哈密顿边值方法(HBVMs)的等谱性质及其与龙格-库塔配置方法的联系
Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their connections with Runge-Kutta collocation methods
AI总结：
研究哈密顿边值方法(HBVMs)在龙格-库塔形式下具有与高斯-勒让德方法相同谱（除零特征值外）的等谱性质，从而证明其完美A稳定性，并阐明与经典龙格-库塔配置方法的联系。
摘要：
在数值积分自治哈密顿系统时，一个主要问题是其某些不变量的长期守恒，其中哈密顿函数本身尤为重要。最近，一类名为哈密顿边值方法(HBVMs)的新方法被引入并分析，该方法能够精确保持任意高次的多项式哈密顿量。我们在此研究这类方法的另一个性质：当表示为龙格-库塔方法时，其Butcher表矩阵具有与相应高斯-勒让德方法相同的谱（除零特征值外），而与所考虑的横坐标无关。因此，HBVMs始终是完美的A稳定方法。这进而阐明了它们与经典龙格-库塔配置方法之间的现有联系。
英文摘要：
One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. Recently, a new class of methods, named Hamiltonian Boundary Value Methods (HBVMs) has been introduced and analysed, which are able to exactly preserve polynomial Hamiltonians of arbitrarily high degree. We here study a further property of such methods, namely that of having, when cast as a Runge-Kutta method, a matrix of the Butcher tableau with the same spectrum (apart from the zero eigenvalues) as that of the corresponding Gauss-Legendre method, independently of the considered abscissae. Consequently, HBVMs are always perfectly A-stable methods. This, in turn, allows to elucidate the existing connections with classical Runge-Kutta collocation methods.
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