Grid approximation of singularly perturbed parabolic equations with moving boundary layers

Abstract

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation in a domain with boundaries moving along the x‐axis in the positive direction. For small values of the parameter ϵ (that is the coefficient of the highest‐order derivative in the equation, ϵ ∈ (0,1]), a moving boundary layer appears in a neighbourhood of the left lateral boundary S^L ₁. It turns out that, in the class of difference schemes on rectangular grids condensing in a neighbourhood of S^L ₁ with respect to x and t, there do not exist schemes that converge even under the condition P ₀ ⁻¹ Â ϵ^1/2, where P ₀ is the total number of nodes in the meshes used, that is, P ₀ Â N N ₀, where the values N and N ₀ define the numbers of mesh points in x and t. On such meshes, convergence under the condition N ⁻¹ + N ₀ ⁻¹ ≤ ϵ^1/4 cannot be achieved. Examination of widths similar to Kolmogorov's widths allows us to establish necessary and sufficient conditions for the ϵ‐uniform convergence of approximations to the solution of the boundary value problem. Using these conditions, a scheme is constructed on a mesh being piece‐wise uniform in a coordinate system adapted to the moving boundary. This scheme converges ϵ‐uniformly at the rate O(N ⁻¹ ln N + N₀ ⁻¹).

Key words:

• boundary value problem
• perturbation parameter ϵ
• parabolic reaction‐diffusion equation
• finite difference approximation
• moving boundary layer
• Kolmogorov's widths
• ϵ‐uniform convergence

Notes

This research was supported by the Russian Foundation for Basic Research under grant No. 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.