The Associated Hermite Orthogonal Expansion in Time-Domain (AH-OETD) Method: A New Tool for Solvig Multi-scale and Multi-physical Fields

This paper provides a comprehensive review of the Associated Hermite orthogonal expansion in time-domain (AH-OETD) method. Initially designed as a solution tool for Maxwell's equations, this method employs a paralleling-in-order scheme for the analysis of electromagnetic (EM) problems. Over time, it has evolved beyond its original scope, and has relevant applications in fundamental schemes for absorption boundaries, plane wave incidents, periodic structures, “alternating direction” iteration technology, and applications of the AH FDTD method in frequency-dependent cases. Together with other polynomials, it forms the OETD method family, which can be classified into two categories: marching-on-in-order scheme and paralleling-in-order scheme. The AH FDTD is based on time-frequency bridge-based operators to address partial differential equations, and this expansion has enabled its application in analyzing multi-physical fields such as thermal and acoustic fluids, thereby enhancing computational efficiency through a multi-scale approach.