Multivalue approximation of second order differential problems: a review

We present a collection of recent results on the numerical approximation of second order differential problems of the type y'' = f(y(t)) by means of family of multivalue numerical methods, here denoted as generalized Nystr¨om methods. These methods can be thought as a general family of formulae for the numerical approximation of second order problems, which properly include classical formulae, such as linear multistep methods and Runge-Kutta-Nystr¨om methods, but also enable to find new methods which provide better balances between accuracy and stability demandings. This is made possible because generalized Nystr¨om methods rely on a larger number of degrees of freedom than classical methods, which can be employed for the mentioned purposes. We provide the formulation of the family of methods, showing that existing methods can be regarded according to the new formalism, study the main properties and give examples of highly stable genuine multivalue methods whose order is higher than that of existing methods. In particular, we aim to inherit the best stability properties known in the literature, i.e. those coming from Gauss-Legendre points leading to P-stable methods, by introducing generalized Nystr¨om methods having with the same stability polynomial of Gauss-Legendre methods but higher order of convergence. We show that it is possible to obtain P-stable methods with order 4 relying on one single internal stage (in the classical case, the maximum attainable order is only 2, requiring the same computational cost). A numerical experiment shows the effectiveness of the approach on a periodic stiff problem, also in comparison with existing methods.