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

摘要

One main issue, when numerically integrating autonomous Hamiltonian systems,is the long-term conservation of some of its invariants, among which theHamiltonian function itself. Recently, a new class of methods, namedHamiltonian Boundary Value Methods (HBVMs) has been introduced and analysed,which are able to exactly preserve polynomial Hamiltonians of arbitrarily highdegree. We here study a further property of such methods, namely that ofhaving, when cast as a Runge-Kutta method, a matrix of the Butcher tableau withthe same spectrum (apart from the zero eigenvalues) as that of thecorresponding 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-Kuttacollocation methods.