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

  • Evgenii V. VorozhtsovEmail author
  • Vasily P. Shapeev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)

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.

Keywords

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

References

  1. 1.
    Binzubair, H.: Efficient multigrid methods based on improved coarse grid correction techniques. Thesis. Delft Univ. of Technology, Delft, The Netherlands (2009)Google Scholar
  2. 2.
    Botella, O., Peyret, R.: Benchmark spectral results on the lid-driven cavity flow. Comput. Fluids 27, 421–433 (1998)CrossRefGoogle Scholar
  3. 3.
    Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  4. 4.
    Fedorenko, R.P.: The speed of convergence of one iterative process. USSR Comput. Math. Math. Phys. 4(3), 227–235 (1964)CrossRefGoogle Scholar
  5. 5.
    Gartling, D.K.: A test problem for outflow boundary conditions - flow over a backward-facing step. Int. J. Numer. Methods Fluids 11, 953–967 (1990)CrossRefGoogle Scholar
  6. 6.
    Isaev, V.I., Shapeev, V.P.: High-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations. Comput. Math. Math. Phys. 50, 1670–1681 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Knupp, P., Steinberg, S.: Fundamentals of Grid Generation. CRC Press, Boca Raton (1994)zbMATHGoogle Scholar
  8. 8.
    Krylov, A.N.: On the numerical solution of the equation, which determines in technological questions the frequencies of small oscillations of material systems. Izv. AN SSSR, Otd. matem. i estestv. nauk 4, 491–539 (1931). (in Russian)Google Scholar
  9. 9.
    Kudryavtseva, I.V., Rykov, S.A., Rykov, S.V., Skobov, E.D.: Optimization Methods in the Examples in the MathCAD 15 Package. Part I, NIU ITMO, St. Petersburg (2014). (in Russian)Google Scholar
  10. 10.
    Shapeev, A.V., Lin, P.: An asymptotic fitting finite element method with exponential mesh refinement for accurate computation of corner eddies in viscous flows. SIAM J. Sci. Comput. 31, 1874–1900 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Shapeev, V.: Collocation and least residuals method and its applications. EPJ Web of Conferences 108, 01009.  https://doi.org/10.1051/epjconf/201610801009CrossRefGoogle Scholar
  12. 12.
    Shapeev, V.P., Vorozhtsov, E.V.: CAS application to the construction of the collocations and least residuals method for the solution of 3D Navier–Stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 381–392. Springer, Cham (2013).  https://doi.org/10.1007/978-3-319-02297-0_31CrossRefGoogle Scholar
  13. 13.
    Shapeev, V.P., Vorozhtsov, E.V.: Symbolic-numeric implementation of the method of collocations and least squares for 3D Navier–Stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 321–333. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32973-9_27CrossRefGoogle Scholar
  14. 14.
    Shapeev, V.P., Vorozhtsov, E.V.: Symbolic-numerical optimization and realization of the method of collocations and least residuals for solving the Navier–Stokes equations. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 473–488. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-45641-6_30CrossRefGoogle Scholar
  15. 15.
    Shapeev, V.P., Vorozhtsov, E.V.: The method of collocations and least residuals combining the integral form of collocation equations and the matching differential relations at the solution of PDEs. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 346–361. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-66320-3_25CrossRefGoogle Scholar
  16. 16.
    Sleptsov, A.G.: Collocation-grid solution of elliptic boundary-value problems. Modelirovanie v mekhanike 5(22), 101–126 (1991). (in Russian)Google Scholar
  17. 17.
    Thompson, J.F., Warsi, Z.U.A., Mastin, C.W.: Numerical Grid Generation - Foundations and Applications. Elsevier Science Publishing Co., New York (1985)zbMATHGoogle Scholar
  18. 18.
    Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media, Inc., Champaign (2003)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk National Research UniversityNovosibirskRussia

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