General linear methods for y'' = f(y(t))

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. © 2012 Springer Science+Business Media, LLC.