Constructive and formal difference schemes for singularly perturbed parabolic equations in unbounded domains in the case of solutions growing at infinity

Abstract

An initial boundary value problem for a singular perturbed parabolic reaction-diffusion equation is considered in a domain unbounded in x on the real axis; the leading derivative of the equation contains the parameter ε 2 ; ε ∈ (0, 1]. The right-hand side of the equation and the initial function indefinitely grow as O x 2 for x → ∞, which leads to an indefinite growth of the solution at infinity as O Ψ(x) , where Ψ(x) = x 2 + 1. For small values of the parameter ε a parabolic boundary layer appears in the neighbourhood of the lateral part of the boundary. In this problem for fixed values of the parameter ε the error of the grid solution indefinitely grows in the uniform norm for x → ∞. The closeness of the solutions to the initial boundary value problem and to its grid approximations is considered in this paper in the weighted uniform norm · w with the weight function Ψ −1 (x); the solution to the initial boundary value problem is ε-uniformly bounded in this norm. Using special grids condensing in the neighbourhood of the boundary layer, we construct formal difference schemes (schemes on grids with an infinite number of nodes) convergent ε-uniformly in the weighted norm. Based on the notion of the domain of essential dependence of the solution, we propose constructive difference schemes (schemes on grids with a finite number of nodes) convergent ε-uniformly in the weighted norm on given bounded subdomains.Special numerical methods for solving boundary value problems for singularly perturbed equations in bounded domains are well elaborated for the case of sufficiently smooth data (see, e.g., [2 -4, 8, 9, 13, 20] and references therein). Such numerical methods allow one to approximate solutions with an error not depending on the perturbation parameter ε, as the methods converge ε-uniformly.Only a few publications have considered ε-uniformly convergent methods for problems in unbounded domains. Special difference schemes for singularly perturbed problems for elliptic and parabolic equations in unbounded domains were studied in [14,18,19] and [1], respectively; the solutions to those problems were ε-uniformly bounded.