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

TL;DR

This paper investigates the spectral properties of Hamiltonian Boundary Value Methods (HBVMs), revealing their spectral similarity to Gauss-Legendre methods, which ensures their A-stability and enables efficient blended implementation for long-term Hamiltonian system integration.

Contribution

It demonstrates that HBVMs share the same spectrum as Gauss-Legendre methods, establishing their A-stability and facilitating an efficient blended implementation.

Findings

HBVMs have the same spectrum as Gauss-Legendre methods (excluding zero eigenvalues).

HBVMs are always perfectly A-stable methods.

The spectral property enables efficient 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.