1999.
[8] H. O. Kreiss and J. Oliger, "Stability of the Fourier Method," SIAM Journal on Numerical Analysis, vol. 16, pp. 421-433, 1979.
[9] N. N. Bojarski, "The k-space formulation of the scattering problem in the time domain," J. Acoust. Soc. Am., vol. 72, pp. 570-584, 1982.
[10] Q.H.Liu, "Transient Electromagnetic Modeling with the Generalized k-Space (GKS) Method," Microwae Opt. Technol. Lett., vol. 7, pp. 842-848, 1994.
[11] Q. H. Liu, "Generalization of the k-space formulation to elastodynamic scattering problems," J. Acoust. Soc. Am., vol. 97, pp. 1373-1379, 1995.
[12] Q.H.Liu, "The PSTD algorithm: A time-domain method requiring only two cells per wavelength," Microwave Opt. Technol. Lett., vol. 15, pp. 158-165, June. 1997.
[13] Q.H.Liu, "The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media," IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 45, pp. 1044-1055, 1998.
[14] M. Krumpholz and L. P. B. Katehi, "MRTD: new time-domain schemes based on multiresolution analysis," IEEE Trans. Microw. Theory Tech., vol. 44, pp. 555-571, Apr. 1996.
[15] Q.H.Liu, "Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm," IEEE Trans. Geosci. Remote Sens., vol. 37, pp. 917-926, Feb. 1999.
[16] A. Taflove and M. E. Brodwin, "Numerical Solution of Steady-State Electromagnetic Scattering Problems Using the Time-Dependent Maxwell's Equations," IEEE Trans. Micro. Theory Tech., vol. 23, pp. 623-630, Aug. 1975.
[17] D. S. Katz, E. T. Thiele and A. Taflove, "Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FDTD meshes," IEEE Microwave and Guided Wave Letters, vol. 4, pp. 268-270, Aug. 1994.
[18] Q. Li, Y. Chen and D. Ge, "Comparison study of the PSTD and FDTD methods for scattering analysis," Microwave Opt. Technol. Lett., vol. 25, pp. 220-226, May. 2000.
[19] Y.F.Leung and C.H.Chan, "Combining the FDTD and PSTD methods," Microwave Opt. Technol. Lett., pp. 249-254, July. 1999.