AI 中文总结
研究二维欧拉方程矩阵离散化的向后误差分析，引入李-泊松约化形式的Butcher级数，通过森林动量映射分析等谱辛龙格-库塔方法，得到修正哈密顿量守恒的指数小误差界，结果扩展到无限维情况。
AI 中文摘要
我们引入了Butcher级数的李-泊松约化形式。相应的森林动量映射可用于描述应用于球面上二维欧拉方程的Zeitlin矩阵离散化的等谱辛龙格-库塔方法的向后误差分析。在此基础上，我们得到了修正哈密顿量守恒的指数小误差界，在指数长的时间间隔内有效。关键是，当不同\(n\)（矩阵大小）的时间步长按\(h = \mathcal{O}(n^{-1})\)缩放时，误差界和时间间隔的长度与空间离散化参数\(n\)无关。因此，我们的结果将有限维哈密顿系统的经典向后误差分析结果扩展到了通过矩阵流体动力学离散化的二维欧拉方程的无限维情况。
英文摘要
We introduce a formalism of Lie--Poisson reduction of Butcher series. The corresponding forest momentum map allows for describing backward error analysis of isospectral symplectic Runge--Kutta methods applied to Zeitlin's matrix discretization of the 2-D Euler equations on the sphere. Based thereon, we obtain exponentially small error bounds for the conservation of modified Hamiltonians, valid for exponentially long time intervals. Crucially, the error bounds and the length of the time intervals are independent of the spatial discretization parameter $n$ (the matrix size) when the time step for different $n$ is scaled as $h = \mathcal{O}(n^{-1})$. Our results thus extend the classical backward error analysis result for finite-dimensional Hamiltonian systems to the infinite-dimensional case of the 2-D Euler equations discretized via matrix hydrodynamics.
