Rendiconti di Matematica, Serie VII Volume 23, Roma (2003), 277-288 Two step Runge-Kutta-Nystr¨om methods based on algebraic polynomials B. PATERNOSTER Abstract: We consider the new family of two step Runge-Kutta-Nystr¨om methods for the numerical integration of y ′′ = f (x, y). We derive the conditions to obtain two step Runge-Kutta-Nystr¨om methods which integrate algebraic polynomials exactly and analyze the one-stage case. 1 – Introduction We are concerned with second order Ordinary Differential Equations, in which the first derivative does not appear explicitly, (1.1) y ′′ (t)= f (t, y(t)), y(t 0 )= y 0 , y ′ (t 0 )= y ′ 0 , y(t),f (t, y) ∈ R n , having a periodic or an oscillatory solution. These initial value problems often arise in applications of molecular dynamics, orbital mechanics, seis- mology, and they are usually considered as a difficult integration problem. Indeed standard numerical methods can require a huge number of time- steps to track the oscillations. In many situations, when the problem has a large dimension, or the evaluation of the right-hand side function is Key Words and Phrases: Numerical methods for Ordinary Differential Equations – Runge-Kutta-Nystr¨om methods – Two step Runge-Kutta methods. A.M.S. Classification: 65L06