Symplecticity-preserving continuous-stage Runge-Kutta-Nystr\"{o}m methods

We develop continuous-stage Runge-Kutta-Nystr\"{o}m (csRKN) methods for solving second order ordinary differential equations (ODEs) in this paper. The second order ODEs are commonly encountered in various fields and some of them can be reduced to the first order ODEs with the form of separable Hamiltonian systems. The symplecticity-preserving numerical algorithm is of interest for solving such special systems. We present a sufficient condition for a csRKN method to be symplecticity-preserving, and by using Legendre polynomial expansion we show a simple way to construct such symplectic RKN type method.