p-stable general Nyström methods for y″=f(y(t))

We focus our attention on the family of General Linear Methods (GLMs), for the numerical solution of second order ordinary differential equations (ODEs). These are multivalue methods introduced in (D'Ambrosio et al., 2012) [3] with the aim to provide an unifying approach for the analysis of the properties of accuracy of numerical methods for second order ODEs. Our investigation is addressed to providing the building blocks useful to analyze the linear stability properties of GLMs for second order ODEs: thus, we present the extension of the classical notions of stability matrix, stability polynomial, stability and periodicity interval, A-stability and P-stability to the family of GLMs. Special attention will be given to the derivation of highly stable GLMs, whose stability properties depend on the stability polynomial of indirect Runge-Kutta-Nystrom methods based on Gauss-Legendre collocation points, which are known to be P-stable. In this way, we are able to provide P-stable GLMs whose order of convergence is greater than that of the corresponding RKN method, without heightening the computational cost. We finally provide and discuss examples of P-stable irreducible GLMs satisfying the mentioned features.

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