The finite-difference scheme for one-dimensional Maxwell's equations

A. F. Mastryukov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 23 (2020) no. 1, pp. 69-82

Résumé

This paper deals with a difference scheme of second order of approximation for one-dimensional Maxwell’s equations using the Laquerre transform. Supplementary parameters are introduced into this difference scheme. These parameters are obtained by minimizing the difference approximation error of the Helmholtz equation. The values of these optimal parameters are independent of the step size and the number of nodes in the difference scheme. It is shown that application of the Laguerre decomposition allows obtaining a higher accuracy of approximation of the equations in comparison with similar difference schemes when using the Fourier decomposition. The finite difference scheme of second order with parameters was compared to the difference scheme of fourth order in two cases. The use of an optimal difference scheme when solving the problem of electromagnetic impulse propagation in an inhomogeneous medium yields the accuracy of the solution compatible with that of the difference scheme of fourth order. When solving the inverse problem, the second order difference scheme makes possible to attain a higher accuracy of the solution as compared to the difference scheme of fourth order. In the considered problems, the application of the difference scheme of second order with supplementary parameters has decreased the calculation time of a problem by 20–25 percent as compared to the fourth order difference scheme.