TITLE

NUMERICAL METHOD FOR SINGULARLY PERTURBED PARABOLIC EQUATIONS IN UNBOUNDED DOMAINS IN THE CASE OF SOLUTIONS GROWING AT INFINITY

AUTHOR(S)
Shishkin, G. I.
PUB. DATE
January 2009
SOURCE
Computational Methods in Applied Mathematics;2009, Vol. 9 Issue 1, p100
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
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.
ACCESSION #
39231327

 

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