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Zh. Vychisl. Mat. Mat. Fiz., 2014, Volume 54, Number 7, Pages 1136–1148 (Mi zvmmf10063)  

Construction and study of high-order accurate schemes for solving the one-dimensional heat equation

S. Yu. Komarova, V. P. Shapeevb

a Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia
b Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Institutskaya ul. 4/1, Novosibirsk, 630090, Russia

Abstract: The method of undetermined coefficients on multipoint stencils with two time levels was used to construct compact difference schemes of $O(\tau^3,h^6)$ accuracy intended for solving boundary value problems for the one-dimensional heat equation. The schemes were examined for von Neumann stability, and numerical experiments were conducted on a sequence of grids with mesh sizes tending to zero. One of the schemes was proved to be absolutely stable. It was shown that, for smooth solutions, the high order of convergence of the numerical solution agrees with the order of accuracy; moreover, solutions accurate up to $\sim10^{-12}$ are obtained on grids with spatial mesh sizes of $\sim10^{-2}$. The formulas for the schemes are rather simple and easy to implement on a computer.

Key words: numerical methods, difference schemes, method of undetermined coefficients, higher order of accuracy, von Neumann stability, heat equation.

DOI: https://doi.org/10.7868/S0044466914070096

Full text: PDF file (339 kB)
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English version:
Computational Mathematics and Mathematical Physics, 2014, 54:7, 1110–1121

Bibliographic databases:

UDC: 519.633
MSC: 80A20,65M06
Received: 30.08.2013
Revised: 05.02.2014

Citation: S. Yu. Komarov, V. P. Shapeev, “Construction and study of high-order accurate schemes for solving the one-dimensional heat equation”, Zh. Vychisl. Mat. Mat. Fiz., 54:7 (2014), 1136–1148; Comput. Math. Math. Phys., 54:7 (2014), 1110–1121

Citation in format AMSBIB
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  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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