A New Unconditionally Stable Scheme for FDTD Method Using Associated Hermite Orthogonal Functions

Abstract

An unconditionally stable solution using associated Hermite (AH) functions is proposed for the finite-difference time-domain (FDTD) method. The electromagnetic fields and their time derivatives in time-domain Maxwell's equations are expanded by these orthonormal basis functions. By applying Galerkin temporal testing procedure to these expanded equations the time variable can be eliminated from the calculations. A set of implicit equations is derived to calculate the magnetic filed expansion coefficients of all orders of AH functions for the temporal variable. And the electrical field coefficients can be obtained respectively. With the appropriate translation and scale parameters, we can find a minimum-order basis functions subspace to approach a particular electromagnetic field. The numerical results have shown that the proposed method can reduce the CPU time to 0.59% of the traditional FDTD method while maintaining good accuracy.

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using Paralleling-in-Order Scheme

“…From [7], if a causal function ur ; t ðÞ , such as the electric or magnetic field function, can be expanded by…”

“…To overcome the numerical stability constraints of conventional finitedifference time-domain (FDTD) method [1,2], many unconditionally stable methods to reduce or eliminate requirements of the stability condition have been proposed and developed, such as alternating-direction implicit method [2,3] and locally one-dimensional schemes [3], explicit and unconditionally stable FDTD method [4], and orthogonal expansions in time domain [5][6][7][8]. For the orthogonal expansions schemes, field-versus-time variations in the FDTD space lattice are expanded using an appropriate set of orthogonal temporal basis and testing functions, such as weighted Laguerre polynomials (WLP) and associated Hermite (AH) functions, which leads to two different solution schemes: marching-on-in-order and paralleling-in-order, respectively.…”

New unconditionally stable scheme for solving two-dimensional acoustic wave problems

“…Hence, to overcome the conflict in the calculative stability of the FDS, some unconditionally stable schemes [4][5][6] containing the alternating direction implicit (ADI) difference scheme have been proposed. Huang et al [7] and Fu et al [8] proposed a new orthogonal decomposition scheme using the associated Hermite polynomials to eliminate the CFL stability condition (AH-FDS). However, the AH-FDS has the drawback of a need for large internal storage and heavy computation.…”

An Explicit and Unconditionally Stable FDTD Method for the Analysis of General 3-D Lossy Problems

“…In the past decade, a number of implicit unconditionally stable methods have been developed such as the alternating-direction implicit (ADI) method [3], [4], the Manuscript Crank-Nicolson (CN) method [5], the CN-based Split Step (SS) scheme [6], the pseudo-spectral time domain (PSTD) method [7], the locally one dimensional (LOD) FDTD [8], [9], the Laguerre FDTD method [10], [11], the Associated Hermite (AH) type FDTD [13], a series of fundamental schemes [14], a recent one-step unconditionally stable method [24], and others. In these methods, the time discretization scheme is different from that of an explicit FDTD.…”