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Recursive form of Sobolev gradient method for ODEs on long intervals

Published: 01 November 2008 Publication History

Abstract

The Sobolev gradient method has been shown to be effective at constructing finite-dimensional approximations to solutions of initial-value problems. Here we show that the efficiency of the algorithm as often used breaks down for long intervals. Efficiency is recovered by solving from left to right on subintervals of smaller length. The mathematical formulation for Sobolev gradients over non-uniform one-dimensional grids is given so that nodes can be added or removed as required for accuracy. A recursive variation of the Sobolev gradient algorithm is presented which constructs subintervals according to how much work is required to solve them. This allows efficient solution of initial-value problems on long intervals, including for stiff ODEs. The technique is illustrated by numerical solutions for the prototypical problem u'=u, equation for flame-size, and the van der Pol's equation.

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Published In

cover image International Journal of Computer Mathematics
International Journal of Computer Mathematics  Volume 85, Issue 11
November 2008
126 pages
ISSN:0020-7160
EISSN:1029-0265
Issue’s Table of Contents

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Taylor & Francis, Inc.

United States

Publication History

Published: 01 November 2008

Author Tags

  1. ODEs
  2. Sobolev gradients
  3. initial-value problems
  4. numerical solutions

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