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 Zh. Vychisl. Mat. Mat. Fiz., 2006, Volume 46, Number 1, Pages 52–76 (Mi zvmmf533)

Grid approximation of singularly perturbed parabolic convection-diffusion equations with a piecewise-smooth 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 convection-diffusion 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 half-open 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 convection-diffusion equation, piecewise smooth initial condition, finite difference approximation, convergence, special grids, additive separation of singularities.

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English version:
Computational Mathematics and Mathematical Physics, 2006, 46:1, 49–72

Bibliographic databases:

UDC: 519.633

Citation: G. I. Shishkin, “Grid approximation of singularly perturbed parabolic convection-diffusion equations with a piecewise-smooth 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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\paper Grid approximation of singularly perturbed parabolic convection-diffusion equations with a~piecewise-smooth initial condition
\jour Zh. Vychisl. Mat. Mat. Fiz.
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\vol 46
\issue 1
\pages 52--76
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2239726}
\zmath{https://zbmath.org/?q=an:05200886}
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\jour Comput. Math. Math. Phys.
\yr 2006
\vol 46
\issue 1
\pages 49--72
\crossref{https://doi.org/10.1134/S0965542506010076}
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. I. Shishkin, “The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids”, Comput. Math. Math. Phys., 46:9 (2006), 1539–1559
2. G. I. Shishkin, “Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions”, Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S213–S230
3. Comput. Math. Math. Phys., 47:3 (2007), 442–462
4. Shishkin G.I., “Grid approximation of singularly perturbed parabolic reaction-diffusion equations with piecewise smooth initial-boundary conditions”, Math. Model. Anal., 12:2 (2007), 235–254
5. Shishkin G.I., “Using the technique of majorant functions in approximation of a singular perturbed parabolic convection-diffusion equation on adaptive grids”, Russian J. Numer. Anal. Math. Modelling, 22:3 (2007), 263–289
6. G. I. Shishkin, “The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition”, Comput. Math. Math. Phys., 49:8 (2009), 1348–1368
7. 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
8. 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
9. Shishkina L., Shishkin G., “Conservative numerical method for a system of semilinear singularly perturbed parabolic reaction-diffusion equations”, Math. Model. Anal., 14:2 (2009), 211–228
10. Shishkin G., “Improved Difference Scheme for a Singularly Perturbed Parabolic Reaction-Diffusion Equation with Discontinuous Initial Condition”, Numerical Analysis and its Applications - 4th International Conference, NAA 2008, Lecture Notes in Computer Science, 5434, 2009, 116–127
11. 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
12. 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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