Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables

In an unbounded (with respect to $x$ and $t$) domain (and in domains that can be arbitrarily large), an initial-boundary value problem for singularly perturbed parabolic reaction-diffusion equations with the perturbation parameter $\varepsilon^2$ multiplying the higher order derivative is considered. The parameter $\varepsilon$ takes arbitrary values in the half-open interval (0, 1]. To solve this problem, difference schemes on grids with an infinite number of nodes (formal difference schemes) are constructed that converge $\varepsilon$-uniformly in the entire unbounded domain. To construct these schemes, the classical grid approximations of the problem on the grids that are refined in the boundary layer are used. Schemes on grids with a finite number of nodes (constructive difference schemes) are also constructed for the problem under examination. These schemes converge for fixed values of $\varepsilon$ in the prescribed bounded subdomains that can expand as the number of grid points increases. As $\varepsilon\to0$, the accuracy of the solution provided by such schemes generally deteriorates and the size of the subdomains decreases. Using the condensing grid method, constructive difference schemes that converge $\varepsilon$-uniformly are constructed. In these schemes, the approximation accuracy and the size of the prescribed subdomains (where the schemes are convergent) are independent of $\varepsilon$ and the subdomains may expand as the number of nodes in the underlying grids increases.