Solving frequency dependent losses in the time domain without the time variable using the associated Laguerre functions

In a traditional finite difference time-domain (FDTD) method, one needs to approximate the time derivatives by finite differences and the time domain convolutions through finite summations, when addressing wave propagation in a general Debye, Drude or a Lorentz medium. In this paper, we present a marching-on-in-degree (MOD) method in a FDTD framework for analyzing transient electromagnetic responses in a general dispersive media. The two issues related to the finite difference approximation of the time derivatives and the time consuming convolution operations are handled analytically using the properties of the associated Laguerre functions. The basic idea here is that we fit the transient nature of the fields, the flux densities, the permittivity and the permeability with a finite sum of orthogonal associated Laguerre functions. Through this novel approach. Not only the time variable can be decoupled analytically from the temporal variations but also the final computational form of the equations is transformed from FDTD to a FD formulation through a Galerkin testing. Representative numerical examples are presented for transient wave propagation in general Debye, Drude or a Lorentz dispersive medium. Numerical results demonstrate that this methodology can result in accurate and stable solutions. Examples will also be presented to illustrate how a similar methodology can be used for the analysis transient skin effect losses in an integral equation methodology in the transient analysis of large conducting structures.