On the recalculation of ellipsoids in estimating the error of the implicit Stormer method for a second-order linear differential equation

In the paper, a new method is proposed for constructing an error estimate for the numerical solution of the Cauchy problem for a second-order differential equation obtained using the implicit Stormer method. Unlike the previously proposed methods, it allows one to take into account the signs of small terms when recalculating ellipsoids containing the exact solution in the case of implicit multistep numerical method. This leads to a more accurate estimation of the error of the numerical solution and the applicability of the ellipsoid method over large intervals. A numerical experiment is presented demonstrating the effectiveness of the proposed method for obtaining a guaranteed error estimate of the implicit Stormer method.