Finite Difference Scheme for a Singularly Perturbed Parabolic Equations in the Presence of Initial and Boundary Layers
Journal Title: Mathematical Modelling and Analysis - Year 2008, Vol 13, Issue 4
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 ε[sub]1[/sub][sup]2 [/sup]and ε[sub]2[/sub][sup]2[/sup], 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. [i]A priori [/i]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 [i]x [/i]and [i]t, [/i] a difference scheme is constructed that converges ε[sup]--[/sup]uniformly at the rate [i]O[/i]([i]N[/i][sup]-2[/sup]ln[sup]2 [/sup][i]N [/i]+[i]N[/i][sub]0[/sub][sup]-1[/sup] ln [i]N[/i][sub]0[/sub]), where ([i]N[/i]+1) and ([i]N[/i][sub]0[/sub]+1) are the numbers of mesh points in [i]x[/i] and [i]t [/i]respectively.
Authors and Affiliations
N. Cordero, K. Cronin, G. Shishkin, L. Shishkina, M. Stynes
Keywords
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- EP ID EP82936
- DOI 10.3846/1392-6292.2008.13.483-49
- Views 105
- Downloads 0
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Finite Difference Scheme for a Singularly Perturbed Parabolic Equations in the Presence of Initial and Boundary Layers
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 different...