Lie–Butcher Series, Geometry, Algebra and Computation

Munthe-Kaas, Hans; K., Føllesdal, Kristoffer

doi:10.1007/978-3-030-01397-4_3

Cited by 8 publications

(7 citation statements)

References 22 publications

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“…In particular, we will consider abstract analysis of higher-order Padé approximations and quadrature techniques, including Runge-Kutta and exponential integration methods, with the goal of constructing methods of arbitrary order. This will also allow for the consideration of the underlying algebraic structures of these methods, as has been done with Butcher and Lie-Butcher series for classical problems [24]. We also intend to extend these results to abstract problems posed on abstract manifold structures, allowing for the consideration of novel problems in differential geometry.…”

Section: Discussionmentioning

confidence: 96%

Analysis of an approximation to a fractional extension problem

Padgett

2019

Bit Numer Math

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The purpose of this article is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework herein. The proposed method differs from many standard techniques, as we approximate the true solution to the abstract problem, rather than solve an associated discrete problem. The numerical method is shown to be consistent, stable, and convergent in an appropriate Banach space. These results are built upon well understood results from semigroup theory. Numerical experiments are provided to demonstrate the theoretical results.

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Section: Discussionmentioning

confidence: 96%

Analysis of an approximation to a fractional extension problem

Padgett

2019

Bit Numer Math

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“…Order conditions for LB-series methods have been studied in [2,12,33,37,38]. Implementations of LB-series methods in Haskel were considered in [31].…”

Section: B-series and Lb-series In Numerical Analysismentioning

confidence: 99%

An operadic approach to substitution in Lie–Butcher series

Rahm¹

2022

Forum of Mathematics, Sigma

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The paper follows an operadic approach to provide a bialgebraic description of substitution for Lie–Butcher series. We first show how the well-known bialgebraic description for substitution in Butcher’s B-series can be obtained from the pre-Lie operad. We then apply the same construction to the post-Lie operad to arrive at a bialgebra $\mathcal {Q}$ . By considering a module over the post-Lie operad, we get a cointeraction between $\mathcal {Q}$ and the Hopf algebra $\mathcal {H}_{N}$ that describes composition for Lie–Butcher series. We use this coaction to describe substitution for Lie–Butcher series.

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“…The enveloping algebra of P C identifies as the tensor algebra T (OT C ). It was introduced and studied in [24], see also [22] for more on the computational aspect in this algebra. Its completion identifies aŝ…”

Section: 1mentioning

confidence: 99%

Pre- and Post-Lie Algebras: The Algebro-Geometric View

Fløystad¹,

Munthe-Kaas²

2018

Abel Symposia

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We relate composition and substitution in pre-and post-Lie algebras to algebraic geometry. The Connes-Kreimer Hopf algebras, and MKW Hopf algebras are then coordinate rings of the infinite-dimensional affine varieties consisting of series of trees, resp. Lie series of ordered trees. Furthermore we describe the Hopf algebras which are coordinate rings of the automorphism groups of these varieties, which govern the substitution law in pre-and post-Lie algebras.

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Lie–Butcher Series, Geometry, Algebra and Computation

Cited by 8 publications

References 22 publications

Analysis of an approximation to a fractional extension problem

Analysis of an approximation to a fractional extension problem

An operadic approach to substitution in Lie–Butcher series

Pre- and Post-Lie Algebras: The Algebro-Geometric View

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