This paper presents a long-term analysis of one-stage extended Runge–Kutta–Nyström (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both types of integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for explicit integrators, a relationship between ERKN integrators and trigonometric integrators is established. For the long-term analysis of implicit integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion. By taking some adaptations of this technology for implicit methods, we derive the modulated Fourier expansion and show the near energy conservation.

本文介绍了对高振荡哈密顿系统的单级扩展 Runge-Kutta-Nyström (ERKN) 积分器的长期分析。我们不仅研究对称积分器的长期数值能量守恒，还研究辛积分器。在分析中，我们既不假设对称方法的辛性，也不假设辛方法的对称性。事实证明，从长期来看，这两种类型的积分器的总能量和振荡能量接近守恒。为了证明显式积分器的结果，建立了ERKN积分器和三角积分器之间的关系。对于隐式积分器的长期分析，上述方法不再适用，我们使用调制傅立叶展开技术。