General Linear Methods for Special Second Order ODEs

In this talk we introduce the family of General Linear Methods for the numerical solution of special second order Ordinary Differential Equations (ODEs) of the type y’’=f(x,y), with the aim to provide an unifying approach for the analysis of the properties of convergence, consistency and stability. This class of methods includes all the classical methods already considered in the literature, such as linear multistep methods, Runge-Kutta-Nyström methods, two-step hybrid methods and two-step Runge-Kutta-Nyström methods as special cases. The family of methods we aim to consider is wider and more general with respect to the ones already considered in the literature: in fact, our new methods depend on more parameters which can be exploited, for instance, in order to provide a better balance between order of convergence and stability properties. We present the re-formulation of the classical methods according to the new approach and the main results regarding consistency, zero-stability, convergence, order conditions and linear stability theory. As initial step, we focus our attention on methods having a rank two zerostability matrix, covering a widespread class of methods already considered in the literature. The approach we will use is the natural extension of the General Linear Methods theory developed for first order ODEs (Butcher, 2008, Jackiewicz, 2009, and references therein contained)