连续性的缺失与无穷小在常微分方程数值方法中的作用：以辛性为例
The Lack of Continuity and the Role of Infinite and Infinitesimal in Numerical Methods for ODEs: the Case of Symplecticity
AI总结：
本文通过构造一系列共享本质谱的数值方法，解决了哈密顿系统数值积分中能量守恒与辛性之间的连续性缺失问题，实现了有限步后的能量保持。
摘要：
在对正则哈密顿系统进行数值积分时，其某些不变量（包括哈密顿函数本身）的长期守恒起着核心作用。解决这一问题的经典方法导致了辛方法的定义，其中我们提到高斯-勒让德配置公式。实际上，在连续设定中，能量守恒是通过无穷多次无穷小接触变换从辛性推导出来的。然而，这个无穷过程不能直接转移到离散设定中。通过采用不同的方法，本文描述了一系列方法，它们共享相同的本质谱（因此具有相同的本质性质），并且从序列中的某个元素开始（即在有限步之后）保持能量守恒。
英文摘要：
When numerically integrating canonical Hamiltonian systems, the long-term conservation of some of its invariants, among which the Hamiltonian function itself, assumes a central role. The classical approach to this problem has led to the definition of symplectic methods, among which we mention Gauss-Legendre collocation formulae. Indeed, in the continuous setting, energy conservation is derived from symplecticity via an infinite number of infinitesimal contact transformations. However, this infinite process cannot be directly transferred to the discrete setting. By following a different approach, in this paper we describe a sequence of methods, sharing the same essential spectrum (and, then, the same essential properties), which are energy preserving starting from a certain element of the sequence on, i.e., after a finite number of steps.
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