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, zerostability and convergence. The approach we use is the natural extension of the GLMs theory developed for first order ODEs.