A general framework for the numerical solution of second order ODEs

In this paper the authors consider the family of general linear methods (GLMs) for special second order ordinary differential equations (ODEs) of the type y″ = f(y(t)), recently introduced with the aim to provide an unifying formulation for numerical methods solving such problems and achieve a general strategy for the analysis of the minimal demandings in terms of accuracy and stability to be asked for, such as consistency, zero-stability and convergence. They emphasize the generality of this approach, by showing that the family of GLMs for second order ODEs recovers classical numerical formulae known in the literature and allows to easily obtain new methods by proving their convergence in a simple, straightforward way.