Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)

TL;DR

This paper develops a general theoretical framework for Hamiltonian Boundary Value Methods, which are integrators that conserve energy in polynomial Hamiltonian systems and approximate energy conservation in non-polynomial cases.

Contribution

It formalizes the definition and theory of Hamiltonian Boundary Value Methods within a broader framework, analyzing their behavior and optimal configurations.

Findings

Methods precisely conserve energy for polynomial Hamiltonians.

Analysis of optimal polynomial basis and node distribution.

Framework applicable to general Hamiltonian systems.

Abstract

Recently, a new family of integrators (Hamiltonian Boundary ValueMethods) has been introduced, which is able to precisely conserve the energy function of polynomial Hamiltonian systems and to provide a practical conservation of the energy in the non-polynomial case. We settle the definition and the theory of such methods in a more general framework. Our aim is on the one hand to give account of their good behavior when applied to general Hamiltonian systems and, on the other hand, to find out what are the optimal formulae, in relation to the choice of the polynomial basis and of the distribution of the nodes. Such analysis is based upon the notion of extended collocation conditions and the definition of discrete line integral, and is carried out by looking at the limit of such family of methods as the number of the so called silent stages tends to infinity.