Recursive form of Sobolev gradient method for ODEs on long intervals

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 o...

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Bibliographic Details
Published in:International journal of computer mathematics Vol. 85; no. 11; pp. 1727 - 1740
Main Authors: Mujeeb, D., Neuberger, J. W., Sial, S.
Format: Journal Article
Language:English
Published: Taylor & Francis 01.11.2008
Subjects:
initial-value problems
numerical solutions
ODEs
Sobolev gradients
ISSN:0020-7160, 1029-0265
Online Access:Get full text
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Summary: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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ISSN:0020-7160
1029-0265
DOI:10.1080/00207160701558465