哈密尔顿系统的辛几何算法是由我国数学家冯康院士倡导并系统发展起来的一个研究方向，已经成为当前微分方程数值计算领域的热点之一。著名的KAM理论和Nekhoroshev理论揭示了哈密尔顿系统不论在运动稳定性还是结构稳定性方面都与不稳定性相伴随，表现为某种程度的稳定行为，也伴随着很大程度的不稳定行为。因此针对这类系统的数值方法，其稳定性分析具有特别重要的意义，也有很大的挑战性。本项目重点研究辛算法的稳定性，如：（1）把辛算法的数值KAM理论由可积哈密尔顿系统扩展到近可积系统，研究相应的步长共振问题和共振步长探测方法；（2）在广义函数的框架内研究辛算法的形式能量（向后误差分析），证明在相空间的某种大测度集合上形式能量是一个“好的函数”，并且探讨当时间步长趋于零时的极限行为；（3）基于向后误差分析理论，研究辛算法和其它保结构算法的非线性稳定性及其与线性稳定性的联系。

The symplectic algorithm of Hamiltonian systems was initiated independently by Feng Kang in China and systematically developed by Feng Kang and his group during last 30 years. The subject has become one of the hottest topics in the area of numerical analysis of differential equations. The famous KAM theory and the Nekhoroshev theory reveal the fact that the stability of Hamiltonian systems are accompanied by instability in the sense of either motion stability or structural stability. Therefore, the stability analysis of numerical methods for such systems, is of particular significance, and also represents a great challenge. This project focuses on the stability issue of symplectic algorithms, e.g. (1) Extend the numerical KAM theory from integrable Hamiltonian systems to nearly integrable ones, and study the corresponding resonance problem of step sizes and give a method of detecting resonant step size; (2) Study the formal energy of symplectic algorithms within the framework of generalized functions, prove that the formal energy is a “good function” in a large measure set of the phase space, and explore the limit behavior when the time step tends to zero; (3) Based on the backward error analysis, study the relationships between the nonlinear stability and the linear stability of symplectic algorithms and other structure-preserving algorithms.