arXiv:0910.3621v5 [math.NA] 7 Jun 2010 Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods) ∗ Luigi Brugnano † Felice Iavernaro ‡ Donato Trigiante § To apper in: Journal of Numerical Analysis, Industrial and Applied Mathematics (JNAIAM). Received: October 25, 2009. Accepted (in revised form): April 15, 2010. Abstract Recently, a new family of integrators (Hamiltonian Boundary Value Meth- ods) has been introduced, which is able to precisely conserve the energy function of polynomial Hamiltonian systems and to provide a practical con- servation 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 behav- ior 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 dis- crete 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. Keywords: Hamiltonian problems, exact conservation of the Hamiltonian, energy conservation, Hamiltonian Boundary Value Methods, HBVMs, dis- crete line integral. MSC 65P10, 65L05. * Work developed within the project “Numerical methods and software for differential equations”. † Dipartimento di Matematica “U.Dini”, Viale Morgagni 67/A , I-50134 Firenze, Italy. (luigi.brugnano@unifi.it). ‡ Dipartimento di Matematica, Via Orabona 4, I-70125 Bari, Italy. (felix@dm.uniba.it). § Dipartimento di Energetica “S.Stecco”, Via Lombroso 6/17, I-50134 Firenze, Italy. (trigiant@unifi.it). 1