Differential–Difference Wave Equation

This paper discusses difference schemes for the wave equation. The wave equation is replaced by a differential–difference equation, where the spatial derivatives are replaced by finite differences of the fourth order of approximation and the time derivative is replaced by the average value over several spatial points with weight factors–parameters. In this equation, the Laguerre transform is carried out and it is reduced to the Helmholtz equation. The optimal values of the parameters are obtained by minimizing the error of the difference approximation of the Helmholtz equation for the zero harmonic of Laguerre. This differential difference equation is solved numerically using iterations over small optimal parameters. The results of the solution for several methods of averaging the derivative over time are presented. It is shown that the accuracy of the obtained solutions significantly depends on the method of averaging the derivative over time. The use of difference schemes with optimal parameters leads to an increase in the accuracy of solving equations. Accounting for optimal parameters comes down to a simple modernization of a nonoptimal difference scheme.