Paper Title
Progress on the Paralleling-In-Order Based Orthogonal Expansion in Time-Domain Methods: A Review

Abstract
In recent years, the orthogonal expansions in time-based finite-difference time-domain (FDTD) methods have emerged as an important branch of unconditionally stable algorithms. By expanding electromagnetic fields and their time derivatives using various orthogonal basis functions—such as Associated Hermite (AH), Chebyshev (CS), and Legendre (LD)—a number of enhanced FDTD schemes have been developed. These methods overcome the stability constraints of traditional FDTD by enabling larger time steps, and they demonstrate excellent accuracy and efficiency in solving multiscale and complex electromagnetic problems. This paper reviews the theoretical foundations and advances of these methods, with a particular focus on the development of CS-FDTD and AH-FDTD. Their advantages and limitations are discussed, and potential future research directions and applications are outlined.