Difference Schemes Based on the Laguerre Transform

Optimal difference schemes based on the Laguerre transform are proposed for solving the wave equation. Additional parameters are introduced into the difference scheme used for the equations of harmonics. Numerical values of these parameters are obtained by minimizing the error in the difference approximation of the Helmholtz equation. The optimal parameter values thus obtained are used to construct optimal difference schemes. Optimal difference schemes of second- and fourth-order accuracy are considered. The optimal parameters of the difference schemes are presented. Their values depend only on the ratio of the spatial step sizes. It is shown that the use of optimal difference schemes improves the accuracy of solutions. The efficiency of the algorithm is enhanced by applying a simple modification of the difference scheme.