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 Zh. Vychisl. Mat. Mat. Fiz., 2013, Volume 53, Number 4, Pages 575–599 (Mi zvmmf9870)

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.

DOI: https://doi.org/10.7868/S0044466913040133

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English version:
Computational Mathematics and Mathematical Physics, 2013, 53:4, 431–454

Bibliographic databases:

UDC: 519.633

Citation: G. I. Shishkin, “Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation”, Zh. Vychisl. Mat. Mat. Fiz., 53:4 (2013), 575–599; Comput. Math. Math. Phys., 53:4 (2013), 431–454

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. G. I. Shishkin, “Computer difference scheme for a singularly perturbed convection-diffusion equation”, Comput. Math. Math. Phys., 54:8 (2014), 1221–1233
2. G. I. Shishkin, “Difference scheme for a singularly perturbed parabolic convection–diffusion equation in the presence of perturbations”, Comput. Math. Math. Phys., 55:11 (2015), 1842–1856
3. G. I. Shishkin, “Computer difference scheme for a singularly perturbed elliptic convection-diffusion equation in the presence of perturbations”, Comput. Math. Math. Phys., 57:5 (2017), 815–832
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