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Sib. Zh. Ind. Mat., 2014, Volume 17, Number 3, Pages 48–58 (Mi sjim846)  

Study of iterative algorithm convergence for numerical solving of problems of thermal convection in variables “stream function-vorticity”

N. T. Danaeva, F. S. Amenovab

a Al-Farabi Kazakh National University, 71 Al-Farabi av., 050040 Almaty, Kazakhstan
b Serikbayev East Kazakhstan State Technical University, 69 Protazanov st., 070004 Ust-Kamenogorsk, Kazakhstan

Abstract: The operator-difference equations, approximating the differential problems of thermal convection for an incompressible fluid in variables “stream function-vorticity” are examined; the issues of iterative schemes convergence for the implementation of their decisions are considered. Research is conducted by the priori estimates method. Limit values for vorticity are selected as Thom formulas. Evaluation of boundedness and the condition of uniqueness for solution of a problem are provided. By introducing auxiliary function of vorticity, given grid equations reduced to relationship with homogeneous boundary conditions. Implicit iterative algorithms for numerical implementation of solving of grid equations for which, when executing conditions equivalent to the uniqueness condition, estimations of convergence speed are received. The behavior of iterations in the case of a Stokes linear problem is analyzed. To illustrate the possibilities of considered iterative algorithms, the problem in a closed area with heated side is reviewed. Calculations on the iterative algorithm of variable directions are performed. The results of numerical calculations are analyzed.

Keywords: problem of thermal convection in the variables “stream function-vorticity”, difference problem, stability, convergence, priori estimates, iteration algorithm.

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English version:
Journal of Applied and Industrial Mathematics, 2014, 8:4, 500–509

Bibliographic databases:

UDC: 519.6
Received: 02.04.2014

Citation: N. T. Danaev, F. S. Amenova, “Study of iterative algorithm convergence for numerical solving of problems of thermal convection in variables “stream function-vorticity””, Sib. Zh. Ind. Mat., 17:3 (2014), 48–58; J. Appl. Industr. Math., 8:4 (2014), 500–509

Citation in format AMSBIB
\Bibitem{DanAme14}
\by N.~T.~Danaev, F.~S.~Amenova
\paper Study of iterative algorithm convergence for numerical solving of problems of thermal convection in variables ``stream function-vorticity''
\jour Sib. Zh. Ind. Mat.
\yr 2014
\vol 17
\issue 3
\pages 48--58
\mathnet{http://mi.mathnet.ru/sjim846}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3364405}
\transl
\jour J. Appl. Industr. Math.
\yr 2014
\vol 8
\issue 4
\pages 500--509
\crossref{https://doi.org/10.1134/S1990478914040061}


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