An algebraic theory of order
ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 607-630.

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
Mots-clés : 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: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 607-630. doi : 10.1051/m2an/2009029. http://www.numdam.org/articles/10.1051/m2an/2009029/

[1] 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. | MR | Zbl

[2] N. Bourbaki, Lie groups and Lie algebras. Springer-Verlag, Berlin-New York (1989). | MR

[3] J.C. Butcher, An algebraic theory of integration methods. Math. Comput. 26 (1972) 79-106. | MR | Zbl

[4] P. Cartier, A primer of Hopf algebras, in Frontiers in number theory, physics, and geometry II. Springer, Berlin (2007) 537-615. | MR | Zbl

[5] P. Chartier and A. Murua, Preserving first integrals and volume forms of additively split systems. IMA J. Numer. Anal. 27 (2007) 381-405. | MR | Zbl

[6] P. Chartier, E. Faou and A. Murua, An algebraic approach to invariant preserving integators: the case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575-590. | MR | Zbl

[7] A. Connes and H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem. Commun. Math. Phys. 198 (1998). | MR | Zbl

[8] A. Dür, Möbius functions, incidence algebras and power-series representations, in Lecture Notes in Mathematics 1202, Springer-Verlag (1986). | MR | Zbl

[9] 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). | MR | Zbl

[10] G.P. Hochschild, Basic theory of algebraic groups and Lie algebras. Springer-Verlag (1981). | MR | Zbl

[11] M.E. Hoffman, Quasi-shuffle products. J. Algebraic Comb. 11 (2000) 49-68. | MR | Zbl

[12] D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2 (1998) 303-334. | MR | Zbl

[13] J. Milnor and J. Moore, On the structure of Hopf algebras. Ann. Math. 81 (1965) 211-264. | MR | Zbl

[14] H. Munthe-Kaas and W. Wright, On the Hopf algebraic structure of Lie group integrators. Found. Comput. Math. 8 (2008) 227-257. | MR | Zbl

[15] A. Murua, Formal series and numerical integrators, Part i: Systems of ODEs and symplectic integrators. Appl. Numer. Math. 29 (1999) 221-251. | MR | Zbl

[16] A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math. 6 (2006) 387-426. | MR | Zbl

[17] A. Murua and J.M. Sanz-Serna, Order conditions for numerical integrators obtained by composing simpler integrators. Phil. Trans. R. Soc. A 357 (1999) 1079-1100. | MR | Zbl

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