ERKN integrators for systems of oscillatory second-order differential equations Original Research Article

For systems of oscillatory second-order differential equations image with image, a symmetric positive semi-definite matrix, X. Wu et al. have proposed the multidimensional ARKN methods [X. Wu, X. You, J. Xia, Order conditions for ARKN methods solving oscillatory systems, Comput. Phys. Comm. 180 (2009) 2250–2257], which are an essential generalization of J.M. Francoʹs ARKN methods for one-dimensional problems or for systems with a diagonal matrix image [J.M. Franco, Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm. 147 (2002) 770–787]. One of the merits of these methods is that they integrate exactly the unperturbed oscillators image. Regretfully, even for the unperturbed oscillators the internal stages image of an ARKN method fail to equal the values of the exact solution image at image, respectively. Recently H. Yang et al. proposed the ERKN methods to overcome this drawback [H.L. Yang, X.Y. Wu, Xiong You, Yonglei Fang, Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777–1794]. However, the ERKN methods in that paper are only considered for the special case where M is a diagonal matrix with nonnegative entries. The purpose of this paper is to extend the ERKN methods to the general case with image, and the perturbing function f depends only on y. Numerical experiments accompanied demonstrates that the ERKN methods are more efficient than the existing methods for the computation of oscillatory systems. In particular, if image is a symmetric positive semi-definite matrix, it is highly important for the new ERKN integrators to show the energy conservation in the numerical experiments for problems with Hamiltonian image in comparison with the well-known methods in the scientific literature. Those so called separable Hamiltonians arise in many areas of physical sciences, e.g., macromolecular dynamics, astronomy, and classical mechanics.