High-order closed Newton–Cotes trigonometrically-fitted formulae for long-time integration of orbital problems

In this paper we investigate the connection between closed Newton–Cotes, trigonometrically-fitted differential methods and symplectic integrators. From the literature we can see that several one step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. The well-known open Newton–Cotes differential methods presented as multilayer symplectic integrators by Zhu et al. [W. Zhu, X. Zhao, Y. Tang, Journal of Chem. Phys. 104 (1996) 2275]. The construction of multistep symplectic integrators based on the open Newton–Cotes integration methods is investigated by Chiou and Wu [J.C. Chiou, S.D. Wu, J. Chem. Phys. 107 (1997) 6894]. In this paper we investigate the closed Newton–Cotes formulae and we write them as symplectic multilayer structures. After this we construct trigonometrically-fitted symplectic methods which are based on the closed Newton–Cotes formulae. We apply the symplectic schemes in order to solve Hamilton's equations of motion which are linear in position and momentum. We observe that the Hamiltonian energy of the system remains almost constant as integration proceeds.