The study of the long time conservation for numerical methods poses interesting and challenging questions from the point of view of geometric integration. In this paper, we analyze the long time energy and magnetic moment conservations of two-step symmetric methods for charged-particle dynamics. A two-step symmetric method is proposed and its long time behaviour is shown not only in a normal magnetic field but also in a strong magnetic field. The approaches to dealing with these two cases are based on the backward error analysis and modulated Fourier expansion, respectively. It is obtained from the analysis that the method has better long time conservations than the variational method which was researched recently in the literature.

中文翻译：

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在正常或强磁场中带电粒子动力学的两步对称方法

从几何积分的角度出发，对数值方法的长期守恒性的研究提出了有趣且具有挑战性的问题。在本文中，我们分析了带电荷粒子动力学的两步对称方法的长时间能量和磁矩守恒。提出了一种两步对称方法，其不仅在正常磁场中而且在强磁场中都表现出长时间的行为。处理这两种情况的方法分别基于后向误差分析和调制傅立叶展开。从分析中可以看出，该方法比文献中最近研究的变分方法具有更好的长期保存性。