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.
Mots-clés : order conditions, Hopf algebra, group of abstract integration schemes, Lie algebra, composition
@article{M2AN_2009__43_4_607_0, author = {Chartier, Philippe and Murua, Ander}, title = {An algebraic theory of order}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {607--630}, publisher = {EDP-Sciences}, volume = {43}, number = {4}, year = {2009}, doi = {10.1051/m2an/2009029}, mrnumber = {2542867}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2009029/} }
TY - JOUR AU - Chartier, Philippe AU - Murua, Ander TI - An algebraic theory of order JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 607 EP - 630 VL - 43 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2009029/ DO - 10.1051/m2an/2009029 LA - en ID - M2AN_2009__43_4_607_0 ER -
%0 Journal Article %A Chartier, Philippe %A Murua, Ander %T An algebraic theory of order %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 607-630 %V 43 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2009029/ %R 10.1051/m2an/2009029 %G en %F M2AN_2009__43_4_607_0
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] 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
and ,[2] Lie groups and Lie algebras. Springer-Verlag, Berlin-New York (1989). | MR
,[3] An algebraic theory of integration methods. Math. Comput. 26 (1972) 79-106. | MR | Zbl
,[4] A primer of Hopf algebras, in Frontiers in number theory, physics, and geometry II. Springer, Berlin (2007) 537-615. | MR | Zbl
,[5] Preserving first integrals and volume forms of additively split systems. IMA J. Numer. Anal. 27 (2007) 381-405. | MR | Zbl
and ,[6] An algebraic approach to invariant preserving integators: the case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575-590. | MR | Zbl
, and ,[7] Hopf algebras, cyclic cohomology and the transverse index theorem. Commun. Math. Phys. 198 (1998). | MR | Zbl
and ,[8] Möbius functions, incidence algebras and power-series representations, in Lecture Notes in Mathematics 1202, Springer-Verlag (1986). | MR | Zbl
,[9] Geometric Numerical Integration - Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics 31. Springer, Berlin (2006). | MR | Zbl
, and ,[10] Basic theory of algebraic groups and Lie algebras. Springer-Verlag (1981). | MR | Zbl
,[11] Quasi-shuffle products. J. Algebraic Comb. 11 (2000) 49-68. | MR | Zbl
,[12] On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2 (1998) 303-334. | MR | Zbl
,[13] On the structure of Hopf algebras. Ann. Math. 81 (1965) 211-264. | MR | Zbl
and ,[14] On the Hopf algebraic structure of Lie group integrators. Found. Comput. Math. 8 (2008) 227-257. | MR | Zbl
and ,[15] Formal series and numerical integrators, Part i: Systems of ODEs and symplectic integrators. Appl. Numer. Math. 29 (1999) 221-251. | MR | Zbl
,[16] The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math. 6 (2006) 387-426. | MR | Zbl
,[17] Order conditions for numerical integrators obtained by composing simpler integrators. Phil. Trans. R. Soc. A 357 (1999) 1079-1100. | MR | Zbl
and ,- Splitting methods for differential equations, Acta Numerica, Volume 33 (2024), pp. 1-161 | DOI:10.1017/s0962492923000077 | Zbl:1546.65048
- Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations, Journal of Computational and Applied Mathematics, Volume 381 (2021), p. 13 (Id/No 113006) | DOI:10.1016/j.cam.2020.113006 | Zbl:1484.65146
- The BCH-formula and order conditions for splitting methods, Lie groups, differential equations, and geometry. Advances and surveys, Cham: Springer; Palermo: Università degli Studi di Palermo, 2017, pp. 71-83 | DOI:10.1007/978-3-319-62181-4_4 | Zbl:1476.65082
- B-series methods are exactly the affine equivariant methods, Numerische Mathematik, Volume 133 (2016) no. 3, pp. 599-622 | DOI:10.1007/s00211-015-0753-2 | Zbl:1364.65145
- The tridendriform structure of a discrete Magnus expansion, Discrete and Continuous Dynamical Systems, Volume 34 (2014) no. 3, pp. 1021-1040 | DOI:10.3934/dcds.2014.34.1021 | Zbl:1292.17029
- The Magnus expansion, trees and Knuth's rotation correspondence, Foundations of Computational Mathematics, Volume 14 (2014) no. 1, pp. 1-25 | DOI:10.1007/s10208-013-9172-x | Zbl:1331.17016
- Backward error analysis and the substitution law for Lie group integrators, Foundations of Computational Mathematics, Volume 13 (2013) no. 2, pp. 161-186 | DOI:10.1007/s10208-012-9130-z | Zbl:1323.65090
- Optimized high-order splitting methods for some classes of parabolic equations, Mathematics of Computation, Volume 82 (2013) no. 283, pp. 1559-1576 | DOI:10.1090/s0025-5718-2012-02657-3 | Zbl:1278.65075
- Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series., Advances in Applied Mathematics, Volume 47 (2011) no. 2, pp. 282-308 | DOI:10.1016/j.aam.2009.08.003 | Zbl:1235.16032
Cité par 9 documents. Sources : zbMATH