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

Luigi Brugnano3
Dipartimento di Matematica “U.Dini”, Universit`a di Firenze
Viale Morgagni 67/A, I-50134 Firenze, Italy
Felice Iavernaro4
Dipartimento di Matematica, Universit`a di Bari
Via Orabona 4, I-70125 Bari, Italy
Donato Trigiante5
Dipartimento di Energetica “S.Stecco”, Universit`a di Firenze
Via Lombroso 6/17, I-50134 Firenze, Italy
Received October 25, 2009; accepted in revised form April 15, 2010.
Abstract: Recently, a new family of integrators (Hamiltonian Boundary Value Methods) 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

2010 European Society of Computational Methods in Sciences and Engineering
Keywords: Hamiltonian problems, exact conservation of the Hamiltonian, energy conservation,
Hamiltonian Boundary Value Methods, HBVMs, discrete line integral.


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