Grid Approximation of Multiscale Problem Arising in Singularly Perturbed Parabolic Reaction-Diffusion Equations in an Unbounded Domain

On a semiaxis we consider an initial boundary value problem for singularly perturbed parabolic reaction-diffusion equations. The highest derivative in the equations is multiplied by a small parameter $\eps$, $\,\eps \in (0,1]$. The solution of such a problem exhibits multiple scales. Besides the usual (natural) scale related to a variation of the problem data, one can observe a resolution scale, which is specified by the width of the domain on which the numerical solutions are being computed, and a boundary-layer scale controlled by the parameter $\eps$. In this paper we solve the multiscale problem using the renormalization method, that is, we construct the following (normalized and renormalized) finite difference schemes: (a) {\it formal} (nonconstructive) {\it schemes}, i.e., schemes on meshes with an {\it infinite number} of nodes, which lead to approximate solutions converging $\eps$-uniformly at each node; and (b) {\it constructive schemes}, i.e., schemes on meshes with a {\it finite number} of the nodes, which lead to approximate solutions converging for fixed values of the parameter $\eps$ at each node of arbitrarily chosen bounded subdomains whose widths increase as the number of nodes grows. With standard constructive schemes, in general, the accuracy of the approximate solutions deteriorates and the widths of subdomains decrease when $\eps \to 0$. Here, conditions are given under which the approximate solutions generated by the constructive schemes converge $\eps$-uniformly, i.e., the accuracy of the numerical approximations and the widths of the subdomains on which the schemes converge are independent of the parameter~$\eps$. To construct the schemes, we use classical finite difference approximations on piecewise uniform meshes which are refined in a neighbourhood of the boundary layer.