An unconditionally stable scheme for the finite-difference time-domain method

Abstract

In this work, we propose a numerical method to obtain an unconditionally stable solution for the finite-difference time-domain (FDTD) method for the TE/sub z/ case. This new method does not utilize the customary explicit leapfrog time scheme of the conventional FDTD method. Instead we solve the time-domain Maxwell's equations by expressing the transient behaviors in terms of weighted Laguerre polynomials. By using these orthonormal basis functions for the temporal variation, the time derivatives can be handled analytically, which results in an implicit relation. In this way, the time variable is eliminated from the computations. By introducing the Galerkin temporal testing procedure, the marching-on in time method is replaced by a recursive relation between the different orders of the weighted Laguerre polynomials if the input waveform is of arbitrary shape. Since the weighted Laguerre polynomials converge to zero as time progresses, the electric and magnetic fields when expanded in a series of weighted Laguerre polynomials also converge to zero. The other novelty of this approach is that, through the use of the entire domain-weighted Laguerre polynomials for the expansion of the temporal variation of the fields, the spatial and the temporal variables can be separated.