**数学与系统科学研究院**

**计算数学所学术报告**

**报告人**：
Prof. Hans Munthe-Kaas

University of Bergen, Norway

**报告题目**
On the Algebraic Structure of Lie-butcher Series

**Abstract:**
The modern theory of order conditions for Runge–Kutta RK-methods was developed
by John Butcher in his seminal papers [2, 3], and further developed by
Hairer and Wanner [6]. Butcher B-series is a particular Taylor series development
of curves in Rn based on rooted trees. The algebraic order conditions of RK methods
are obtained by matching terms in the B-series of the analytical and the numerical
solution. The composition of curves in Rn corresponds to a convolution product of
B-series, yielding the Butcher group.

Recently Brouder [1] has explained the algebraic structure of B-series in terms of
a (cocommutative) Hopf algebra, and important connections has been made between
B-series and Connes and Kreimer’s Hopf-algebraic formulation of renormalization
theory [4].

In the last decade much research has been devoted to the study of time integration
of dierential equations on manifolds [5, 8, 10, 12, 7]. Order conditions for particular
classes of Lie Group Integrators (LGI) is studied by Crouch/Grossman [5],
Munthe-Kaas [10] and Owren/Marthinsen [12], using Lie techniques. The Lie–
Butcher LB-series [9, 10, 11] genearlizes B-series from the classical case, which can
be understood via the commutative action of Rn on itself, to the general noncommutative
case where a general Lie group acts transitively on a homogeneous
manifold.

In this paper will we further develop the theory of LB-series and establish the
convolution formula and Hopf algebraic structure of LB-series. Furthermore, we
will discuss the relationship between various approaches to order theory of LGI. It
is interesting to note that the series of Owren/Marthinsen is a particular representation
of Chen-Fliess series (used in control theory), while the LB-series correspond
to logarithms of CF-series.

REFERENCES

[1] C. Brouder, Runge-Kutta methods and renormalization, Euro. Phys. J. C, 12 (2000),
pp. 521–534.

[2] J. C. Butcher, Coecients for the study of Runge-Kutta integration processes, J. Austral.
Math. Soc., 3 (1963), pp. 185–201.

[3] An algebraic theory of integration methods, Math. Comput., 26 (1972), pp. 79–106.

[4] A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry,
Commun. Math. Phys., (1998), pp. 203–242.

[5] P. E. Crouch and R. Grossman, Numerical integration of ordinary dierential equations
on manifolds, J. Nonlinear Sci., 3 (1993), pp. 1–33.

[6] E. Hairer and G. Wanner, On the Butcher group and general multi-value methods, Computing,
13 (1974), pp. 1–15.

[7] A. Iserles, H. Z. Munthe-Kaas, S. P. N?rsett, and A. Zanna, Lie-group methods, Acta
Numerica, (2000), pp. 215–365.

[8] D. Lewis and J. C. Simo, Conserving algorithms for the dynamics of Hamiltonian systems
on Lie groups, J. Nonlinear Sci., 4 (1994), pp. 253–299.

[9] H. Munthe-Kaas, Lie Butcher theory for Runge-Kutta methods, BIT, 35 (1995), pp. 572–
587.

[10] Runge-Kutta methods on Lie groups, BIT, 38 (1998), pp. 92–111.

[11] H. Z. Munthe-Kaas and S. Krogstad, On enumeration problems in Lie–Butcher theory.
To appear in Special Issue of FCGS. Preprint: http://www.focm.net/gi/gips, 2002.

[12] B. Owren and A. Marthinsen, Runge-Kutta methods adapted to manifolds and based on
rigid frames, BIT, 39 (1999), pp. 116–142.

**报告时间**：2004年5月13日 下午4：00

**报告地点**：科技综合楼三层报告厅