A note on stability of multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems

For the general multidimensional oscillatory systems y″ + Ky = f(y, y  ′) with K∈Rd×dK∈Rd×d, a positive semi-definite matrix, the order conditions for the ARKN methods are presented by Wu et al. [X. Wu, X.You, J. Xia, Order conditions for ARKN methods solving oscillatory systems, Computer Physics Communications 180 (2009) 2250–2257]. The effective multidimensional ARKN methods are proposed based on the order conditions obtained by Wu et al. [X. Wu, B. Wang, Multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems, Comput. Phys. Comm. 181 (2010) 1955–1962]. These methods integrate exactly the multidimensional unperturbed oscillators and are highly efficient when the perturbing function is small. In this note, we are concerned with the analysis of stability for multidimensional adapted Runge–Kutta–Nyström methods for the oscillatory systems. We give a complete stability analysis for the multidimensional ARKN methods based on the revised linear test equation y″(t) + ω2y(t) = −ϵy(t) with ω2 + ϵ > 0.