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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Vydáno v:International journal of computer mathematics Ročník 85; číslo 11; s. 1727 - 1740
Hlavní autoři: Mujeeb, D., Neuberger, J. W., Sial, S.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Taylor & Francis 01.11.2008
Témata:
initial-value problems
numerical solutions
ODEs
Sobolev gradients
ISSN:0020-7160, 1029-0265
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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.
AbstractList 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.
Author Mujeeb, D.
Sial, S.
Neuberger, J. W.
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  organization: Department of Mathematics , Lahore University of Management Sciences, Sector U, DHA
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CitedBy_id crossref_primary_10_1002_num_23121
crossref_primary_10_1007_s40314_016_0319_7
crossref_primary_10_1016_j_jcp_2011_03_056
crossref_primary_10_1016_j_jcp_2009_10_048
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10.1088/0953-2048/18/5/015
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10.1016/S0378-4754(02)00190-8
10.1016/S0021-9991(03)00202-X
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Snippet The Sobolev gradient method has been shown to be effective at constructing finite-dimensional approximations to solutions of initial-value problems. Here we...
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SubjectTerms initial-value problems
numerical solutions
ODEs
Sobolev gradients
Title Recursive form of Sobolev gradient method for ODEs on long intervals
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