On the recalculation of ellipsoids when estimating the error of the Stormer interpolation method for a nonlinear second-order differential equation

Abstract

In this paper, we consider a numerical solution of the Cauchy problem for a second-order differential equation calculated using the interpolation (implicit) multistep Stormer method. A guaranteed error estimate is found using ellipsoids that contain exact and numerical solutions. The use of ellipsoids avoids the wrapping effect that occurs when using interval analysis methods. It is proposed to modernize the method of obtaining a more accurate estimate of small terms in the recalculation of ellipsoids, which was previously proposed for solving linear differential equations, to the case of nonlinear equations. Formulas for recalculating ellipsoids have been obtained to obtain a guaranteed estimate of the error of the Stormer interpolation multistep method. The results of estimating the error in the numerical solution of nonlinear second-order differential equations on large intervals are presented, demonstrating the effectiveness of the proposed method.