On the convergence of high-accuracy difference schemes for solving nonstationary second-order equations

In this article, fourth-order accurate difference schemes were proposed and analyzed for sys-tems of second-order ordinary differential equations within the class of non-smooth solutions. Stability conditions and a priori estimates were obtained and theorems on the accuracy of the constructed difference schemes were proven. Additive difference schemes were proposed and the results were applied to the study of multidimensional hyperbolic second-order partial differential equations. Accuracy estimates for spatial and temporal variables were obtained. An algorithm for implementing the method was developed and the scheme was tested. The results of a computational experiment illustrated the effectiveness of the constructed numerical methods for solving hyperbolic equations with non-smooth solutions.