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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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