Multidimensional adapted Runge-Kutta-Nyström methods for oscillatory systems

For the general multidimensional perturbed oscillators y″+Ky=f(y, y′) with K∈Rd×d, the order conditions for the ARKN methods are presented by X. Wu et al. [Order conditions for ARKN methods solving oscillatory systems, Comput. Phys. Comm. 180 (2009) 2250-2257]. These methods integrate exactly the multidimensional unperturbed oscillators and are highly efficient when the perturbing function is small. In this paper, we are concerned with the analysis of the concrete multidimensional ARKN methods based on the order conditions for the multidimensional ARKN methods. We extent the matrix K to a general case, in which K is not required to be symmetric. Numerical experiments demonstrate that the novel multidimensional ARKN methods presented in this paper are more efficient compared with some well-know RKN methods in the scientific literature.