Symmetric Positive Semi-Definite FDTD Subgridding Algorithms in Both Space and Time for Accurate Analysis of Inhomogeneous Problems
Kaiyuan Zeng, Dan Jiao
Abstract
In this article, we develop a systematic approach to derive symmetric positive semi-definite (SPD) finite-difference time-domain (FDTD) subgridding operators in both space and time for analyzing general inhomogeneous problems in an accurate fashion. The operators are SPD by construction and independent of the grid ratio. The resultant explicit time marching is guaranteed to be stable because such subgridding operators have only nonnegative real eigenvalues. Furthermore, the use of a time step local to the base grid and the subgrid is permitted without sacrificing stability and accuracy. Moreover, the algorithm takes the subgrid information into account to accurately analyze inhomogeneous problems. In addition, we provide an interpretation of the proposed subgridding operators and show how to implement them in the original difference equation-based FDTD framework. Extensive numerical experiments involving both 2- and 3-D subgrids with various grid ratios and inhomogeneities have demonstrated the stability, accuracy, and efficiency of the proposed new SPD subgridding algorithms.