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

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.

12 pages

References in corpus (4)

• Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)
• Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
• The Hamiltonian BVMs (HBVMs) Homepage
• Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their blended implementation

Cited by in corpus (6)

• A unifying framework for the derivation and analysis of effective classes of one-step methods for ODEs
• A note on the efficient implementation of Hamiltonian BVMs
• On the Existence of Energy-Preserving Symplectic Integrators Based upon Gauss Collocation Formulae
• The Lack of Continuity and the Role of Infinite and Infinitesimal in Numerical Methods for ODEs: the Case of Symplecticity
• The Hamiltonian BVMs (HBVMs) Homepage
• Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems