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

Annotation

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.

Keywords

ellipsoid method, error estimation, Numerov’s method, numerical solution of the Cauchy problem for second-order ODEs

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DOI

10.20310/2686-9667-2022-27-139-261-269

UDC

519.622

Pages

261-269

References

[1] R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to Interval Analysis, 1st ed., SIAM, Philadelphia, 2009, 184 с. [2] N.D. Zolotareva, “Ellipsoid method for estimating the global error of the Stormer method”, Moscow University Computational Mathematics and Cybernetics, 2002, №1, 20-26. [3] A.F. Philippov, N.D. Zolotareva, “Estimates of local and global errors of the Stormer method for systems of equations”, Computational Mathematics and Mathematical Physics, 44:1 (2004), 100-111. [4] N.D. Zolotareva, “Estimates of local and global errors of the implicit Stormer method for systems of equations”, Computational Mathematics and Mathematical Physics, 45:2 (2005), 256-260. [5] N.D. Zolotareva, “New approach to Shtermer's method guaranteed error assessment using ellipsoids”, Moscow University Computational Mathematics and Cybernetics, 2002, №3, 3-9. [6] N.S. Bakhvalov, N.P. Zhidkov, G.M. Kobelkov, Numerical Methods, Nauka Publ., Moscow, 1987. [7] F.L. Chernousko, Estimation of the Phase State of Dynamical Systems, Nauka Publ., Moscow, 1988. [8] Yu.N. Reshetnyak, “Summation of ellipsoids in the guaranteed estimation problem”, Applied. Math. and Mechan., 53:2 (1989), 249-254.

Received

2022-06-21

Section of issue

Scientific articles

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