Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their connections with Runge-Kutta collocation methods
For researchers in numerical integration of Hamiltonian systems, this theoretical result deepens understanding of HBVMs' stability and links to established methods, but is incremental.
The paper proves that Hamiltonian Boundary Value Methods (HBVMs) have the same non-zero spectrum as Gauss-Legendre methods when cast as Runge-Kutta methods, ensuring perfect A-stability regardless of abscissae. This clarifies connections between HBVMs and classical Runge-Kutta collocation methods.
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.