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An algebraic theory of order

Philippe Chartier 1 Ander Murua 2
1 IPSO - Invariant Preserving SOlvers
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
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.
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Philippe Chartier, Ander Murua. An algebraic theory of order. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2009, 43 (4), pp.607-630. ⟨10.1051/m2an/2009029⟩. ⟨hal-00777190⟩

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