Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their blended implementation. (arXiv:1002.1387v1 [math.NA])

Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their blended implementation. (arXiv:1002.1387v1 [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 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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