The solution to the 2D Maxwell equations by Laguerre spectral method

In this paper, a spectral method for solving 2D Maxwell equations with relaxation of electromagnetic parameters is presented. The method proposed is based on the expansion of equations solution in the Laguerre functions in the temporal domain.
The operation of functions convolution that is a part of formulas, describing relaxation processes is reduced to the sum of harmonics products. Maxwell's equations transform to a system of linear algebraic equations for harmonics of the solution. In the algorithm, the inner parameter of the Laguerre transform is used. With large values of this parameter, the solution is shifted to the field of high harmonics. This is done to simplify the numerical algorithm and to increase the efficiency of the problem solution.
The results of comparison between the accuracy of the Laguerre method and a finite-difference method both for 2D medium structure and for a layered medium are given. The results of comparison of efficiency of the spectral and the finite difference methods for the axial and for the plane geometries of the problem are presented.