Difference scheme for an initial-boundary value problem for a singularly perturbed transport equation

An initial-boundary value problem for a singularly perturbed transport equation with a perturbation parameter $\varepsilon$ multiplying the spatial derivative is considered on the set $\overline{G}=G\cup S$, where $\overline{G}=\overline{D}\times[0\leqslant t\leqslant T]$, $\overline{D}=\{0\leqslant x\leqslant d\}$, $S = S^l\cup S$, $S^l$ and $S_0$ are the lateral and lower boundaries. The parameter $\varepsilon$ takes arbitrary values from the half-open interval $(0,1]$. In contrast to the well-known problem for the regular transport equation, for small values of $\varepsilon$, this problem involves a boundary layer of width $O(\varepsilon)$ appearing in the neighborhood of $S^l$; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge $\varepsilon$-uniformly in the maximum norm. Convergence occurs only if $h=dN^{-1}\ll\varepsilon$, $N_0^{-1}\ll 1$, where $N$ and $N_0$ are the numbers of grid intervals in $x$ and $t$, respectively, and $h$ is the mesh size in $x$. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in $x$ and uniform in $t$. On such a grid, a monotone difference scheme for the initial-boundary value problem for the singularly perturbed transport equation converges $\varepsilon$-uniformly in the maximum norm at an $\mathcal{O}(N^{-1}+N_0^{-1})$ rate.