Singular perturbation methods for ordinary differential equations
Singular perturbation methods for ordinary differential equations
Steepest descent using smoothed gradients
Applied Mathematics and Computation
A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates
Mathematics and Computers in Simulation - Nonlinear waves: computation and theory II
Energy minimization using Sobolev gradients: application to phase separation and ordering
Journal of Computational Physics
On the recovery of transport parameters in groundwater modelling
Journal of Computational and Applied Mathematics - Special issue: On the occasion of the eightieth birthday of prof. W.M. Everitt
Preconditioning operators and Sobolevgradients for nonlinear elliptic problems
Computers & Mathematics with Applications
Sobolev gradient preconditioning for the electrostatic potential equation
Computers & Mathematics with Applications
Journal of Computational Physics
Approximate solutions to Poisson-Boltzmann systems with Sobolev gradients
Journal of Computational Physics
Hi-index | 0.01 |
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.