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

Abstract

Fourth-order accuracy difference schemes are proposed and studied in this article, for systems
of second-order ordinary differential equations in the class of non-smooth solutions. Stability conditions
and a priori estimates are obtained. Theorems on the accuracy of the constructed difference schemes
are proven. Additive difference schemes are proposed. The results obtained are applied to the study
of multidimensional hyperbolic second-order partial differential equations, where accuracy estimates are
obtained for spatial and temporal variables. An algorithm for implementing the method was developed
and the scheme was tested. The results of a computational experiment illustrate the effectiveness of the
constructed numerical methods for solving hyperbolic equations with non-smooth solutions.