Parallelization Methods for a Characteristic’s Pressure Correction Scheme

  • S. Blazy
  • W. Borchers
  • U. Dralle
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NONUFM, volume 48)


Pressure correction schemes and the method of characteristics are combined to obtain numerical approximation procedures for incompressible Navier-Stokes flows on massive parallel computers. The projection step is carried out with a new more efficient parallel preconditioned conjugate gradient method and for the computation of the characteristics we use a fast local Crank-Nicholson solver for linear finite elements over unstructered grids.


Domain Decomposition Multigrid Method Pressure Correction Artificial Boundary Preconditioned Conjugate Gradient Method 
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]
    Beale, J.T.: The approximation of the Navier-Stokes equations by fractional time steps, Lecture at the conference: The Navier-Stokes equations, theory and numerical methods, Oberwolfach 18.-24.8.91 [IBM Research Report RCI 18072 (79337) 6-1192].Google Scholar
  2. [2]
    Bern, M., Eppstein, D., Gilbert, J.: Provably good mesh generation. Proc. 31st Symp. on Foundations of Computer Science (FOCS), (1990), 231–241Google Scholar
  3. [3]
    Blazy, S., Dralle, U., Simon, J.: Parallel CG Poisson Solver for PowerPC 601, Power Explorer User Report. Application and Projects on Parsytec PowerXplorer Parallel Computers, 1st Edition, May (1995).Google Scholar
  4. [4]
    Borchers, W.: A splitting algorithm for incompressible Navier-Stokes equations, in: H. Niki, M. Kawahara (eds.): Int. Conf. on computational methods in flow analysis, Okayama, Japan (1988), 454–461.Google Scholar
  5. [5]
    Borchers, W.: On the characteristic’s method for the incompressible Navier-Stokes-equations, In: Finite Approximations in Fluid Mechanics II; E. H. Hirschel (Ed.), Notes on Numerical Fluid Mechanics, Vol. 25, Vieweg, Braunschweig 1989, 43–50.Google Scholar
  6. [6]
    Borchers, W.: Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes Gleichungen inkompressibler viskoser Flüssigkeiten. Habilitationsschrift für das Fach Mathematik im Fachbereich Mathematik-Informatik der Universität-GH Paderborn (1992).Google Scholar
  7. [7]
    Borchers, W.: A new conjugate gradient boundary iteration method for parallel numerical computation of elliptic boundary value problems. In preparation.Google Scholar
  8. [8]
    Bramble, J. H., Pasciac, J.E., Schatz, A.H.: The construction of preconditioners for elliptic problems by substructering I, Math. Comp. 47, (1986).Google Scholar
  9. [9]
    Bramble, J. H., Pasciac, J.E., Schatz, A.H.: The construction of preconditioners for elliptic problems by substructering II, Math. Comp. 49, (1987).Google Scholar
  10. [10]
    Chorin, A. J., Hughes, T.J.R., Mc Cracken, M.F., Marsden, J.E.: Product formulas and numerical algorithms, Comm. Pure Appl. Math. XXXI (1978) 205–256.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Chorin, A. J.: Numerical solution of the Navier-Stokes equation. Math. Comp. 22, (1968), 745–762.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Chorin, A. J.: On the convergence of the discrete approximations of the Navier-Stokes equations. Math. Comp. 23, (1969), 341–353.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Diekmann, R., Meyer, D., Monien, B.: Parallel Decomposition of Unstructured FEM-Meshes. Proc. of IRREGULAR’ 95, Springer LNCS 980, (1995), 199–215.Google Scholar
  14. [14]
    Diekmann, R., Dralle, U., Neugebauer, F., Römke, T.: PadFem: A portable parallel FEM-Tool. Draft, Univ. of Paderborn, (1995) (submitted).Google Scholar
  15. [15]
    Douglas, J., Russel, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with Finite Element or Finite Difference procedures. Siam J. Numer. Anal. Vol.19, No. 5, October (1982)Google Scholar
  16. [16]
    Güntsch, E. Friedrich, R.: DNS of Turbulence Compressed in a Cylinder. In this publication.Google Scholar
  17. [17]
    Hackbusch, W., Trottenberg, M. (eds.): Multigrid methods. Lecture Notes in Mathematics. Vol. 960. Springer Verlag, Berlin-Heidelberg-New York (1982).Google Scholar
  18. [18]
    Hebeker, F.-K.: Analysis of a characteristics method for some incompressible and compressible Navier-Stokes problems. Preprint 1126 des Fb. Mathematik, TH Darmstadt (1988).Google Scholar
  19. [19]
    Langer, U., Haase, G.: The approximate Dirichlet domain decomposition method. II. Application to 2nd-order elliptic BVPs. Comp. Arch. f. Inf. a. Numer. Comp. 47, (1991), 153–167.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Lions, J. L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I, Springer Verlag, Berlin-Heidelberg-New York (1982).Google Scholar
  21. [21]
    K. Masuda, Rautmann, R.: H 2-convergent approximation schemes to the Navier-Stokes equations. Commentarii Mathematici Universitatis Sancti Pauli 43, (1994), 55–108.MathSciNetzbMATHGoogle Scholar
  22. [22]
    Pironneau, O.: On the transport diffusion algorithm and its applications to the Navier-Stokes equations, Num. Math. 38, (1982), 309–332.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Rautmann, R.: Ein Näherungsverfahren für spezielle parabolische Anfangsrandwertaufgaben mit Operatoren. Springer Lecture Notes in Math. 267, (1972), 187–231 (Habilitationsschrift).Google Scholar
  24. [24]
    Rautmann, R.: Navier-Stokes approximations in high order norms. In this publication.Google Scholar
  25. [25]
    Rannacher, R.: On Chorin’s projection method for the incompressible Navier-Stokes equations, Lecture Notes in Math. Vol. 1530, (Heywood, J.G., Masuda, K., Raut-mann, R., Solonnikov, S.A. Eds.), The Navier-Stokes Equations II-Theory and Numerical Methods, Proceedings, Oberwolfach (1991), Springer-Verlag.Google Scholar
  26. [26]
    Shen, J.: On error estimates of projection methods for the Navier-Stokes equations: First order schemes, SIAM J. Numer. Anal. 29 (1992), no 1, 57–77.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    Süli, E., Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math. 53 (1988) 459–483.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    Ternani, R.: Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionaires (I), Arch. Rational Mech. Anal., 32 (No 2), 1969, 135–153.MathSciNetGoogle Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1996

Authors and Affiliations

  • S. Blazy
    • 1
  • W. Borchers
    • 1
  • U. Dralle
    • 1
  1. 1.Fachbereich 17Universität PaderbornPaderbornDeutschland

Personalised recommendations