On a new method for obtaining a guaranteed error estimate for Numerov's method using ellipsoids

In this article, we consider a numerical solution of the Cauchy problem for a second-order differential equation calculated by the means of the Numerov method. A new method for obtaining a guaranteed error estimate using ellipsoids is proposed. The numerical solution is enclosed in an ellipsoid containing both the exact and the numerical solutions of the problem, which is recalculated at each step. In contrast to the previously proposed method for recalculating ellipsoids, a more accurate estimate of small terms in the difference equation for the error is proposed. This leads to a more accurate estimate of the error of the numerical solution and the applicability of the proposed method to estimating the error on longer intervals. The results of estimating the error of Numerov's method in solving the two-body problem over a large interval are presented. This numerical experiment demonstrates the effectiveness of the proposed method.