Abstract
Continuous-stage extended Runge-Kutta-Nyström (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.
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