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arXiv · 1002.4394

Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their connections with Runge-Kutta collocation methods

Luigi Brugnano·Felice Iavernaro·Donato Trigiante
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Abstract

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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Luigi Brugnano, Felice Iavernaro, Donato Trigiante. 2010-02-23. Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their connections with Runge-Kutta collocation methods. https://arxiv.org/abs/1002.4394

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