New closed Newton-Cotes type formulae as multilayer symplectic integrators

In this paper, we introduce new integrators of Newton-Cotes type and investigate the connection between these new methods, 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. In this paper, we introduce a new numerical method of closed Newton-Cotes type and we write it as a symplectic multilayer structure. 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.