Journal ArticleDOI
Numerical integration of ordinary differential equations on manifolds
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TLDR
It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field.Abstract:
This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method “decouples” the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from “obstructions” defined by nonvanishing Lie brackets.read more
Citations
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The Physics of Granular Materials
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References
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Book
Solving Ordinary Differential Equations I: Nonstiff Problems
TL;DR: In this paper, the authors describe the historical development of the classical theory of linear methods for solving nonstiff ODEs and present a modern treatment of Runge-Kutta and extrapolation methods.
Book
Analysis of Numerical Methods
TL;DR: Reference Record created on 2005-11-18, modified on 2016-08-08 as discussed by the authors, created on 2011-11 -18, created on 2006-11 18 and modified on 2008-08 -08,08
Journal ArticleDOI
Nonlinear controllability and observability
R. Hermann,Arthur J. Krener +1 more
TL;DR: The properties of controllability, observability, and the theory of minimal realization for linear systems are well-understood and have been very useful in analyzing such systems as discussed by the authors.
Journal ArticleDOI
Canonical transformations depending on a small parameter
TL;DR: In this paper, the concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicity on the small parameter, and the formalism generates nonconservative as well as conservative transformations.