High-Order Unconditionally-Stable Four-Step Adi-FDTD Methods and Numerical Analysis
Yong-Dan Kong
Qing-Xin Chu
Rong-Lin Li
High-order unconditionally-stable three-dimensional (3-D) four-step alternating direction implicit finite-difference time-domain (ADI-FDTD) methods are presented. 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, 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. Finally, in order to demonstrate the validity of the proposed methods, numerical experiments are presented. Furthermore, the effects of the order of schemes, the propagation angle, the time step, and the mesh size on the dispersion are illustrated through numerical results. Specifically, the normalized numerical phase velocity error (NNPVE) and the maximum NNPVE of the proposed schemes are lower than that of the traditional ADI-FDTD method.
