A new FDTD solution method without the time variable

This paper presents a new FDTD formulation where the time variable has completely been eliminated and one solves the problem in space only. We propose a numerical method to obtain an unconditionally stable solution for the finite difference time domain (FDTD) method for the TE _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 proposed method is 100 times faster than a conventional FDTD procedure without the undesired effects.