An algebraic theory of order
INRIA Rennes and École Normale Supérieure de Cachan, Antenne de Bretagne, Avenue Robert Schumann, 35170 Bruz, France. Philippe.Chartier@inria.fr
2 Konputazio Zientziak eta A. A. saila, Informatika Fakultatea, University of the Basque Country Donostia/San Sebastiàn, Spain.
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.
Mathematics Subject Classification: 05E99 / 17B99 / 93B25 / 65L99
Key words: Order conditions / Hopf algebra / group of abstract integration schemes / Lie algebra / composition.
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