Numerical study of an initial-boundary value Neumann problem for a singularly perturbed parabolic equation

For a singularly perturbed one-dimensional parabolic equation with a perturbation parameter $\varepsilon$ multiplying the highest-order derivative in the equation, $\varepsilon \in (0,1]$, an initial-boundary value Neumann problem is considered on a segment. In this problem, when the parameter $\varepsilon$ tends to zero, boundary layers appear in neighborhoods of the lateral boundary. In the paper, convergence of the problem solution and its regular and singular components are studied. It is shown that standard finite difference schemes on uniform grids used to numerically solve this problem do not converge $\varepsilon$-uniformly. The error in the grid solution grows unboundedly when the parameter $\varepsilon \rightarrow 0$. The use of a special difference scheme on Shishkin grid which is a piecewise-uniform mesh with respect to $x$ condensing in neighborhoods of boundary layers and a uniform mesh in $t$, constructed by using monotone grid approximations of the differential problems, allows us to find a numerical solution of this problem convergent in the maximum norm $\varepsilon$-uniformly. The results of the numerical experiments confirm the theoretical results.