Reseach On An Unconditionally Stable And Fast Time-domain Algorithm And Its Applications

A new unconditionally stable and fast time-domain algorithm in computational electromagnetics, the finite-difference time-domain(FDTD) method based on weighted Laguerre polynomials(WLPs), is studied with its applications in this dissertation.WLP-FDTD uses the weighted Laguerre polynomials as the temporal basis functions and the Galerkin’s method as the temporal testing procedure to eliminate the time variable, and employs Yee’s cell and central difference discretization in space domain. Thus, with an order-marching procedure, the calculation of both electric and magnetic fields is separated in spatial domain and temporal domain. WLP-FDTD does not deal with time steps and is more efficient than the conventional FDTD method when solving broadband electromagnetic problems with complex and multiscale structures.Based on the existing framework of the WLP-FDTD algorithm, the basic theory and technique are further improved, and the scope of its application is extended.Firstly, a theoretical analysis of the numerical dispersion and key parameter selection of the WLP-FDTD method is presented. The numerical dispersion relation of WLP-FDTD is derived from the Fourier analysis of a monochromatic wave. The relationship between the time-scale factor and operating frequency is derived. The number of the marching order can be calculated by analyzing the maximum zero root of the Laguerre polynomial. Then, the numerical dispersion analysis of two-dimensional(2D) WLP-FDTD is extended to three-dimensional case, which enriches the basic theory of WLP-FDTD. The unconditional stability for marching-on-in-order WLPFDTD, with the amplification factor, is proved from Fourier analysis of a monochromatic wave.Then, the high-order WLP-FDTD method with fourth-order central difference in space domain is presented. Its numerical dispersion relationship and unconditional stability are analyzed and the quantitative analysis of key parameter selection is presented. Compared with low-order WLP-FDTD, the high-order method has low numerical dispersion error, high precision and high computational efficiency characteristics.Third, an auxiliary differential equation(ADE) WLP-FDTD method is presented to simulate electromagnetic wave propagation in general dispersive materials. In addition, a nearly perfectly matched layer(NPML), which is more effective than traditional PML, is presented for the ADE-WLP-FDTD method. The ADE-WLP-FDTD method with its NPML, which is used to simulate electromagnetic issues of complex and dispersive medium, effectively extends the application range of traditional WLP-FDTD.Fourth, both factorization-splitting and domain decomposition techniques are introduced to solve huge sparse matrix equations for ADE-WLP-FDTD. These two techniques for solving a matrix equation are more efficient than the direct calculation.Finally, the WLP-FDTD method in a generalized coordinate system is presented to solve electromagnetic problems with arbitrary curved surface structures. For electromagnetic problems with complex structures, the computational precision is improved while no extra grids are involved in nonorthogonal WLP-FDTD.