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An algebraic theory of order

Published online by Cambridge University Press:  08 July 2009

Philippe Chartier  and
Ander Murua
Philippe Chartier
Affiliation:
INRIA Rennes and École Normale Supérieure de Cachan, Antenne de Bretagne, Avenue Robert Schumann, 35170 Bruz, France. Philippe.Chartier@inria.fr
Ander Murua
Affiliation:
Konputazio Zientziak eta A. A. saila, Informatika Fakultatea, University of the Basque Country Donostia/San Sebastiàn, Spain.
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Abstract

In this paper, we present an abstract framework which describes algebraically the derivation of order conditions independently of the nature of differential equations considered or the type of integrators used to solve them. Our structure includes a Hopf algebra of functions, whose properties are used to answer several questions of prime interest in numerical analysis. In particular, we show that, under some mild assumptions, there exist integrators of arbitrarily high orders for arbitrary (modified) vector fields.

Keywords

Order conditionsHopf algebragroup of abstract integration schemesLie algebracomposition.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 4: Special issue on Numerical ODEs today , July 2009 , pp. 607 - 630
Copyright
© EDP Sciences, SMAI, 2009

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References

H. Berland and B. Owren, Algebraic structures on ordered rooted trees and their significance to Lie group integrators, in Group theory and numerical analysis, CRM Proc. Lecture Notes, Amer. Math. Soc., Providence R.I. (2005) 49–63.
N. Bourbaki, Lie groups and Lie algebras. Springer-Verlag, Berlin-New York (1989).
Butcher, J.C., An algebraic theory of integration methods. Math. Comput. 26 (1972) 79–106. CrossRef
P. Cartier, A primer of Hopf algebras, in Frontiers in number theory, physics, and geometry II. Springer, Berlin (2007) 537–615.
Chartier, P. and Murua, A., Preserving first integrals and volume forms of additively split systems. IMA J. Numer. Anal. 27 (2007) 381–405. CrossRef
Chartier, P., Faou, E. and Murua, A., An algebraic approach to invariant preserving integators: the case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575–590. CrossRef
A. Connes and H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem. Commun. Math. Phys. 198 (1998).
A. Dür, Möbius functions, incidence algebras and power-series representations, in Lecture Notes in Mathematics 1202, Springer-Verlag (1986).
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration – Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics 31. Springer, Berlin (2006).
G.P. Hochschild, Basic theory of algebraic groups and Lie algebras. Springer-Verlag (1981).
Hoffman, M.E., Quasi-shuffle products. J. Algebraic Comb. 11 (2000) 49–68. CrossRef
Kreimer, D., On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2 (1998) 303–334. CrossRef
Milnor, J. and Moore, J., On the structure of Hopf algebras. Ann. Math. 81 (1965) 211–264. CrossRef
Munthe-Kaas, H. and Wright, W., On the Hopf algebraic structure of Lie group integrators. Found. Comput. Math. 8 (2008) 227–257. CrossRef
Murua, A., Formal series and numerical integrators, Part i: Systems of ODEs and symplectic integrators. Appl. Numer. Math. 29 (1999) 221–251. CrossRef
Murua, A., The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math. 6 (2006) 387–426. CrossRef
Murua, A. and Sanz-Serna, J.M., Order conditions for numerical integrators obtained by composing simpler integrators. Phil. Trans. R. Soc. A 357 (1999) 1079–1100. CrossRef