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

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

Keyword : 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