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Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation
G. I. Shishkin Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
Abstract:
For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not
$\varepsilon$-uniformly well conditioned or
$\varepsilon$-uniformly stable to perturbations of the data of the grid problem (here,
$\varepsilon$ is a perturbation parameter,
$\varepsilon\in(0,1]$). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges
$\varepsilon$-uniformly in the maximum norm at an
$O(N^{-1}\ln N+N_0^{-1})$ rate, where
$N+1$ and
$N_0+1$ are the numbers of grid nodes in
$x$ and
$t$, respectively. This scheme is
$\varepsilon$-uniformly well conditioned and
$\varepsilon$-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order
$O(\delta^{-2}\ln\delta^{-1}+\delta_0^{-1})$; i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here,
$\delta=N^{-1}\ln N$ and
$\delta_0=N_0^{-1}$ are the accuracies of the discrete solution in
$x$ and
$t$, respectively.
Key words:
singularly perturbed initial-boundary value problem, parabolic convection-diffusion equation, boundary layer, finite difference schemes on uniform meshes, solution decomposition scheme, $\varepsilon$-uniform convergence, maximum norm, $\varepsilon$-uniform stability of a scheme to perturbations, $\varepsilon$-uniformly well conditioned scheme.
UDC:
519.633 Received: 27.10.2012
DOI:
10.7868/S0044466913040133
Fulltext:
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