Open Newton-Cotes differential methods as multilayer symplectic integrators

Open Newton-Cotes differential methods that possess the characteristics of multilayer symplectic structures are shown in this paper. In numerical simulation, volume-preservation plays an important role in solving the Hamiltonian system. In this regard, developing a numerical integrator that preserves the volume in the phase space of a Hamiltonian system is a great challenge to the researchers in this field. Symplectic integrators were proven to be good candidates for volume-preserving integrators (VPIs) in the past ten years. Several one-step (single-stage or multistages) symplectic integrators have been developed based on the symplectic geometric theory. However, multistep VPIs have seldom been investigated by other researchers for the lack of an advanced theory. Recently, Zhu et al. converted open Newton-Cotes differential methods into a multilayer symplectic structure so that multistep VPIs of a Hamiltonian system are obtained. Mainly, their work has concentrated on the issue of achieving both the accuracy and efficiency by solving the quantum systems. But, there exist some unclear aspects in deriving this result in their paper. In this regard, we resolve their problem and provide a different aspect in connecting the relationship between open Newton-Cotes differential methods and symplectic integrators. A numerical example has been carried out to show the effectiveness of the present differential method.