Shishkin, G. I.
January 2009
Computational Methods in Applied Mathematics;2009, Vol. 9 Issue 1, p100
Academic Journal
An initial-boundary value problem is considered in an unbounded domain on the x-axis for a singularly perturbed parabolic reaction-diffusion equation; the highest-order derivative of the equation is multiplied by the parameter ε², where ε ϵ (0, 1]. When x tends to ∞, the right-hand side of the equation and the initial function increase unboundedly (as O (x²)), which leads to an unbounded growth of the solution at infinity (as O (Ψ(x) , where Ψ(x) = x² + 1). For small values of the parameter ", a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound as x → ∞ even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered in the weight maximum norm ∥ · ∥w with the weight function Ψ-1(x); in this norm the solution of the initial-boundary value problem is ε-uniformly bounded. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.


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