On extended RKN integrators for multidimensional perturbed oscillators with applications

Recently, Fang and Ming [Y.L. Fang, Q.H. Ming, Embedded pair of extended Runge–Kutta–Nyström type methods for perturbed oscillators, Appl. Math. Modelling 34 (2010) 2665–2675] constructed an embedded pair of extended Runge–Kutta–Nyström type methods for perturbed oscillators based on the order conditions of extended Runge–Kutta–Nyström type methods proposed by Yang et al. [H.L. Yang, X.Y. Wu, X. You, Y.L. Fang, Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Commun. 180 (2009) 1777–1794]. The authors applied their embedded pair to one-dimensional and two-dimensional problems in numerical experiments. However, the extended Runge–Kutta–Nyström type methods by Yang et al. are designed for one-dimensional perturbed oscillators or systems of perturbed oscillators with a diagonal and positive semi-definite matrix M and a function f(y). For multidimensional perturbed oscillators y″ + My = f(y) with M ∈ Rm×m, a symmetric positive semi-definite matrix, the order conditions of the extended RKN-type methods must be reanalyzed. In this paper, the order conditions for the multidimensional perturbed oscillators are stated and accordingly Fang et al.’s ERKN method of order five for systems of perturbed oscillators is reconsidered. The numerical experiments of the fifth order ERKN method for multidimensional perturbed oscillators are accompanied in comparison with some existing well-known methods in the scientific literature.