Hamiltonian边界值方法(HBVMs)及其混合实现的等谱性质
Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their blended implementation
AI总结：
本文研究Hamiltonian边界值方法(HBVMs)在作为Runge-Kutta方法时，其Butcher表矩阵与对应Gauss-Legendre方法具有相同非零谱的性质，从而保证方法始终完美A稳定，并支持高效的混合实现。
摘要：
在数值积分自治Hamiltonian系统时，一个主要问题是其某些不变量（包括Hamiltonian函数本身）的长期守恒。最近，一类名为“Hamiltonian边界值方法（HBVMs）”的新方法被引入和分析，该方法能够精确保持任意高次多项式Hamiltonian。我们在此研究这类方法的另一个性质，即当将其视为Runge-Kutta方法时，Butcher表矩阵与对应的Gauss-Legendre方法具有相同的谱（除了零特征值），而与所考虑的横坐标无关。因此，HBVMs始终是完美A稳定的方法。此外，这使得它们能够通过高效的“混合”实现来求解生成的离散问题。
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 Runge-Kutta methods, a matrix of the Butcher tableau with the same spectrum (apart the zero eigenvalues) as that of the corresponding Gauss-Legendre method, independently of the considered abscissae. Consequently, HBVMs are always perfectly A-stable methods. Moreover, this allows their efficient "blended" implementation, for solving the generated discrete problems.
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