The lack of continuity and the role of infinite and infinitesimal in numerical methods for ODEs: The case of symplecticity
The lack of continuity and the role of infinite and infinitesimal in numerical methods for ODEs: The case of symplecticity
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DOI：
10.1016/j.amc.2011.03.022
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发表时间：
2010-10
期刊：
影响因子：
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通讯作者：
L. Brugnano;F. Iavernaro;D. Trigiante
中科院分区：
文献类型：
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作者：
L. Brugnano;F. Iavernaro;D. Trigiante
当数值积分正则哈密顿系统时，它的一些不变量的长期守恒，例如哈密顿函数本身，承担了核心作用。这个问题的经典方法导致辛方法的定义，其中我们提到高斯-勒让德配置公式。事实上，在连续的设置，能量守恒是来自辛通过无限数量的无穷小接触变换。然而，这个无限的过程不能直接转移到离散设置。通过遵循不同的方法，在本文中，我们描述了一系列方法，这些方法共享相同的基本谱（并且然后具有相同的基本属性），这些方法从序列的某个元素开始保持能量，即，在有限的步骤之后。
When numerically integrating canonical Hamiltonian systems, the long-term conservation of some of its invariants, for example 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.