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 Trudy Inst. Mat. i Mekh. UrO RAN, 2007, Volume 13, Number 2, Pages 218–233 (Mi timm101)

Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions

G. I. Shishkin

Abstract: The Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise continuous initial-boundary conditions in a rectangular domain. The highest derivative in the equation is multiplied by a parameter $\varepsilon^2$, $\varepsilon\in (0,1]$. For small values of the parameter $\varepsilon$, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the limit equation passing through the point of discontinuity of the initial function, there arise a boundary layer and an interior layer (of characteristic width $\varepsilon$), respectively, which have bounded smoothness for fixed values of the parameter $\varepsilon$. Using the method of additive splitting of singularities (generated by discontinuities of the boundary function and its low-order derivatives), as well as the method of condensing grids (piecewise uniform grids condensing in a neighborhood of boundary layers), we construct and investigate special difference schemes that converge $\varepsilon$-uniformly with the second order of accuracy in $x$ and the first order of accuracy in $t$, up to logarithmic factors.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2007, 259, suppl. 2, S213–S230

Document Type: Article
UDC: 519.633

Citation: G. I. Shishkin, “Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions”, Trudy Inst. Mat. i Mekh. UrO RAN, 13, no. 2, 2007, 218–233; Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S213–S230

Citation in format AMSBIB
\Bibitem{Shi07} \by G.~I.~Shishkin \paper Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2007 \vol 13 \issue 2 \pages 218--233 \mathnet{http://mi.mathnet.ru/timm101} \elib{http://elibrary.ru/item.asp?id=12040781} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2007 \vol 259 \issue , suppl. 2 \pages S213--S230 \crossref{https://doi.org/10.1134/S0081543807060156} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-38949177520} 

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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, “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
2. Shishkin G.I., “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
3. Shishkin G.I., Shishkina L.P., “Finite difference schemes for the singularly perturbed reaction-diffusion equation in the case of spherical symmetry”, Comput. Math. Math. Phys., 49:5 (2009), 810–826
4. 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
5. Shishkina L., Shishkin G., “Grid Approximation of a Singularly Perturbed Parabolic Reaction-Diffusion Equation on a Ball”, Numerical Analysis and its Applications - 4th International Conference, NAA 2008, Lecture Notes in Computer Science, 5434, 2009, 501–508
6. G. I. Shishkin, L. P. Shishkina, “A conservative difference scheme for a singularly perturbed elliptic reaction-diffusion equation: approximation of solutions and derivatives”, Comput. Math. Math. Phys., 50:4 (2010), 633–645
7. I. V. Popov, “O monotonnykh raznostnykh skhemakh”, Matem. modelirovanie, 31:8 (2019), 21–43
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