Abstract
In this paper we consider the family of General Linear Methods (GLMs) for the numerical solution of special second order Ordinary Differential Equations (ODEs) of the type y′′ = f(y(t)), with the aim to provide a unifying approach for the analysis of the properties of consistency, zero-stability and convergence. This class of methods properly includes all the classical methods already considered in the literature (e.g. linear multistep methods, Runge–Kutta–Nyström methods, two-step hybrid methods and two-step Runge–Kutta–Nyström methods) as special cases. We deal with formulation of GLMs and present some general results regarding consistency, zero-stability and convergence. The approach we use is the natural extension of the GLMs theory developed for first order ODEs.
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References
Burrage, K., Butcher, J.C.: Non-linear stability for a general class of differential equation methods. BIT 20, 185–203 (1980)
Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Runge–Kutta and General Linear Methods. Wiley, New York (1987)
Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)
Coleman, J.P.: Order Conditions for a class of two–step methods for y′′ = f(x,y). IMA J. Numer. Anal. 23, 197–220 (2003)
D’Ambrosio, R., Ferro, M., Paternoster, B.: Two-step hybrid collocation methods for y′′ = f(x,y). Appl. Math. Lett. 22, 1076–1080 (2009)
D’Ambrosio, R., Paternoster, B.: Runge–Kutta–Nyström stability for a class of general linear methods for y′′ = f(x,y). In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.) Numerical Analysis and Applied Mathematics. AIP Conference Proceedings, vol. 1168(1), pp. 444–447 (2009)
Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations I—nonstiff problems. In: Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (2000)
Hairer, E., Wanner, G.: Solving ordinary differential equations II—stiff and differential–algebraic problems. In: Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (2002)
Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York-London (1962)
Ixaru, L.Gr., Vanden Berghe, G.: Exponential Fitting. Kluwer Academic Publishers, Dordrecht (2004)
Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, Hoboken, New Jersey (2009)
Paternoster, B.: Two step Runge–Kutta–Nystrom methods for y′′ = f(x,y) and P–stability. In: Sloot, P.M.A., Tan, C.J.K., Dongarra, J.J., Hoekstra, A.G. (eds.) Computational Science – ICCS 2002. Lecture Notes in Computer Science, vol. 2331, Part III, pp. 459–466. Springer Verlag, Amsterdam (2002)
Wright, W.M.: General linear methods with inherent Runge–Kutta stability. Ph.D. thesis, The University of Auckland, New Zealand (2002)
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D’Ambrosio, R., Esposito, E. & Paternoster, B. General linear methods for y′′ = f (y (t)). Numer Algor 61, 331–349 (2012). https://doi.org/10.1007/s11075-012-9637-z
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DOI: https://doi.org/10.1007/s11075-012-9637-z