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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Waveform relaxation as a dynamical system


Authors: Morten Bjørhus and Andrew M. Stuart
Journal: Math. Comp. 66 (1997), 1101-1117
MSC (1991): Primary 65L05, 34C35, 65Q05
MathSciNet review: 1415796
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Abstract: In this paper the properties of waveform relaxation are studied when applied to the dynamical system generated by an autonomous ordinary differential equation. In particular, the effect of the waveform relaxation on the invariant sets of the flow is analysed. Windowed waveform relaxation is studied, whereby the iterative technique is applied on successive time intervals of length $T$ and a fixed, finite, number of iterations taken on each window. This process does not generate a dynamical system on $\mathbb {R}^+$ since two different applications of the waveform algorithm over different time intervals do not, in general, commute. In order to generate a dynamical system it is necessary to consider the time $T$ map generated by the relaxation process. This is done, and $C^1$-closeness of the resulting map to the time $T$ map of the underlying ordinary differential equation is established. Using this, various results from the theory of dynamical systems are applied, and the results discussed.


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Additional Information

Morten Bjørhus
Affiliation: Department of Mathematical Sciences, The Norwegian Institute of Technology, N-7034 Trondheim, Norway
Address at time of publication: Forsvarets Forskningsinstitutt, PO Box 25, N-2007 Kjeller, Norway
Email: mbj@ffi.no

Andrew M. Stuart
Affiliation: Department of Mechanical Engineering, Division of Mechanics and Computation, Stanford University, Durand Building, Room 257, Stanford, California 94305
Email: stuart@sccm.stanford.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-97-00847-8
PII: S 0025-5718(97)00847-8
Received by editor(s): December 19, 1994
Received by editor(s) in revised form: October 16, 1995
Additional Notes: The first author was supported by the Research Council of Norway
The second author was supported by the National Science Foundation and the Office for Naval Research
Article copyright: © Copyright 1997 American Mathematical Society