Finite difference scheme for a singularly perturbed parabolic equations in the presence of initial and boundary layers

    Nicolas Cordero Info
    Kevin Cronin Info
    Grigorii Shishkin Info
    Lida Shishkina Info
    Martin Stynes Info

Abstract

The grid approximation of an initial‐boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation. The second‐order spatial derivative and the temporal derivative in the differential equation are multiplied by parameters å 2 1 and å 2 2, respectively, that take arbitrary values in the open‐closed interval (0,1]. The solutions of such parabolic problems typically have boundary, initial layers and/or initial‐boundary layers. A priori estimates are constructed for the regular and singular components of the solution. Using such estimates and the condensing mesh technique for a tensor‐product grid, piecewise‐uniform in xand t, a difference scheme is constructed that converges å‐uniformly at the rate O(N−2 ln2 N + N0 −1 ln N0 ), where (N + 1) and (N0 + 1) are the numbers of mesh points in x and t respectively.

First Published Online: 14 Oct 2010

Keywords:

initial-boundary value problem, parabolic reaction-diffusion equation, perturbation vector-parameter ε, finite difference approximation, boundary layer, initial layer, initial-boundary layer, piecewise-uniform grids, ε-uniform convergence

How to Cite

Cordero, N., Cronin, K., Shishkin, G., Shishkina, L., & Stynes, M. (2008). Finite difference scheme for a singularly perturbed parabolic equations in the presence of initial and boundary layers. Mathematical Modelling and Analysis, 13(4), 483-492. https://doi.org/10.3846/1392-6292.2008.13.483-492

Share

Published in Issue
December 31, 2008
Abstract Views
512

View article in other formats

CrossMark check

CrossMark logo

Published

2008-12-31

Issue

Section

Articles

How to Cite

Cordero, N., Cronin, K., Shishkin, G., Shishkina, L., & Stynes, M. (2008). Finite difference scheme for a singularly perturbed parabolic equations in the presence of initial and boundary layers. Mathematical Modelling and Analysis, 13(4), 483-492. https://doi.org/10.3846/1392-6292.2008.13.483-492

Share