A Divergence-Free Method for Solving the Incompressible Navier–Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation

Evgenii V. Vorozhtsov; Vasily P. Shapeev

doi:10.1007/978-3-030-26831-2_28

A Divergence-Free Method for Solving the Incompressible Navier–Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation

Evgenii V. Vorozhtsov, Vasily P. Shapeev

Mathematical Modeling Section

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review

Abstract

To increase the accuracy of computations by the method of collocations and least squares (CLS) a generalization of this method is proposed for the case of a non-uniform logically rectangular grid. The main work formulas of the CLS method on non-uniform grid, including the formulas implementing the prolongation operator on a non-uniform grid at the use of a multigrid complex are obtained with the aid of the computer algebra system (CAS) Mathematica. The proposed method has been applied for the numerical solution of two-dimensional stationary Navier–Stokes equations governing the laminar flows of viscous incompressible fluids. On a smooth test solution, the application of a non-uniform grid has enabled a 47-fold reduction of the solution error in comparison with the uniform grid case. At the solution of the problem involving singularities – the lid-driven cavity flow – the error of the solution obtained by the CLS method was reduced by the factors from 2.65 to 3.05 depending on the Reynolds number value.

Original language	English
Title of host publication	Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings
Editors	Matthew England, Timur M. Sadykov, Werner M. Seiler, Wolfram Koepf, Evgenii V. Vorozhtsov
Publisher	Springer-Verlag GmbH and Co. KG
Pages	430-450
Number of pages	21
ISBN (Print)	9783030268305
DOIs	https://doi.org/10.1007/978-3-030-26831-2_28
Publication status	Published - 1 Jan 2019
Event	21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019 - Moscow, Russian Federation Duration: 26 Aug 2019 → 30 Aug 2019

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	11661 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019
Country/Territory	Russian Federation
City	Moscow
Period	26.08.2019 → 30.08.2019

Keywords

Krylov subspaces
Logically rectangular grids
Method of collocations and least squares
Multigrid
Navier–Stokes equations
Non-uniform grids
Preconditioners
Navier-Stokes equations

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10.1007/978-3-030-26831-2_28

Cite this

Vorozhtsov, E. V., & Shapeev, V. P. (2019). A Divergence-Free Method for Solving the Incompressible Navier–Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation. In M. England, T. M. Sadykov, W. M. Seiler, W. Koepf, & E. V. Vorozhtsov (Eds.), Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings (pp. 430-450). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11661 LNCS). Springer-Verlag GmbH and Co. KG. https://doi.org/10.1007/978-3-030-26831-2_28

Vorozhtsov, Evgenii V. ; Shapeev, Vasily P. / A Divergence-Free Method for Solving the Incompressible Navier–Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation. Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings. editor / Matthew England ; Timur M. Sadykov ; Werner M. Seiler ; Wolfram Koepf ; Evgenii V. Vorozhtsov. Springer-Verlag GmbH and Co. KG, 2019. pp. 430-450 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "To increase the accuracy of computations by the method of collocations and least squares (CLS) a generalization of this method is proposed for the case of a non-uniform logically rectangular grid. The main work formulas of the CLS method on non-uniform grid, including the formulas implementing the prolongation operator on a non-uniform grid at the use of a multigrid complex are obtained with the aid of the computer algebra system (CAS) Mathematica. The proposed method has been applied for the numerical solution of two-dimensional stationary Navier–Stokes equations governing the laminar flows of viscous incompressible fluids. On a smooth test solution, the application of a non-uniform grid has enabled a 47-fold reduction of the solution error in comparison with the uniform grid case. At the solution of the problem involving singularities – the lid-driven cavity flow – the error of the solution obtained by the CLS method was reduced by the factors from 2.65 to 3.05 depending on the Reynolds number value.",

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author = "Vorozhtsov, {Evgenii V.} and Shapeev, {Vasily P.}",

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Vorozhtsov, EV & Shapeev, VP 2019, A Divergence-Free Method for Solving the Incompressible Navier–Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation. in M England, TM Sadykov, WM Seiler, W Koepf & EV Vorozhtsov (eds), Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11661 LNCS, Springer-Verlag GmbH and Co. KG, pp. 430-450, 21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019, Moscow, Russian Federation, 26.08.2019. https://doi.org/10.1007/978-3-030-26831-2_28

A Divergence-Free Method for Solving the Incompressible Navier–Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation. / Vorozhtsov, Evgenii V.; Shapeev, Vasily P.

Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings. ed. / Matthew England; Timur M. Sadykov; Werner M. Seiler; Wolfram Koepf; Evgenii V. Vorozhtsov. Springer-Verlag GmbH and Co. KG, 2019. p. 430-450 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11661 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review

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N2 - To increase the accuracy of computations by the method of collocations and least squares (CLS) a generalization of this method is proposed for the case of a non-uniform logically rectangular grid. The main work formulas of the CLS method on non-uniform grid, including the formulas implementing the prolongation operator on a non-uniform grid at the use of a multigrid complex are obtained with the aid of the computer algebra system (CAS) Mathematica. The proposed method has been applied for the numerical solution of two-dimensional stationary Navier–Stokes equations governing the laminar flows of viscous incompressible fluids. On a smooth test solution, the application of a non-uniform grid has enabled a 47-fold reduction of the solution error in comparison with the uniform grid case. At the solution of the problem involving singularities – the lid-driven cavity flow – the error of the solution obtained by the CLS method was reduced by the factors from 2.65 to 3.05 depending on the Reynolds number value.

AB - To increase the accuracy of computations by the method of collocations and least squares (CLS) a generalization of this method is proposed for the case of a non-uniform logically rectangular grid. The main work formulas of the CLS method on non-uniform grid, including the formulas implementing the prolongation operator on a non-uniform grid at the use of a multigrid complex are obtained with the aid of the computer algebra system (CAS) Mathematica. The proposed method has been applied for the numerical solution of two-dimensional stationary Navier–Stokes equations governing the laminar flows of viscous incompressible fluids. On a smooth test solution, the application of a non-uniform grid has enabled a 47-fold reduction of the solution error in comparison with the uniform grid case. At the solution of the problem involving singularities – the lid-driven cavity flow – the error of the solution obtained by the CLS method was reduced by the factors from 2.65 to 3.05 depending on the Reynolds number value.

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KW - Non-uniform grids

KW - Preconditioners

KW - Navier-Stokes equations

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Vorozhtsov EV, Shapeev VP. A Divergence-Free Method for Solving the Incompressible Navier–Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation. In England M, Sadykov TM, Seiler WM, Koepf W, Vorozhtsov EV, editors, Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings. Springer-Verlag GmbH and Co. KG. 2019. p. 430-450. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-030-26831-2_28

A Divergence-Free Method for Solving the Incompressible Navier–Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation

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