Order conditions for ARKN methods solving oscillatory systems

For the perturbed oscillators in one-dimensional case, J.M. Franco designed the so-called Adapted Runge-Kutta-Nyström (ARKN) methods and derived the sufficient order conditions as well as the necessary and sufficient order conditions for ARKN methods based on the B-series theory [J.M. Franco, Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm. 147 (2002) 770-787]. These methods integrate exactly the unperturbed oscillators and are highly efficient when the perturbing function is small. Unfortunately, some critical mistakes have been made in the derivation of order conditions in that paper. On the basis of the results from that paper, Franco extended directly the ARKN methods and the corresponding order conditions to multidimensional case where the perturbed function f does not depend on the first derivative y′ [J.M. Franco, New methods for oscillatory systems based on ARKN methods, Appl. Numer. Math. 56 (2006) 1040-1053]. In this paper, we present the order conditions for the ARKN methods for the general multidimensional perturbed oscillators where the perturbed function f may depend on only y or on both y and y′.