Difference scheme for the wave equation

The paper deals with a numerical solution of the wave equation. The solution algorithm uses optimal parameters which are obtained by using Laguerre transform in time for the wave equation. Additional parameters are introduced into a difference scheme of 2nd-order approximation for the equation. The optimal values of these parameters are obtained by minimizing the error of a difference approximation of the Helmholtz equation. Applying the inverse Laguerre transform in the equation for harmonics, a differential-difference wave equation with the optimal parameters is obtained. This equation is difference in the spatial variables and differential in time. An iterative algorithm for solving the differential-difference wave equation with the optimal parameters is proposed. 2-dimensional and 1-dimensional equations are considered. The results of numerical calculations of the differential-difference equations are presented. It is shown that the difference schemes with the optimal parameters give an increase in the accuracy of solving the equations.