Abstract:
We propose and implement new, more general versions of the method of collocations and least squares (the CLS method) and, for a system of linear algebraic equations, an orthogonal method for accelerating the convergence of an iterative solution. The use of the latter method and the proper choice of values of control parameters, based on the results of investigating the dependence of the properties of the CLS method on these parameters, as well as some other improvements of the CLS method suggested in this paper, allow one to solve numerically problems for Navier–Stokes equations in a reasonable time using a single-processor computer even for grids as large as 1280×1280. In this case, the total number of unknown variables is ∼25⋅106. The numerical results for the problem of the lid-driven cavity flow of a viscous fluid are in good agreement with known results of other authors, including those obtained by means of schemes of higher approximation order with a small artificial viscosity. This and some other facts prove that the new versions of the CLS method make it possible to obtain an approximate solution with high accuracy.
Citation:
V. I. Isaev, V. P. Shapeev, “Development of the collocations and least squares method”, Trudy Inst. Mat. i Mekh. UrO RAN, 14, no. 1, 2008, 41–60; Proc. Steklov Inst. Math. (Suppl.), 261, suppl. 1 (2008), S87–S106
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\by V.~I.~Isaev, V.~P.~Shapeev
\paper Development of the collocations and least squares method
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2008
\vol 14
\issue 1
\pages 41--60
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\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2008
\vol 261
\issue , suppl. 1
\pages S87--S106
\crossref{https://doi.org/10.1134/S0081543808050088}
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Linking options:
https://www.mathnet.ru/eng/timm6
https://www.mathnet.ru/eng/timm/v14/i1/p41
This publication is cited in the following 11 articles:
V. A. Belyaev, V. P. Shapeev, AIP Conference Proceedings, 2027, 2018, 030094
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
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
Belyaev V.A., Shapeev V.P., “Versions of the Collocation and Least Squares Method For Solving Biharmonic Equations in Non-Canonical Domains”, Proceedings of the XXV Conference on High-Energy Processes in Condensed Matter (HEPCM 2017), AIP Conference Proceedings, 1893, ed. Fomin V., Amer Inst Physics, 2017, UNSP 030102
A. V. Chernov, “O kusochno postoyannoi approksimatsii v raspredelennykh zadachakh optimizatsii”, Tr. IMM UrO RAN, 21, no. 1, 2015, 264–279
O. R. Michuta, A. P. Vlasyuk, P. N. Martynyuk, “Modelirovanie vliyaniya khimicheskoi suffozii na filtratsionnuyu konsolidatsiyu zasolennykh gruntov v neizotermicheskikh usloviyakh”, Matem. modelirovanie, 25:2 (2013), 3–18
Isaev V.I., Shapeev V.P., “High-order accurate collocations and least squares method for solving the Navier–Stokes equations”, Dokl. Math., 85:1 (2012), 71–74
M. A. Komarov, “Structure of the local controllability set for a family of $2$-systems on a plane near the zero indicatrix point”, Journal of Mathematical Sciences, 199:6 (2014), 667–686
V. I. Isaev, V. P. Shapeev, “High-accuracy versions of the collocations and least squares method for the numerical solution of the Navier–Stokes equations”, Comput. Math. Math. Phys., 50:10 (2010), 1670–1681
Isaev V.I., Shapeev V.P., Cherepanov A.N., “Numerical simulation of laser welding of thin metallic plates taking into account convection in the welding pool”, Thermophysics and Aeromechanics, 17:3 (2010), 419–434
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