Research On Split-Step Scheme-Based Unconditionally Stable Finite-Difference Time-Domain Methods

Recently, to resolve the limitation of the Courant-Friedrichs-Lewy (CFL) condition on the time step size of the finite-difference time-domain (FDTD) method, an unconditionally-stable FDTD method has been developed, which can reduce the CPU time and improve the computational efficiency. Due to the great development of communication industry, components with characteristics of miniaturization are required. The unconditionally-stable FDTD method plays an important role in the numerical technique, and many researches on it have been carried out. A detailed study on the analytical theory and numerical method of the unconditionally-stable FDTD methods is presented in this dissertation. Some novel unconditionally-stable FDTD methods with low dispersion error and high-order accuracy are proposed, and the numerical analyses of the unconditionally-stable FDTD methods are given.The main contributions of this dissertation can be summarized as follows.(?) Based on the split-step (SS) scheme and Crank-Nicolson (CN) scheme, a novel three-dimensional (3D) split-step unconditionally-stable FDTD method is proposed. The proposed method provides simple algorithm implementation and new split forms. Along the x, y and z coordinate directions, the Maxwell's matrix is split into three sub-matrices. Accordingly, the time step is divided into three sub-steps, and the proposed method is denoted as SSCN3-FDTD. Therefore, a 3D problem is converted to three 1D problems, which reduces the computational complexity. Furthermore, when the Maxwell's matrix is split into six sub-matrices, another unconditionally-stable SSCN6-FDTD method is presented. The SSCN3-FDTD method and SSCN6-FDTD method have second-order accuracy both in time and space. The numbers of arithmetic operators of the SSCN3-FDTD method and the SSCN6-FDTD method are fewer than that of the alternating direction implicit (ADI) FDTD method. Thus, the CPU time is reduced and the computational efficiency is improved. In addition, based on the high-order central finite-difference operators, the high-order SSCN6-FDTD methods are proposed. To reduce the numerical dispersion error, three controlling parameters are introduced when splitting the matrix, and an improved SSCN6-FDTD method is proposed.(?) A new split-step unconditionally-stable FDTD method with high-order accuracy in 2-D domains is proposed, which has four sub-steps and is denoted as SS4-FDTD-2D. The SS4-FDTD-2D method has new splitting form, which is different from the SS1-FDTD-2D method. Besides, the normalized numerical phase velocity error and the normalized numerical phase velocity anisotropic error of the SS4-FDTD-2D method are lower than those of the ADI-FDTD-2D method, the SS1-FDTD-2D method and the SS2-FDTD-2D method. Applied to open region FDTD applications, an unconditionally-stable SS4-FDTD-2D method with nearly perfectly matched layer (NPML) absorbing boundary conditions is proposed. To reduce the numerical dispersion error further, two parameters, which can be controlled, are introduced when the matrix is split, and an improved SS4-FDTD-2D method is proposed.(?) Based on the split-step scheme, three novel unconditionally-stable FDTD methods with low numerical dispersion and high-order accuracy in 2D domains are proposed. In the first method, symmetric operator and uniform splitting are adopted simultaneously to split the Maxwell's matrix into six sub-matrices, and meanwhile the time step is divided into six sub-steps. For that the second and third proposed methods are obtained by adjusting the order of the sub-matrices deduced from the first novel method, all the novel methods have the similar formulations, and the normalized numerical phase velocity errors and the normalized numerical phase velocity anisotropic errors of three novel methods are lower than those of the ADI-FDTD-2D method and the SS1-FDTD-2D method.(?) The SS4-FDTD-2D method is extended into 3D domains, and a novel 3D unconditionally-stable FDTD method with high-order accuracy is proposed, which is denoted as SS4-FDTD-3D. Symmetric operator and uniform splitting are adopted simultaneously to split the Maxwell's matrix into four sub-matrices. Accordingly, the time step is divided into four sub-steps. The normalized numerical phase velocity error and the normalized numerical phase velocity anisotropic error of the SS4-FDTD-3D method are lower than that of the ADI-FDTD method. The SS4-FDTD-3D method with coarser mesh achieves the same level of accuracy as the ADI-FDTD method with finer mesh. In addition, based on the high-order central finite-difference operators, high-order SS4-FDTD-3D methods are proposed.