Recursive form of Sobolev gradient method for ODEs on long intervals

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

2000 AMS Subject Classification:

• 65L09
• 34G20

Acknowledgements

We are grateful to the School of Arts and Sciences, Lahore University of Management Sciences for a research grant for this project.