Recursive form of Sobolev gradient method for ODEs on long intervals

Authors: Mujeeb, D.¹; Neuberger, J. W.²; Sial, S.³

Source: International Journal of Computer Mathematics, Volume 85, Number 11, November 2008 , pp. 1727-1740(14)

Buy & download fulltext article:

Price: $56.94 plus tax (Refund Policy)

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.

Keywords: Sobolev gradients; ODEs; numerical solutions; initial-value problems

Document Type: Research article

DOI: http://dx.doi.org/10.1080/00207160701558465

Affiliations: 1: Department of Computer Science, Cornell University, Ithaca, NY, USA 2: Department of Mathematics, University of North Texas, Denton, TX, USA 3: Department of Mathematics, Lahore University of Management Sciences, Sector U, DHA, Lahore Cantt, Pakistan

Publication date: 2008-11-01

More about this publication?