General linear methods for y′′ = f (y (t))

• DOI: 10.1007/s11075-012-9637-z
• Publication Date: 2012-08-23T07:30:44Z
• Authors: R. D’Ambrosio; E. Esposito; B. Paternoster
• Keywords: Runge–Kutta methods; Linear multistep method; Ode; Theory of computation; Zero (linguistics)

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---Nystrom methods, two-step hybrid methods and two-step Runge---Kutta---Nystrom 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.