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 Mat. Sb. (N.S.), 1985, Volume 128(170), Number 4(12), Pages 530–544 (Mi msb2174)

Sharp error estimates of some two-level methods of solving the three-dimensional heat equation

A. A. Zlotnik, I. D. Turetaev

Abstract: The initial-boundary value problem $\partial u/\partial t-\Delta u=f$ in $Q=\Omega\times(0,T)$, $u|_{\partial\Omega\times(0,T)}=0$, $u|_{t=0}=u_0$, is solved, where $\Omega$ is a three-dimensional rectangular parallelepiped. Two-level methods of second-order approximation are considered: families of projection and finite-difference schemes with a splitting operator as well as Crank–Nicolson schemes. Error estimates in $L_2(Q)$ of order $O(\tau^{1+\alpha}+h^2)$ for all $0\leqslant\alpha\leqslant1$ are derived. It is shown that the inclusion of values $0<\alpha\leqslant1$ yields sharpened estimates when $f$ is discontinuous. Accuracy of the estimates with respect to order – and in the case of Crank–Nicolson schemes their unimprovability – is proved. It is found that for difference schemes with splitting operator when $0<\alpha\leqslant1$, $f$ must have in $Q$ not only order $\alpha$ smoothness with respect to $t$ (as in the case of Crank–Nicolson schemes) but also order $2\alpha$ smoothness (in a certain weak sense) in the space variables. Only one scheme with splitting operator out of each family constitutes an important exception, a scheme equivalent to one proposed by J. Douglas and its projective analogue, and that only for $0<\alpha\leqslant1/2$. The situation described is qualitatively different from those studied previously in the literature.
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 56:2, 529–544

Bibliographic databases:

UDC: 519.633
MSC: Primary 65M20; Secondary 35K05

Citation: A. A. Zlotnik, I. D. Turetaev, “Sharp error estimates of some two-level methods of solving the three-dimensional heat equation”, Mat. Sb. (N.S.), 128(170):4(12) (1985), 530–544; Math. USSR-Sb., 56:2 (1987), 529–544

Citation in format AMSBIB
\Bibitem{ZloTur85} \by A.~A.~Zlotnik, I.~D.~Turetaev \paper Sharp error estimates of some two-level methods of solving the three-dimensional heat equation \jour Mat. Sb. (N.S.) \yr 1985 \vol 128(170) \issue 4(12) \pages 530--544 \mathnet{http://mi.mathnet.ru/msb2174} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=820401} \zmath{https://zbmath.org/?q=an:0613.65102} \transl \jour Math. USSR-Sb. \yr 1987 \vol 56 \issue 2 \pages 529--544 \crossref{https://doi.org/10.1070/SM1987v056n02ABEH003050} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Zaytseva S., Zlotnik A., “Sharp Error Analysis of Locally One-Dimensional Methods for Heat Equation with Right-Hand Side in l(2)”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1996, no. 6, 40–43
2. S. B. Zaitseva, A. A. Zlotnik, “On some properties of the alternating triangular vector method for the heat equation”, Russian Math. (Iz. VUZ), 43:7 (1999), 1–9
3. S. B. Zaitseva, A. A. Zlotnik, “Sharp error estimates of vector splitting methods for the heat equation”, Comput. Math. Math. Phys., 39:3 (1999), 448–467
4. V. V. Smagin, “Mean-square estimates for the error of a projection-difference method for parabolic equations”, Comput. Math. Math. Phys., 40:6 (2000), 868–879
5. Smagin, VV, “Energy error estimates for the projection-difference method with the Crank-Nicolson scheme for parabolic equations”, Siberian Mathematical Journal, 42:3 (2001), 568
6. A. Zlotnik, A. Romanova, “On a Numerov–Crank–Nicolson–Strang scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip”, Applied Numerical Mathematics, 2014
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