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

This article is cited in 7 scientific papers (total in 7 papers)

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
Received: 19.03.2007

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  mathnet  crossref  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  elib  scopus
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  elib  scopus
    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  crossref  zmath  isi
    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  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  adsnasa  isi
    7. I. V. Popov, “O monotonnykh raznostnykh skhemakh”, Matem. modelirovanie, 31:8 (2019), 21–43  mathnet  crossref  elib
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