Newton–Cotes formulae for long-time integration

The connection between closed Newton–Cotes differential methods and symplectic integrators is considered in this paper. Several one step symplectic integrators have been developed based on symplectic geometry. However, multistep symplectic integrators have seldom been investigated. Zhu et al. (J. Chem. Phys. 104 (1996) 2275) converted open Newton–Cotes differential methods into a multilayer symplectic structure. Also, Chiou and Wu (J. Chem. Phys. 107 (1997) 6894) have written on the construction of multistep symplectic integrators based on the open Newton–Cotes integration methods. In this work we examine the closed Newton–Cotes formulae and we write them as symplectic multilayer structures. 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.