ERKN integrators for systems of oscillatory second-order differential equations

For systems of oscillatory second-order differential equations y″+My=f with M″Rm×m, 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. Franco's ARKN methods for one-dimensional problems or for systems with a diagonal matrix M=w2I [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 y″+My=0. Regretfully, even for the unperturbed oscillators the internal stages Yi of an ARKN method fail to equal the values of the exact solution y(t) at tn+cih, 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 M″Rm×m, 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 M″Rm×m 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 H(p,q)=12pTp+12qTMq+V(q) 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.