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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  13. J. Milnor and J. Moore, On the structure of Hopf algebras. Ann. Math.81 (1965) 211–264.  Zbl0163.28202
  14. H. Munthe-Kaas and W. Wright, On the Hopf algebraic structure of Lie group integrators. Found. Comput. Math.8 (2008) 227–257.  Zbl1147.16028
  15. A. Murua, Formal series and numerical integrators, Part i: Systems of ODEs and symplectic integrators. Appl. Numer. Math.29 (1999) 221–251.  Zbl0929.65126
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  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.  Zbl0946.65056

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