Constructing the Numerical Method for Navier — Stokes Equations Using Computer Algebra System

  • Leonid Semin
  • Vasily Shapeev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)


The present study demonstrates a very helpful role of computer algebra systems (CAS) for deriving and testing new numerical methods. We use CAS to construct and test a new numerical method for solving boundary – value problems for the 2D Navier — Stokes equations governing steady incompressible viscous flows. We firstly describe the core of the method and the algorithm of its construction, then we describe the implementation in CAS for deriving formulas of the method and for testing them, and finally we give some numerical results and concluding remarks.


Reynolds Number Stokes Equation Linear Algebraic Equation Computer Algebra System Convergence Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Semin, L.G., Sleptsov, A.G., Shapeev, V.P.: Collocation and least - squares method for Stokes equations. Computational technologies 1(2), 90–98 (1996) (in Russian) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Valiullin, A.N., Shapeev, V.P., et al.: Symbolic manipulations in the methods of mathematical physics. In: Symposium: Mathematics for Computer Science, Paris, pp. 431–438 (1982)Google Scholar
  3. 3.
    Steinberg, S.: A problem solving environment for numerical partial differential equations. In: 6th IMACS Conference on Application of Computer Algebra, St. Petersburg, June 25–28, pp. 98–99 (2000)Google Scholar
  4. 4.
    Karasözen, B., Tsybulin, V.G.: Conservative finite difference schemes for cosymmetric systems. Computer Algebra in Scientific Computing, pp. 363–375. Springer, Berlin (2001)Google Scholar
  5. 5.
    Schacht, W., Vorozhtsov, E.V.: Implementation of Roe’s method for numerical solution of three-dimensional fluid flow problems with the aid of computer algebra systems. In: 7th workshop on Computer Algebra in Scientific Computing, St. Petersburg, July 12–19, pp. 409–422 (2004)Google Scholar
  6. 6.
    Ascher, U., Christiansen, J., Russell, R.D.: A collocation solver for mixed order systems of boundary value problems. Mathematics of Computation 33, 659–679 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Barton, I.E.: The entrance effect of laminar flow over a backward - facing step geometry. Int. J. for Numerical Methods in Fluids 25, 633–644 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kim, J., Moin, P.: Application of a fractional - step method to incompressible Navier — Stokes equations. J. Comput. Phys. 59, 308–323 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Armaly, B.F., Durst, F., Pereira, J.C.F., Schönung, B.: Experimental and theoretical investigation of backward - facing step flow. J. Fluid Mech. 127, 473–496 (1983)CrossRefGoogle Scholar
  10. 10.
    Oosterlee, C.W.: A GMRES - based plane smoother in multigrid to solve 3D anisotropic fluid flow problems. J. Comput. Phys. 130, 41–53 (1997)zbMATHCrossRefGoogle Scholar
  11. 11.
    Pan, F., Acrivos, A.: Steady flows in rectangular cavities. J. Fluid Mech. 28(4), 643–655 (1967)CrossRefGoogle Scholar
  12. 12.
    Bozeman, J.D., Dalton, C.: Numerical study of viscous flow in a cavity. J. of Comput. Phys. 12, 348–363 (1973)CrossRefGoogle Scholar
  13. 13.
    Bruneau, C.H., Jouron, C.: An efficient scheme for solving steady incompressible Navier — Stokes equations. J. Comput. Phys. 89, 389–413 (1990)zbMATHCrossRefGoogle Scholar
  14. 14.
    Ghia, U., Ghia, K.N., Shin, C.T.: High – Re solutions for incompressible flow using the Navier — Stokes equations and a multigrid method. J. Comput. Phys. 48, 387–411 (1982)zbMATHCrossRefGoogle Scholar
  15. 15.
    Deng, G.B., Piquet, J., Queutey, P., Visonneau, M.: A new fully coupled solution of the Navier — Stokes equations. Int. J. Numer. Methods in Fluids 19, 605–639 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Chen, C.J., Chen, H.J.: Finite analytic numerical method for unsteady two-dimensional Navier — Stokes equations. J. Comput. Phys. 53, 209–226 (1984)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Leonid Semin
    • 1
  • Vasily Shapeev
    • 1
  1. 1.Institute of Theoretical and Applied Mechanics SB RASNovosibirskRussia

Personalised recommendations