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Many practical problems can be described by second-order system$\ddot{q}=-M\nabla U(q)$, in which people give special emphasis to someinvariants with explicit physical meaning, such as energy, momentum, angularmomentum, etc. However, conventional numerical integrators for such systemswill fail to preserve any of these quantities which may lead to qualitativelyincorrect numerical solutions. This paper is concerned with the development ofenergy-preserving continuous-stage Runge-Kutta-Nystr\"om (csRKN) methods forsolving second-order systems. Sufficient conditions for csRKN methods to beenergy-preserving are presented and it is proved that all the energy-preservingcsRKN methods satisfying these sufficient conditions can be essentially inducedby energy-preserving continuous-stage partitioned Runge-Kutta methods. Someillustrative examples are given and relevant numerical results are reported.

Energy-preserving continuous-stage Runge-Kutta-Nystr\"om methods