In the last few decades, Runge-Kutta-Nyström (RKN) methods have made significant progress and the study of RKN-type methods for solving highly oscillatory differential equations has received a great deal of attention. In this paper, from the point of view of geometric integration, the oscillation-preserving behaviour of the existing RKN-type methods in the literature are analysed in detail. To this end, it is convenient to introduce the concept of oscillation preservation for RKN-type methods. It turns out that if both the internal stages and the updates of an RKN-type method respect the qualitative and global features of the highly oscillatory solution, then the method is oscillation preserving. Since the internal stages of standard RKN and adapted RKN (ARKN) methods are inimical to the oscillation-preserving structure, neither ARKN methods nor the symplectic and symmetric RKN methods, and standard RKN methods are oscillation preserving. Other concerns relating to oscillation preservation are also considered. In particular, we are concerned with the computational issues for efficiently solving semi-discrete wave equations such as semi-discrete Klein-Gordon (KG) equations and damped sine-Gordon equations. The results of numerical experiments show the importance of the oscillation-preserving property for an RKN-type method and the remarkable superiority of oscillation-preserving integrators when applied to nonlinear multi-frequency highly oscillatory systems.

在过去的几十年中，Runge-Kutta-Nyström（RKN）方法取得了重大进展，解决高振荡微分方程的RKN型方法的研究受到了广泛关注。本文从几何积分的角度，详细分析了文献中现有RKN型方法的振动保持行为。为此，方便地引入RKN型方法的振荡保持的概念。事实证明，如果RKN型方法的内部阶段和更新都尊重高振荡解的定性和全局特征，则该方法将保持振荡。由于标准RKN和调整后的RKN（ARKN）方法的内部阶段对保持振动的结构不利，ARKN方法，辛对称RKN方法和标准RKN方法都不能保持振荡。还考虑了与振荡保持有关的其他问题。特别是，我们关注有效解决半离散波动方程（例如半离散Klein-Gordon（KG）方程和阻尼正弦-Gordon方程）的计算问题。数值实验结果表明，对于非线性多频高振荡系统，RKN型方法保持振动的重要性，以及保持积分器的优越性。我们关注有效解决半离散波动方程（例如半离散Klein-Gordon（KG）方程和阻尼正弦-Gordon方程）的计算问题。数值实验结果表明，对于非线性多频高振荡系统，RKN型方法保持振动的重要性，以及保持积分器的优越性。我们关注有效解决半离散波动方程（例如半离散Klein-Gordon（KG）方程和阻尼正弦-Gordon方程）的计算问题。数值实验结果表明，对于非线性多频高振荡系统，RKN型方法保持振动的重要性，以及保持积分器的优越性。