The error in solving a wave equation based on a scheme with weights

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The error in approximating a Cauchy problem for a two-dimensional wave equation based on a scheme with weights is studied. The dependence of the approximation error on the time step and weight parameter is considered. With this aim in view, the difference of second-order spatial derivatives in the wave equation is approximated, while the time derivative is preserved continuous; and an analytical solution of a Cauchy problem for a system of ordinary differential equations is obtained as the decomposition in the orthonormal basis consisting of the eigenvectors of the operator of the second difference derivative with respect to the spatial variables. Based on this solution, the errors of the approximation of the wave problem by three-layer difference schemes are studied and the conditions for the stability of a three-layer difference scheme are obtained. It is established that, when simulating the propagation of oscillation processes using difference methods, the oscillation frequency values differ from the real ones and depend on the weight parameter and time step. The optimal values of the weight parameter with which the deviation of the oscillation frequency for the difference scheme is minimal are obtained. The dependences of the approximation error on weight and spatial step are derived. The optimal values of the weight parameter with which the schemes are of the second and fourth order of accuracy with respect to the time step are found.