International Journal of Computer Mathematics Vol. 85, No. 11, November 2008, 1727–1740 Recursive form of Sobolev gradient method for ODEs on long intervals D. Mujeeb a , J.W. Neuberger b and S. Sial c * a Department of Computer Science, Cornell University, Ithaca NY, USA; b Department of Mathematics, University of North Texas, Denton TX, USA; c Department of Mathematics, Lahore University of Management Sciences, Sector U, DHA, Lahore Cantt, Pakistan (Received 05 January 2007; revised version received 10 May 2007; accepted 03 July 2007 ) 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 recur- sive 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 1. Introduction One approach to the solution of ODEs is to seek a critical point of an error functional constructed so that the equation can be considered to be solved when the error functional is zero. The recent theory of Sobolev gradients [12] gives a unified point of view on such problems, both in function spaces and in finite-dimensional approximations to such problems. Sobolev gradients have been used for ODE problems [9,12] in a finite-difference setting, PDEs in finite-difference [9] and finite- element settings [1], minimizing energy functionals associated with Landau–Ginzburg models in finite-difference [16] and finite-element [15,17] settings, the electrostatic potential equation [7], non-linear elliptic problems [6], semilinear elliptic systems [5], simulation of Bose–Einstein con- densates [4], and inverse problems in elasticity [2] and groundwater modelling [8]. A preliminary study of the use of the standard Sobolev gradient method as an alternative to numerical solution of a stiff ODE for flame-size via implicit Runge–Kutta or other such time-integration methods has been presented elsewhere [11]. *Corresponding author. Email: sultans@lums.edu.pk ISSN 0020-7160 print/ISSN 1029-0265 online © 2008 Taylor & Francis DOI: 10.1080/00207160701558465 http://www.informaworld.com