High-Order Split-Step Unconditionally-Stable FDTD Methods and Numerical Analysis

High-order split-step unconditionally-stable finite-difference time-domain (FDTD) methods in three-dimensional (3-D) domains are presented. Symmetric operator and uniform splitting are adopted simultaneously to split the matrix derived from the classical Maxwell's equations into four sub-matrices. Accordingly, the time step is divided into four sub-steps. In addition, high-order central finite-difference operators based on the Taylor central finite-difference method are used to approximate the spatial differential operators first, and then the uniform formulation of the proposed high-order schemes is generalized. Subsequently, the analysis shows that all the proposed high-order methods are unconditionally stable. The generalized form of the dispersion relations of the proposed high-order methods is carried out. Moreover, the effects of the mesh size, the time step and the order of schemes on the dispersion are illustrated through numerical results. Specifically, the normalized numerical phase velocity error (NNPVE) and the maximum NNPVE of the proposed second-order scheme are lower than that of the alternating direction implicit (ADI) FDTD method. Furthermore, the analysis of the accuracy of the proposed methods is presented. In order to demonstrate the efficiency of the proposed methods, numerical experiments are presented.