Formules de type Runge-Kutta-Nystrom.

Abstract

Explicit Runge-Kutta-Nystrom pairs, which solve directly second order initial value problems of the form $y\sp{\prime\prime}$ = $f(x,y)$ with the first derivative $y\sp\prime$ absent, are considered. Pairs consisting of formulae of order $p-1$ and p, respectively, can be designed to control the local error in y, in $y\sp\prime$ or in y and $y\sp\prime$. They may also advance the numerical approximations using the lower order formulae or the higher order formulae. These two sets of choices lead to five types of pairs. We establish the minimum number of stages required to form the five types of pairs for p = 2, ...,6, by producing an existing pair and disproving the existence of a similar pair with fewer stages. Notions of Nystrom methods and Nystrom trees are recalled along with the order conditions for these methods.