Highly stable numerical methods for second order differential equations

We consider the numerical solution of second order ordinary differential equations (ODEs) by General Linear Nystr¨om (GLN) methods. GLN methods, introduced in [1], are multi-value numerical methods for second order ODEs, framed into an unifying approach for the analysis of the properties of convergence, consistency and zero-stability. Our investigation is addressed to providing building blocks useful to analyze the linear stability properties of GLN methods: thus, we present the extension of the classical notions of stability matrix, stability polynomial, stability and periodicity interval, P-stability to GLN methods. Special attention will be focused on the practical derivation of highly stable methods, by investigating GLN methods inheriting the same stability properties of highly stable numerical methods existing in literature, e.g. Runge-Kutta-Nystr¨om (RKN) methods based on indirect collocation on Gauss-Legendre points, which are known to be P-stable: this property, in analogy to a similar feature introduced for first order ODEs is called RKN-stability. The stability properties of GLN methods with RKN-stability depend on a quadratic polynomial, which results to be the same stability polynomial of the best RKN method assumed as reference. We finally provide and discuss examples of P-stable irreducible GLN methods with RKN-stability, whose order of convergence is higher than that of the corresponding reference RKN method. Examples of parallel methods for second order ODEs are also considered, and their acceleration on GPUs is also discussed.