]. J. But69 and . Butcher, The effective order of Runge-Kutta methods, Proceedings of Conference on the Numerical Solution of Differential Equations, pp.133-139, 1969.

]. J. But72 and . Butcher, An algebraic theory of integration methods, Math. Comput, vol.26, pp.79-106, 1972.

E. [. Chartier, A. Faou, and . Murua, An algebraic analysis of formal methods preserving structures. in preparation, 2005.

D. [. Connes and . Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun . Math. Phys, 1998.
DOI : 10.1007/s002200050499
URL : http://arxiv.org/abs/hep-th/9808042

E. [. Chartier and . Lapôtre, Reversible B-series, 1221.

M. P. Calvo, A. Murua, and J. M. Sanz-serna, Modified equations for ODEs, Contemporary Mathematics, vol.172, pp.63-74, 1994.
DOI : 10.1090/conm/172/01798

J. [. Channell and . Scovel, Symplectic integration of Hamiltonian systems, Nonlinearity, vol.3, issue.2, pp.231-259, 1990.
DOI : 10.1088/0951-7715/3/2/001

J. [. Calvo and . Sanz-serna, Canonical B-series, Numerische Mathematik, vol.67, issue.2, pp.161-175, 1994.
DOI : 10.1007/s002110050022

]. K. Fen86 and . Feng, Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comp. Math, vol.4, pp.279-289, 1986.

H. [. Feng, M. Wu, D. L. Qin, and . Wang, Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comp. Math, vol.7, pp.71-96, 1989.

]. E. Hai94 and . Hairer, Backward analysis of numerical integrators and symplectic methods, Annals of Numerical Mathematics, vol.1, pp.107-132, 1994.

]. E. Hai99 and . Hairer, Backward error analysis for multistep methods, Numer. Math, vol.84, pp.199-232, 1999.

C. [. Hairer, G. Lubich, and . Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Series in Computational Mathematics 31, 2002.
DOI : 10.4171/owr/2006/14
URL : https://hal.archives-ouvertes.fr/hal-01403326

G. [. Hairer and . Wanner, ??ber die Butchergruppe und allgemeine Multi-Value Methoden, Computing, vol.11, issue.1, pp.1-15, 1974.
DOI : 10.1007/BF02268387

]. A. Mur03 and . Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, 2003.

B. Trees, 4 2.2 Basic tools for trees, 6 2.2.3 Partitions and skeletons

H. Fields and S. , 21 4.5.1 Hamiltonian fields: a subgroup, Campus universitaire de Beaulieu -35042 Rennes Cedex

I. Unité-de-recherche and . Futurs, Technopôle de Nancy-Brabois -Campus scientifique 615, rue du Jardin Botanique -BP 101 -54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l'Europe -38330 Montbonnot-St-Martin (France) Unité de recherche INRIA Rocquencourt, Domaine de Voluceau -Rocquencourt -BP 105 -78153 Le Chesnay Cedex (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles -BP 93 -06902 Sophia Antipolis Cedex, pp.105-78153

I. De-voluceau-rocquencourt, BP 105 -78153 Le Chesnay Cedex (France) http://www.inria.fr ISSN, pp.249-6399