Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their blended implementation

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 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.

17 pages, 2 figures

References in corpus (3)

• 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
• Blended General Linear Methods based on Boundary Value Methods in the GBDF family

Cited by in corpus (3)

• Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)
• A note on the efficient implementation of Hamiltonian BVMs
• Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their connections with Runge-Kutta collocation methods