On Cayley-Transform Methods for the Discretization of Lie-Group Equations |
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Authors: | A Iserles |
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Institution: | (1) Department of Applied Mathematics and Theoretical Physics University of Cambridge Silver Street Cambridge CB3 9EW, England {a.iserles@damtp.cam.ac.uk.}, UK |
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Abstract: | In this paper we develop in a systematic manner the theory of time-stepping methods based on the Cayley transform. Such methods
can be applied to discretize differential equations that evolve in some Lie groups, in particular in the orthogonal group
and the symplectic group. Unlike many other Lie-group solvers, they do not require the evaluation of matrix exponentials.
Similarly to the theory of Magnus expansions in 13], we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals
can be evaluated to high order by using special quadrature formulas similar to the construction in 13]. Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual
advantages associated with time symmetry, e.g., even order of approximation. However, time symmetry (with its attendant benefits)
is attained when exact integrals are replaced by certain quadrature formulas.
March 7, 2000. Final version received: August 10, 2000. Online publication: January 2, 2001. |
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Keywords: | AMS Classification 65L05 22E60 |
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