Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions

Authors:
Natalia Kopteva;Martin Stynes
Affiliations:
Department of Computational Mathematics and Cybernetics, Moscow State University, Vorob'ovy Gory, 119899 Moscow, Russia;Department of Mathematics, National University of Ireland, Cork, Ireland
Venue:
Applied Numerical Mathematics
Year:
2004

Citing 4
Cited 4

Singular perturbation methods for ordinary differential equations

Singular perturbation methods for ordinary differential equations
A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions

Mathematics of Computation
Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes

Computing
Sufficient conditions for uniform convergence on layer-adapted grids

Applied Numerical Mathematics

Solving regularly and singularly perturbed reaction-diffusion equations in three space dimensions

Journal of Computational Physics
A recent survey on computational techniques for solving singularly perturbed boundary value problems

International Journal of Computer Mathematics
Numerical Analysis of a 2d Singularly Perturbed Semilinear Reaction-Diffusion Problem

Numerical Analysis and Its Applications
Numerical analysis of singularly perturbed nonlinear reaction-diffusion problems with multiple solutions

Computers & Mathematics with Applications

Quantified Score

Hi-index	0.01

Visualization

Abstract

A nonlinear reaction-diffusion two-point boundary value problem with multiple solutions is considered. Its second-order derivative is multiplied by a small positive parameter ε, which induces boundary layers. Using dynamical systems techniques, asymptotic properties of its discrete sub- and super-solutions are derived. These properties are used to investigate the accuracy of solutions of a standard three-point difference scheme on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in ε for ε ≤ CN-1, where N is the number of mesh intervals. Numerical experiments are performed to support the theoretical results.