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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Waveform relaxation as a dynamical system
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by Morten Bjørhus and Andrew M. Stuart PDF
Math. Comp. 66 (1997), 1101-1117 Request permission

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
  • 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
  • © Copyright 1997 American Mathematical Society
  • Journal: Math. Comp. 66 (1997), 1101-1117
  • MSC (1991): Primary 65L05, 34C35, 65Q05
  • DOI: https://doi.org/10.1090/S0025-5718-97-00847-8
  • MathSciNet review: 1415796