An algebraic theory of order

Philippe Chartier; Ander Murua

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 4, page 607-630
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Chartier, Philippe, and Murua, Ander. "An algebraic theory of order." ESAIM: Mathematical Modelling and Numerical Analysis 43.4 (2009): 607-630. <http://eudml.org/doc/250616>.

@article{Chartier2009,
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. },
author = {Chartier, Philippe, Murua, Ander},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Order conditions; Hopf algebra; group of abstract integration schemes; Lie algebra; composition.; order conditions; composition},
language = {eng},
month = {7},
number = {4},
pages = {607-630},
publisher = {EDP Sciences},
title = {An algebraic theory of order},
url = {http://eudml.org/doc/250616},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Chartier, Philippe
AU - Murua, Ander
TI - An algebraic theory of order
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/7//
PB - EDP Sciences
VL - 43
IS - 4
SP - 607
EP - 630
AB - 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.
LA - eng
KW - Order conditions; Hopf algebra; group of abstract integration schemes; Lie algebra; composition.; order conditions; composition
UR - http://eudml.org/doc/250616
ER -

References

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  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.  
  2. N. Bourbaki, Lie groups and Lie algebras. Springer-Verlag, Berlin-New York (1989).  
  3. J.C. Butcher, An algebraic theory of integration methods. Math. Comput.26 (1972) 79–106.  
  4. P. Cartier, A primer of Hopf algebras, in Frontiers in number theory, physics, and geometryII. Springer, Berlin (2007) 537–615.  
  5. P. Chartier and A. Murua, Preserving first integrals and volume forms of additively split systems. IMA J. Numer. Anal.27 (2007) 381–405.  
  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.  
  7. A. Connes and H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem. Commun. Math. Phys.198 (1998).  
  8. A. Dür, Möbius functions, incidence algebras and power-series representations, in Lecture Notes in Mathematics1202, Springer-Verlag (1986).  
  9. E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration – Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics31. Springer, Berlin (2006).  
  10. G.P. Hochschild, Basic theory of algebraic groups and Lie algebras. Springer-Verlag (1981).  
  11. M.E. Hoffman, Quasi-shuffle products. J. Algebraic Comb.11 (2000) 49–68.  
  12. D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys.2 (1998) 303–334.  
  13. J. Milnor and J. Moore, On the structure of Hopf algebras. Ann. Math.81 (1965) 211–264.  
  14. H. Munthe-Kaas and W. Wright, On the Hopf algebraic structure of Lie group integrators. Found. Comput. Math.8 (2008) 227–257.  
  15. A. Murua, Formal series and numerical integrators, Part i: Systems of ODEs and symplectic integrators. Appl. Numer. Math.29 (1999) 221–251.  
  16. A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math.6 (2006) 387–426.  
  17. A. Murua and J.M. Sanz-Serna, Order conditions for numerical integrators obtained by composing simpler integrators. Phil. Trans. R. Soc. A357 (1999) 1079–1100.  

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