ERKN integrators for systems of oscillatory second-order differential equations ?
For systems of oscillatory second-order differential equations y?+My=f y ? + M y = f with M?Rmm M ? R m  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) 22502257], 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 M = w 2 I [J.M. Franco, RungeKuttaNystrm methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm. 147 (2002) 770787]. One of the merits of these methods is that they integrate exactly the unperturbed oscillators y?+My=0 y ? + M y = 0 . Regretfully, even for the unperturbed oscillators the internal stages Yi Y i of an ARKN method fail to equal the values of the exact solution y(t) y ( t ) at tn+cih t n + c i h , 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) 17771794]. 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?Rmm M ? R m  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?Rmm M ? R m  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 ) = 1 2 p T p + 1 2 q T M q + 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.
erkn integrators order conditions oscillatory systems b-series hamiltonian systems nonlinear wave equations