Geodesic
An algebraic theory of order
ESAIM: Mathematical Modelling and Numerical Analysis , Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 607-630.

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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.

DOI : 10.1051/m2an/2009029
Classification : 05E99, 17B99, 93B25, 65L99
Keywords: order conditions, Hopf algebra, group of abstract integration schemes, Lie algebra, composition 
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Chartier, Philippe; Murua, Ander. An algebraic theory of order. ESAIM: Mathematical Modelling and Numerical Analysis , Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 607-630. doi : 10.1051/m2an/2009029. http://geodesic.mathdoc.fr/articles/10.1051/m2an/2009029/

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