Energy-Preserving Variant of Collocation Methods

E. Hairer3
Universit´┐Że de Gen`eve,
Section de Math´┐Żematiques, 2-4 rue du Li`evre,
CH-1211 Gen`eve 4, Switzerland
Received 15 October, 2009; accepted in revised form 21 March, 2010
Abstract: We propose a modification of collocation methods extending the ‘averaged vector
field method’ to high order. These new integrators exactly preserve energy for Hamiltonian
systems, are of arbitrarily high order, and fall into the class of B-series integrators. We
discuss their symmetry and conjugate-symplecticity, and we compare them to energypreserving
composition methods.

c 2010 European Society of Computational Methods in Sciences and Engineering
Keywords: Hamiltonian systems, energy-preserving integrators, B-series, Runge-Kutta
methods, collocation, conjugate-symplecticity, symmetry, Gaussian quadrature.
Mathematics Subject Classification: 65P10, 65L06


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