ERKN integrators for systems of oscillatory second-order differential equations
For systems of oscillatory second-order differential equations y '' + My = f with M is an element of R(m x 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 = W(2)l [J.M. Franco, Runge-Kutta-Nystrom 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 Y(i) of an ARKN method fail to equal the values of the exact solution y(t) at 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) 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 is an element of R(m x 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 is an element of R(m x 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) Mq + V(q) in comparison with the well-known methods in the scientific literature. Those so called separable Hamiltonians arise ill many areas of physical sciences, e.g., macromolecular dynamics, astronomy, and classical mechanics. (C) 2010 Elsevier B.V. All rights reserved.
erkn integrators order conditions oscillatory systems b-series hamiltonian systems nonlinear wave equations