About One Improvement of Approximating Functions Method for the Problems with Nonlinear Media

Volterra integral equation method, based on integral equations equivalent to the Maxwell’s equations, is an alternative to the differential formulation of the problem for modeling a wide range of electrodynamics problems. The approximating functions method, a particular case of the finite element method, plays the role of analytical-numerical component of Volterra integral equation method. It based on partitioning the definition region of the problem by cells and on the approximation of the desired solution by orthogonal polynomials. This process leads to constructing a system of nonlinear algebraic equations, which is the result of calculating the original Volterra integral equation at the mesh points. Its computational efficiency can be significantly improved by dividing each equation of the system into a set of blocks that can be calculated in advance. This article presents a modification of the approximating functions method for solving problems of electrodynamics in one spatial dimension and time domain using the approach of Volterra integral equations. The main purpose of the modification is to increase the speed of computations and reduce the consumption of computer resources, which is especially important when considering problems with nonlinear media. It is proved that the proposed modification does not violate the algorithm of the method and does not lead to an increase in the error. The proposed method is applied to the problems of interaction of electromagnetic pulses of three different types: simple Gaussian pulse, single cycle Gaussian pulse and oscillated Gaussian pulse – incident on a layer with a second-order nonlinear medium, placed in a linear environment. The obtained simulation results are analyzed, estimates of the reduction in computation time and errors are presented.