Symplecticity-preserving continuous-stage Runge-Kutta-Nyström methods

In this paper, we develop continuous-stage Runge-Kutta-Nyström (csRKN) methods for numerical integration of second-order ordinary differential equations (ODEs) written in the form q¨=f(t,q). Numerous ODEs in such form can be reduced to first-order ODEs with the separable form of Hamiltonian systems and symplecticity-preserving discretizations of these systems are of interest. For the sake of designing symplectic csRKN methods, we explore the sufficient conditions for symplecticity, and we show a simple way to derive symplectic RKN-type integrators by using Legendre polynomial expansion. Numerical results show the efficiency of the presented methods.