High-Order Unconditionally-Stable Four-Step Adi-FDTD Methods and Numerical Analysis

Abstract

Abstract-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.

A 3-D Unconditionally Stable Laguerre-Rpim Meshless Method for Time-Domain Electromagnetic Computations

“…∂φ i (r) ∂y (6) where N j is the number of H nodes in the support domain of interested E node and N i is the number of H nodes in the support domain of interested E node.…”

“…Unlike conventional simulation algorithms [1][2][3][4][5][6][7], which relies on a grid or a mesh, meshless techniques, in contrast, use scattered nodes to represent the spatial solving area as shown in Figure 1. These nodes can be placed randomly in the solution region, and no strict limitation between adjacent nodes is required, thus make the meshless methods more flexible for solving EM problems, especially for those which have curved and slopped boundaries.…”

Parallel Implementation and Application of the MRTD With an Efficient CFS-PML

“…The Multi-Resolution Time-Domain (MRTD) technique was first published in 1996 by Krumpholz and Katehi [1,2], and has been developed rapidly as an efficient numerical algorithm in the timedomain like the long established Finite Difference Time-Domain (FDTD) technique [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and other time-domain methods [18][19][20]. As the dispersion of the MRTD scheme compared to the conventional FDTD scheme shows an excellent capability to approximate the exact solution with negligible error for sampling rates approaching the Nyquist limit, it becomes possible that larger targets can be simulated without sacrificing accuracy.…”

Implementation and Application of the Spherical MRTD Algorithm

“…The Multi-Resolution Time-Domain (MRTD) technique was first published in 1996 by Krumpholz and Katehi [1], and has been developed rapidly as an efficient numerical algorithm in the timedomain like the long established Finite Difference Time-Domain (FDTD) technique [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and other time-domain methods [17][18][19]. With highly-linear dispersion performance, the MRTD scheme implies that a low sampling rate in space can still provide for a relatively small phase error in the numerical simulation of a wave propagation problem, so it becomes possible that larger targets can be simulated without sacrificing accuracy.…”