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 Num. Meth. Prog., 2013, Volume 14, Issue 3, Pages 306–322 (Mi vmp117)

This article is cited in 5 scientific papers (total in 5 papers)

Вычислительные методы и приложения

The method of collocations and least residuals for three-dimensional Navier-Stokes equations

V. P. Shapeeva, E. V. Vorozhtsova, V. I. Isaevb, S. V. Idimeshevc

a Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University
c Institute of Computing Technologies, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: The method of collocations and least residuals, which was previously proposed for the numerical solution of two-dimensional Navier-Stokes equations, is extended for the three-dimensional case. In the implemented version of the method, the solution is sought in the form of an expansion in basis solenoidal functions. An overdetermined system of linear algebraic equations is obtained in each cell of the computational grid. This system is solved by the method of rotations. To accelerate the iteration process convergence, a new algorithm is proposed on the basis of Krylov's subspaces. The results of the method's verification confirm its second order of convergence for the velocity vector components. The results of solving the benchmark problem of a lid-driven cubic cavity flow for the Reynolds numbers $Re = 100$ and $Re = 1000$ are discussed. It is shown that the obtained results are very close to the most accurate results obtained by other authors with the aid of different numerical high-accuracy methods.

Keywords: three-dimensional Navier-Stokes equations; method of collocations and least residuals; cubic cavity flow; overdetermined linear system; Krylov's subspaces.

Full text: PDF file (664 kB)
UDC: 519.63.4:532.51.5
Received: 12.05.2013

Citation: V. P. Shapeev, E. V. Vorozhtsov, V. I. Isaev, S. V. Idimeshev, “The method of collocations and least residuals for three-dimensional Navier-Stokes equations”, Num. Meth. Prog., 14:3 (2013), 306–322

Citation in format AMSBIB
\Bibitem{ShaVorIsa13} \by V.~P.~Shapeev, E.~V.~Vorozhtsov, V.~I.~Isaev, S.~V.~Idimeshev \paper The method of collocations and least residuals for three-dimensional Navier-Stokes equations \jour Num. Meth. Prog. \yr 2013 \vol 14 \issue 3 \pages 306--322 \mathnet{http://mi.mathnet.ru/vmp117} 

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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. V. A. Balashov, E. B. Savenkov, “Numerical study of a quasi-hydrodynamic system of equations for flow computation at small mach numbers”, Comput. Math. Math. Phys., 55:10 (2015), 1743–1751
2. E. V. Vorozhtsov, V. P. Shapeev, “Chislennoe reshenie uravneniya Puassona v polyarnykh koordinatakh metodom kollokatsii i naimenshikh nevyazok”, Model. i analiz inform. sistem, 22:5 (2015), 648–664
3. V. P. Shapeev, E. V. Vorozhtsov, “O kombinirovanii razlichnykh metodov uskoreniya pri iteratsionnom reshenii uravnenii s chastnymi proizvodnymi metodom kollokatsii i naimenshikh nevyazok”, Model. i analiz inform. sistem, 24:1 (2017), 39–63
4. V. A. Belyaev, V. P. Shapeev, “Varianty metoda kollokatsii i naimenshikh nevyazok dlya resheniya zadach matematicheskoi fiziki v vypuklykh chetyrekhugolnykh oblastyakh”, Model. i analiz inform. sistem, 24:5 (2017), 629–648
5. E. V. Vorozhtsov, V. P. Shapeev, “Bezdivergentnyi metod kollokatsii i naimenshikh kvadratov dlya rascheta techenii neszhimaemoi zhidkosti i ego effektivnaya realizatsiya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 24:3 (2020), 542–573
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