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Difference scheme for an initial-boundary value problem for a singularly perturbed transport equation
G. I. Shishkin Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, Russia
Abstract:
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.
Key words:
transport equation, singularly perturbed initial-boundary value problem, boundary layer, standard difference scheme, uniform mesh, special difference scheme, Shishkin mesh, maximum norm, decomposition of solution.
UDC:
519.63 Received: 01.12.2016
DOI:
10.7868/S004446691711014X
Fulltext:
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