High-Order Unconditionally Stable Two-Step Leapfrog ADI-FDTD Methods and Numerical Analysis

Abstract

High-order unconditionally stable two-step leapfrog alternating direction implicit-finite-difference time-domain (ADI-FDTD) methods in three-dimensional (3-D) domains are presented. Based on the exponential evolution operator (EEO), the Maxwell's equations in a matrix form can be split into four subprocedures first, and then two subprocedures are generated by using the leapfrog scheme. Subsequently, the proposed high-order methods are theoretically proven for unconditional stability, and the numerical dispersion is derived analytically. There are several factors to effect the dispersion: for example, the order of schemes, propagation angle, time step, and mesh size. These four factors are illustrated comprehensively in this paper. Specifically, the normalized numerical phase velocity error (NNPVE) of the proposed schemes is lower than that of the one-step leapfrog ADI-FDTD method. Finally, numerical experiments are presented to demonstrate the validity of the proposed methods. It was found that the proposed second-order method has the same level of relative error as that of the four-step ADI-FDTD method, but it has a higher computational efficiency.