Improved ADI Iterative Algorithm With Gauss-Seidel Ideology for Efficient WLP-FDTD Method in 3-D Cylindrical Coordinate System

Abstract

An improved Alternating-Direction-Implicit iterative algorithm with Gauss-Seidel ideology for efficient weighted-Laguerre-polynomial finite-difference time-domain method is proposed in 3-D cylindrical coordinate system. By transferring the coefficient matrix of the conventional WLP-FDTD equation to the right-side of its equal sign, a linear equation system with ADI characteristic is formed, which makes it more flexible. And then, a correction equation is added to the ADI linear equations and the two-steps Gauss-Seidel procedures are applied to instead of the one-step one in the existing scheme, the purpose of these operations is to speed up the convergence and improve the computation in the term of efficiency and accuracy. Meanwhile, a detailed special treatment scheme for on $z$ -axis and $\varphi $ -direction in 3-D cylindrical coordinate system is introduced and discussed, which demonstrates the importance of on ${z}$ -axis and $\varphi $ -direction treatment in the proposed method. In addition, the choice scheme in the term of the time-scaling factor $s$ and the order of the weighted Laguerre polynomials $q$ is discussed. Finally, we develop the Perfectly-Matched-Layer implementation to verify the advantage of the proposed method in accuracy. To validate the term of accuracy and efficiency of the proposed method, six numerical examples are provided. At the same time, the discussions of the convergence speed and stability of the proposed method are presented.

Keywords

• Finite-difference time-domain (FDTD) method
• weighted Laguerre polynomial (WLP)
• altering-direction implicit (ADI)
• Gauss–Seidel ideology
• correction equation