2011
DOI: 10.1090/conm/539/10634
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Magnus expansions and beyond

Abstract: In this brief review we describe the coming of age of Magnus expansions as an asymptotic and numerical tool in the investigation of linear differential equations in a Lie-group and homogeneous-space setting. Special attention is afforded to the many connections between modern theory of geometric numerical integration and other parts of mathematics: from abstract algebra to differential geometry and combinatorics, all the way to classical numerical analysis. Lie-group equationsNumerical solution of evolutionary… Show more

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Cited by 5 publications
(5 citation statements)
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“…(51). The main paradigm of the Lie group methods is the reformulation of the underlying equation on the Lie group as an algebra action, and since the Lie algebra is a linear space, all reasonable discretization methods can be expected to respect its structure [23,24]. Therefore the user can influence the accuracy of the overall algorithm by choosing the ODE integrator to solve Eq.…”
Section: Explicit Runge-kutta Methods Of Fourth-order (Rk4)mentioning
confidence: 99%
See 1 more Smart Citation
“…(51). The main paradigm of the Lie group methods is the reformulation of the underlying equation on the Lie group as an algebra action, and since the Lie algebra is a linear space, all reasonable discretization methods can be expected to respect its structure [23,24]. Therefore the user can influence the accuracy of the overall algorithm by choosing the ODE integrator to solve Eq.…”
Section: Explicit Runge-kutta Methods Of Fourth-order (Rk4)mentioning
confidence: 99%
“…[3]. The smooth coordinate map φ takes so(3) to SO(3) [22,24]. The most important coordinate map is the matrix exponential 2 exp(•), which may be evaluated on SO(3) efficiently by Rodrigues' formula [3,24].…”
Section: Standard Lie Group Approachmentioning
confidence: 99%
“…In the present paper, Ω is truncated using a sixth-order Magnus series and the integrals are approximated using the three-point Gaussian integration method [41]. Ω [6] is obtained in the following equations.…”
Section: Magnus Expansionmentioning
confidence: 99%
“…which agrees with the formulation (30) with f 1 (g, µ) = ι(g, µ) and f 2 (g, µ) = R * g ∂ ℓ ∂ g (g, ι(g, µ)). (35) Variational integrators are derived by extremising an approximation to (33),…”
Section: Variational Integrators On Lie Groupsmentioning
confidence: 99%
“…Important subjects related to Lie group integrators not covered here include the case of linear differential equations in Lie groups and the methods based on Magnus expansions, Fer expansions, and Zassenhaus splitting schemes. These are methods that could fit well into a survey on Lie group integrators, but for information on these topics we refer the reader to excellent expositions such as [4,33]. Another topic we leave out here is that of stochastic Lie group integrators, see e.g.…”
mentioning
confidence: 99%