Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators

Abstract

New Runge–Kutta–Nyström methods specially adapted to the numerical integration of perturbed oscillators are obtained. Our interest is centered on algorithms which integrate exactly harmonic oscillators with frequency ω whereas for perturbed problems the local truncation error contains the perturbation parameter as a factor. The methods depend upon a parameter ν=ωh>0 (h is the integration step), and they are derived by using the theory of Nyström trees. Based on the B-series theory we derive the sufficient order conditions as well as the necessary and sufficient order conditions for this class of Nyström methods. With the help of these order conditions we construct explicit methods up to order 5 in the sense that y(tn+1)−yn+1=O(h6) and y′(tn+1)−y′n+1=O(h6). The numerical experiments show the efficiency of these methods when they are compared with other Runge–Kutta–Nyström type methods from the scientific literature.