Robust numerical method for a system of singularly perturbed parabolic reaction‐diffusion equations on a rectangle

Abstract

A Dirichlet problem is considered for a system of two singularly perturbed parabolic reaction‐diffusion equations on a rectangle. The parabolic boundary layer appears in the solution of the problem as the perturbation parameter ϵ tends to zero. On the basis of the decomposition solution technique, estimates for the solution and derivatives are obtained. Using the condensing mesh technique and the classical finite difference approximations of the boundary value problem under consideration, a difference scheme is constructed that converges ϵ‐uniformly at the rate O ‘N⁻² ln² N + N₀ ⁻¹) , where N = min_s N_s, s = 1, 2, N_s + 1 and N₀ + 1 are the numbers of mesh points on the axis x_s and on the axis t, respectively.

Key words:

• initial‐boundary value problem
• problem on a rectangle
• perturbation parameter ϵ
• system of parabolic equations
• reaction‐diffusion equations
• finite difference approximation
• parabolic boundary layer
• decomposition solution technique
• a priori estimates
• ϵ‐uniform convergence

Notes

This research was supported by the Russian Foundation for Basic Research under grants Nos. 07–01–00729, the Boole Centre for Research in Informatics at the National University of Ireland in Cork, and by the Mathematics Applications Consortium for Science and Industry in Ireland (MACSI) under the Science Foundation Ireland (SFI) Mathematics Initiative.