A new pair of explicit ARKN methods for the numerical integration of general perturbed oscillators

Abstract

A new embedded pair of explicit RKN methods adapted to the numerical integration of general perturbed oscillators is derived. This pair is based on the RKN methods adapted to the numerical integration of perturbed oscillators constructed by Franco (see Ref. [J.M. Franco, Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Commun. 147 (2002) 770–787]). It not only can be used to deal with the particular problems in which the perturbed functions are independent of y ′ but the general problems. We show that the embedded methods have algebraic order 4 and 3. The numerical experiments show the efficiency of our pair compared with the variable step code proposed by Vanden Berghe et al. (see Ref. [G. Vanden Berghe, H. De Meyer, M. Van Daele, T. Van Hecke, Exponentially-fitted explicit Runge–Kutta methods, J. Comput. Appl. Math. 125 (2000) 107–115]) and the other high order Runge–Kutta(Nyström) pairs (such as the Runge–Kutta 8(7) and 5(4) pair of Dormand and Prince given in Ref. [E. Hairer, S.P. Nørsett, S.P. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer, Berlin, 1993]), when they are used to deal with the special problems with the perturbed functions independent of y ′ as well as the general problems.