
Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity
G. I. Shishkin^{} ^{} Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia
Abstract:
The initialboundary value problem in a domain on a straight line that is unbounded in $x$ is considered for a singularly perturbed reactiondiffusion parabolic equation. The higher order derivative in the equation is multiplied by a parameter $\varepsilon^2$, where $\varepsilon\in(0,1]$. The righthand side of the equation and the initial function grow unboundedly as $x\to\infty$ at a rate of $O(x^2)$. This causes the unbounded growth of the solution at infinity at a rate of $O(\Psi(x))$, where $\Psi(x)=x^2+1$. The initialboundary function is piecewise smooth. When $\varepsilon$ is small, a boundary and interior layers appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristics of the reduced equation passing through the discontinuity points of the initial function. In the problem under examination, the error of the grid solution grows unboundedly in the maximum norm as $x\to\infty$ even for smooth solutions when $\varepsilon$ is fixed. In this paper, the proximity of solutions of the initialboundary value problem and its grid approximations is considered in the weighted maximum norm $\\cdot\^w$ with the weighting
function $\Psi^{1}(x)$; in this norm, the solution of the initialboundary value problem is $\varepsilon$uniformly bounded. Using the method of special grids that condense in a neighborhood of
the boundary layer or in neighborhoods of the boundary and interior layers, special finite difference schemes are constructed and studied that converge $\varepsilon$uniformly in the weighted norm. It is shown that the convergence rate considerably depends on the type of nonsmoothness in the initialboundary conditions. Grid approximations of the Cauchy problem with the righthand side and the initial function growing as $O(\Psi(x))$ that converge $\varepsilon$uniformly in the weighted norm are also considered.
Key words:
parabolic reactiondiffusion equation, unbounded domain, bounded growth of the solution at infinity, piecewise smooth initialboundary function, boundary and interior layers, $\varepsilon$uniform convergence, weighted maximum norm, Cauchy problem.
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Computational Mathematics and Mathematical Physics, 2009, 49:10, 1748–1764
Bibliographic databases:
UDC:
519.633 Received: 12.05.2009
Citation:
G. I. Shishkin, “Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity”, Zh. Vychisl. Mat. Mat. Fiz., 49:10 (2009), 1827–1843; Comput. Math. Math. Phys., 49:10 (2009), 1748–1764
Citation in format AMSBIB
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\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2009
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\pages 18271843
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\jour Comput. Math. Math. Phys.
\yr 2009
\vol 49
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\pages 17481764
\crossref{https://doi.org/10.1134/S0965542509100091}
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http://mi.mathnet.ru/eng/zvmmf4772 http://mi.mathnet.ru/eng/zvmmf/v49/i10/p1827
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