A Novel Four-Step Weakly Conditionally Stable HIE-FDTD Algorithm and Numerical Analysis
A novel four-step weakly conditionally stable hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) algorithm in three-dimensional (3-D) domains is presented in this paper, which is suitable for a finer discretization in one dimension. Based on the exponential evolution operator (EEO), the Maxwell's equations in a matrix form can be split into four sub-procedures. Accordingly, the time step is divided into four sub-steps. In addition, by taking second-order central finite-difference approximation for both the temporal and spatial derivatives, the formulation of the proposed four-step HIE-FDTD method is obtained. The proposed four-step HIE-FDTD algorithm is implemented, in which the implicit scheme was applied only in one direction with a fine grid, and the explicit scheme was applied in two other directions with coarser grids. Compared with the existing HIE-FDTD methods, the proposed method has a weaker Courant-Friedrichs-Lewy (CFL) stability condition (and), which means that the proposed method can improve computational efficiency by taking larger time step size. Since the CFLN stability condition of the proposed method is determined by the smaller grid size of the two coarse grid sizes, the proposed method is suitable for analyzing the electromagnetic objects with fine structures in one direction effectively. Besides, the numerical dispersion analysis is given, and the (Δt ≤ 2Δx/c and Δt ≤ 2Δz/c) comparisons of the numerical dispersion analysis among the proposed method, traditional FDTD method, ADI-FDTD method, and two existing HIE-FDTD methods are given. Finally, to testify the computational accuracy and efficiency, numerical experiments of the five FDTD methods are presented.