Energy-preserving continuous stage extended Runge-Kutta-Nystrom methods for oscillatory Hamiltonian systems

Continuous-stage extended Runge-Kutta-Nystrom (CSERKN) methods are proposed and developed for oscillatory problem q ''(t) + Mq(t) = f (q(t)). These new methods take into account the special feature of the oscillatory problem so that they integrate exactly unperturbed problem q ''(t) + Mq(t) = 0. When this problem can be regarded as a Hamiltonian system, we show sufficient conditions for energy-preservation in terms of the coefficients of the method. We also study the symmetry and stability of the methods. Two symmetric and energy-preserving CSERKN schemes of order two and four, respectively, are constructed. Some numerical experiments are provided to confirm the theoretical expectations. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.