Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their connections with Runge-Kutta collocation methods. (arXiv:1002.4394v1 [math.NA])

Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their connections with Runge-Kutta collocation methods. (arXiv:1002.4394v1 [math.NA]): "

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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