Unconditionally stable finite-difference time-domain methods with high-order accuracy in two and three dimensions

Abstract

Finite-difference time-domain (FDTD) methods with high-order accuracy in two-dimensional (2D) and three-dimensional (3D) cases are presented, which are based on the split-step scheme and the Crank–Nicolson scheme. In the proposed methods, a symmetric operator and a uniform splitting are adopted simultaneously to split the matrix derived from the classical Maxwell's equations into six sub-matrices. Accordingly, the time step is divided into six sub-steps. Subsequently, our analysis results show that the proposed methods in the 2D and 3D cases are unconditionally stable. The dispersion relations of the proposed methods are derived. The normalised numerical phase velocity errors and the numerical dispersion errors of the proposed methods are lower than those of the alternating direction implicit (ADI)-FDTD method and the four-stage split-step (SS4)-FDTD method. Furthermore, the accuracy analysis of the proposed methods is generated. In order to demonstrate the efficiency of the proposed methods, the numerical experiments in the 2D and 3D cases are carried out. With the same level of accuracy, the proposed methods cost less CPU time and lower memory requirement than those of the ADI-FDTD method and the SS4-FDTD method.