
This article is cited in 12 scientific papers (total in 12 papers)
Grid approximation of singularly perturbed parabolic convectiondiffusion equations with a piecewisesmooth initial condition
G. I. Shishkin^{} ^{} Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia
Abstract:
A boundary value problem for a singularly perturbed parabolic convectiondiffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter $\varepsilon$ that can take arbitrary values in the halfopen interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point $x_0$. For small values of $\varepsilon$ a boundary layer with the typical width of $\varepsilon$ appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point $(x_0,0)$, a transient (moving in time) layer with the typical width of $\varepsilon^{1/2}$ appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem $\varepsilon$uniformly on the entire set $\bar G$, approximate the diffusion flow (i.e., the product $\varepsilon(\partial/\partial x)u(x,t))$ on the set $\bar G^*=\bar G\setminus\{(x_0,0)\}$, and approximate the derivative $(\partial/\partial x)u(x,t)$ on the same set outside the $m$neighborhood of the boundary layer. The approximation of the derivatives $\varepsilon^2(\partial^2/\partial x^2)u(x,t))$ and $(\partial/\partial t)u(x, t)$ on the set $\bar G^*$ is also examined.
Key words:
singularly perturbed boundary value problem, parabolic convectiondiffusion equation, piecewise smooth initial condition, finite difference approximation, convergence, special grids, additive separation of singularities.
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Computational Mathematics and Mathematical Physics, 2006, 46:1, 49–72
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UDC:
519.633 Received: 23.08.2005
Citation:
G. I. Shishkin, “Grid approximation of singularly perturbed parabolic convectiondiffusion equations with a piecewisesmooth initial condition”, Zh. Vychisl. Mat. Mat. Fiz., 46:1 (2006), 52–76; Comput. Math. Math. Phys., 46:1 (2006), 49–72
Citation in format AMSBIB
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\jour Zh. Vychisl. Mat. Mat. Fiz.
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\jour Comput. Math. Math. Phys.
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\pages 4972
\crossref{https://doi.org/10.1134/S0965542506010076}
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http://mi.mathnet.ru/eng/zvmmf533 http://mi.mathnet.ru/eng/zvmmf/v46/i1/p52
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This publication is cited in the following articles:

G. I. Shishkin, “The use of solutions on embedded grids for the approximation of singularly perturbed parabolic
convectiondiffusion equations on adapted grids”, Comput. Math. Math. Phys., 46:9 (2006), 1539–1559

G. I. Shishkin, “Grid approximation of singularly perturbed parabolic equations with piecewise continuous initialboundary conditions”, Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S213–S230

Comput. Math. Math. Phys., 47:3 (2007), 442–462

Shishkin G.I., “Grid approximation of singularly perturbed parabolic reactiondiffusion equations with piecewise smooth initialboundary conditions”, Math. Model. Anal., 12:2 (2007), 235–254

Shishkin G.I., “Using the technique of majorant functions in approximation of a singular perturbed parabolic convectiondiffusion equation on adaptive grids”, Russian J. Numer. Anal. Math. Modelling, 22:3 (2007), 263–289

G. I. Shishkin, “The Richardson scheme for the singularly perturbed parabolic reactiondiffusion equation in the case of a discontinuous initial condition”, Comput. Math. Math. Phys., 49:8 (2009), 1348–1368

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”, Comput. Math. Math. Phys., 49:10 (2009), 1748–1764

Shishkin G.I., “Constructive and formal difference schemes for singularly perturbed parabolic equations in unbounded domains in the case of solutions growing at infinity”, Russ. J. Numer. Anal. Math. Model., 24:6 (2009), 591–617

Shishkina L., Shishkin G., “Conservative numerical method for a system of semilinear singularly perturbed parabolic reactiondiffusion equations”, Math. Model. Anal., 14:2 (2009), 211–228

Shishkin G., “Improved Difference Scheme for a Singularly Perturbed Parabolic ReactionDiffusion Equation with Discontinuous Initial Condition”, Numerical Analysis and its Applications  4th International Conference, NAA 2008, Lecture Notes in Computer Science, 5434, 2009, 116–127

Kadalbajoo M.K., Gupta V., “A brief survey on numerical methods for solving singularly perturbed problems”, Applied Mathematics and Computation, 217:8 (2010), 3641–3716

Vrabel R., “On the Approximation of the Boundary Layers for the Controllability Problem of Nonlinear Singularly Perturbed Systems”, Syst. Control Lett., 61:3 (2012), 422–426

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