Ramos, H.; Vigo-Aguiar, J. Variable stepsize Störmer-Cowell methods. (English) Zbl 1092.65060 Math. Comput. Modelling 42, No. 7-8, 837-846 (2005). Authors’ summary: Störmer and Cowell methods are multistep codes for the numerical solution of second-order initial value problems where the first derivative does not appear explicitly. In this paper, we develop a procedure to obtain \(k\)-step Störmer and Cowell methods in their variable step size version, avoiding computation of the coefficients by recurrences or integrals. We offer a strategy for conveniently selecting the stepsize. Considering a pair of explicit and implicit formulae, these may be implemented in a predictor-corrector mode. Reviewer: Zdzislaw Jackiewicz (Tempe) Cited in 19 Documents MSC: 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems Keywords:variable step size; second-order differential equations; multistep methods; predictor-corrector methods; Störmer method; Cowell method; initial value problems PDFBibTeX XMLCite \textit{H. Ramos} and \textit{J. Vigo-Aguiar}, Math. Comput. Modelling 42, No. 7--8, 837--846 (2005; Zbl 1092.65060) Full Text: DOI References: [1] Hairer, E.; Norsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I (1987), Springer: Springer New York · Zbl 0638.65058 [2] Ixaru, L. G., Numerical Methods for Differential Equations and Applications (1984), Editura Academiei: Editura Academiei Berlin [3] Lambert, J. D., Numerical Methods for Ordinary Differential Systems (1991), John Wiley: John Wiley Romania · Zbl 0745.65049 [4] Cano, B.; García-Archilla, B., A generalization to variable stepsizes of Störmer methods for second-order differential equations, Applied Numerical Mathematics, 19, 401-417 (1996) · Zbl 0856.65088 [5] Vigo-Aguiar, J., An approach to variable coefficients multistep methods for special differential equations, Int. J. Appl. Math., 1, 8, 911 (1999) · Zbl 1171.65411 [6] Isaacson, E.; Keller, H. B., Discrete Variable Methods in Ordinary Differential Equations (1966), John Wiley: John Wiley England · Zbl 0168.13101 [7] Ramos, H., Nuevas familias de métodos numéricos para la resolución de problemas de segundo orden de tipo oscilatorio, (Doctorial Dissertation (2004), Salamanca University: Salamanca University New York), (in Spanish) [8] Shampine, L. F.; Gordon, M. K., Computer Solution of Ordinary Differential Equations. The Initial Value Problem (1975), Freeman · Zbl 0347.65001 [9] Willé, D. R., Experiments in stepsize control for Adams linear multistep methods, Advances in Computational Mathematics, 8, 335-344 (1998) · Zbl 0905.65087 [10] Henrici, P., Discrete Variable Methods in Ordinary Differential Equations (1962), John Wiley: John Wiley San Francisco, CA · Zbl 0112.34901 [11] Papageorgiou, G., A P-stable singly diagonally implicit Runge-Kutta-Nyström method, Numerical Algorithms, 17, 345 (1998) · Zbl 0939.65097 [12] Khiyal, M. S.H.; Thomas, R. M., Variable-order, variable-step methods for second-order initial-value problems, Journal of Computational and Applied Mathematics, 79, 263 (1997) · Zbl 0876.65056 [13] Stiefel, E.; Bettis, D. G., Stabilization of Cowell’s method, Numer. Math., 13, 154-175 (1969) · Zbl 0219.65062 [14] Psihoyios, G.; Simos, T. E., The numerical solution of orbital problems with the use of symmetric four-step trigonometrically-fitted methods, Applied Numerical Analysis and Computational Mathematics, 1, 1-2, 217-223 (2004) · Zbl 1064.65056 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.