Invariant connections, lie algebra actions, and foundations of numerical integration on manifolds

Hans Z. Munthe-Kaas; Ari Stern; Olivier Verdier

doi:10.1137/19M1252879

Invariant connections, lie algebra actions, and foundations of numerical integration on manifolds

Hans Z. Munthe-Kaas
, Ari Stern
, Olivier Verdier

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numerical integrators, giving a characterization of the spaces on which Butcher and Lie–Butcher series methods, which generalize Runge–Kutta methods, may be applied.

Original language	English
Pages (from-to)	49-68
Number of pages	20
Journal	SIAM Journal on Applied Algebra and Geometry
Volume	4
Issue number	1
DOIs	https://doi.org/10.1137/19M1252879
State	Published - 2020

Keywords

Connections
Lie algebroids
Lie–Butcher series
Numerical integration on manifolds
Post-Lie algebras
Pre-Lie algebras

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10.1137/19M1252879

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abstract = "Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numerical integrators, giving a characterization of the spaces on which Butcher and Lie–Butcher series methods, which generalize Runge–Kutta methods, may be applied.",

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AU - Stern, Ari

AU - Verdier, Olivier

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N2 - Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numerical integrators, giving a characterization of the spaces on which Butcher and Lie–Butcher series methods, which generalize Runge–Kutta methods, may be applied.

AB - Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numerical integrators, giving a characterization of the spaces on which Butcher and Lie–Butcher series methods, which generalize Runge–Kutta methods, may be applied.

KW - Connections

KW - Lie algebroids

KW - Lie–Butcher series

KW - Numerical integration on manifolds

KW - Post-Lie algebras

KW - Pre-Lie algebras

UR - https://www.scopus.com/pages/publications/85090685230

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DO - 10.1137/19M1252879

M3 - Article

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JO - SIAM Journal on Applied Algebra and Geometry

JF - SIAM Journal on Applied Algebra and Geometry

IS - 1

ER -

Invariant connections, lie algebra actions, and foundations of numerical integration on manifolds

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Keywords

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