Sampling Biorthogonal Time Domain Method (SBTD) The finite difference time domain (FDTD) method was proposed by K. Yee [1] in 1966. The simplicity of the FDTD method in mathematics has proved to be its great advantage. The method does not involve any integral equations, Green’s functions, singularities, nor matrix equations. Neither does it involve functional or variational principles. In addition the FDTD proves to be versatile when used in complicated geometries. The computational issues associated with the FDTD are the radiation boundary conditions or absorption boundary conditions for open structures, numerical dispersion, and stability conditions. Its major drawbacks include its massive. | Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright 2003 John Wiley Sons Inc. ISBN 0-471-41901-X CHAPTER FIVE Sampling Biorthogonal Time Domain Method SBTD The finite difference time domain FDTD method was proposed by K. Yee 1 in 1966. The simplicity of the FDTD method in mathematics has proved to be its great advantage. The method does not involve any integral equations Green s functions singularities nor matrix equations. Neither does it involve functional or variational principles. In addition the FDTD proves to be versatile when used in complicated geometries. The computational issues associated with the FDTD are the radiation boundary conditions or absorption boundary conditions for open structures numerical dispersion and stability conditions. Its major drawbacks include its massive memory consumption and huge computational time. In these regard wavelets offer significant improvements to the FDTD. It will be shown that the Yee-based FDTD is identical to the Galerkin method using Haar wavelets. Since the Haar bases are discontinuous the slow decay of the frequency components and the Gibbs phenomena of the Haar basis prevent the use of a coarse mesh in the FDTD. In contrast the Daubechies-based sampling functions are continuous basis functions with fast decay in both the spatial and spectral domains. Thus a more efficient time domain method can be derived the sampling biorthogonal time domain SBTD algorithm. BASIS FDTD FORMULATION For a lossy medium with a conductivity a we begin with Maxwell s two curl equations -V x E d t e aE V x H. d t 189 190 SAMPLING BIORTHOGONAL TIME DOMAIN METHOD SBTD We obtain by the leapfrog method 2 a set of finite difference equations k i 2 Hx f m 2 n k- 1 2 Hx m 2 n 2 - - kEy m 1 n 1H - kEy m 1 n i z L A 2 2 At r i - z K .Um n-n 2 _zA . 1 1 kEz m n I k 1 2 Hy f 2 m n 2 k- 1 2 Hy 2 til n 2 ---7 kEz f 1 m n k Ez f m n Z-Ax L 2 2 kEx 2 m n 1 - kEx 2 m n Az L A 2 2 k 1 2 Hz f 2 m 2 n k- 1 2 Hz 2 m
