Advertisement

Journal of Scientific Computing

, Volume 38, Issue 1, pp 1–14 | Cite as

A Single-Step Characteristic-Curve Finite Element Scheme of Second Order in Time for the Incompressible Navier-Stokes Equations

  • Hirofumi Notsu
  • Masahisa Tabata
Article

Abstract

In this paper we present a new single-step characteristic-curve finite element scheme of second order in time for the nonstationary incompressible Navier-Stokes equations. After supplying correction terms in the variational formulation, we prove that the scheme is of second order in time. The convergence rate of the scheme is numerically recognized by computational results.

Keywords

Characteristic-curve Second order in time The Navier-Stokes equations Finite Element Method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baba, K., Tabata, M.: On a conservative upwind finite element scheme for convective diffusion equations. RAIRO Anal. 15, 3–25 (1981) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia (1994) Google Scholar
  3. 3.
    Bristeau, M.O., Glowinski, R., Mantel, B., Periaux, J., Perrier, P., Pironneau, O.: A finite element approximation of Navier-Stokes equations for incompressible viscous fluids. Iterative methods of solution. In: Rautmann, R. (ed.) Approximation Methods for Navier-Stokes Problems. Lecture Notes in Mathematics, vol. 771, pp. 78–128. Springer, Berlin (1980) CrossRefGoogle Scholar
  4. 4.
    Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Boukir, K., Maday, Y., Metivet, B., Razafindrakoto, E.: A high-order characteristics/finite element method for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 25, 1421–1454 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991) zbMATHGoogle Scholar
  7. 7.
    Douglas Jr., J., Russell, 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. 19, 871–885 (1982) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Franca, L.P., Frey, S.L.: Stabilized finite element methods: II. The incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 99, 209–233 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
  10. 10.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer, Berlin (1986) zbMATHGoogle Scholar
  11. 11.
    Hughes, T.J.R., Franca, L.P., Hulbert, G.M.: A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Eng. 73, 173–189 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hughes, T.J.R., Tezduyar, T.E.: Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Comput. Methods Appl. Mech. Eng. 45, 217–284 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hansbo, P., Johnson, C.: Adaptive streamline diffusion methods for compressible flow using conservation variables. Comput. Methods Appl. Mech. Eng. 87, 267–280 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge Univ. Press, Cambridge (1987) zbMATHGoogle Scholar
  15. 15.
    Le Beau, G.J., Ray, S.E., Aliabadi, S.K., Tezduyar, T.E.: SUPG finite element computation of compressible flows with the entropy and conservation variables formulations. Comput. Methods Appl. Mech. Eng. 104, 397–422 (1993) zbMATHCrossRefGoogle Scholar
  16. 16.
    Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38, 309–332 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Pironneau, O.: Finite Element Methods for Fluids. Wiley, New York (1989) Google Scholar
  18. 18.
    Pironneau, O., Liou, J., Tezduyar, T.: Characteristic-Galerkin and Galerkin/least-squares space-time formulations for the advection-diffusion equation with time-dependent domains. Comput. Methods Appl. Mech. Eng. 100, 117–141 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rui, H., Tabata, M.: A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math. 92, 161–177 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs (1971) zbMATHGoogle Scholar
  21. 21.
    Süli, E.: Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math. 53, 459–483 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Tabata, M.: A finite element approximation corresponding to the upwind finite differencing. Mem. Numer. Math. 4, 47–63 (1977) zbMATHMathSciNetGoogle Scholar
  23. 23.
    Tabata, M.: Discrepancy between theory and real computation on the stability of some finite element schemes. J. Comput. Appl. Math. 199, 424–431 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Tabata, M., Fujima, S.: Finite-element analysis of high Reynolds number flows past a circular cylinder. J. Comput. Appl. Math. 38, 411–424 (1991) zbMATHCrossRefGoogle Scholar
  25. 25.
    Tabata, M., Fujima, S.: Robustness of a characteristic finite element scheme of second order in time increment. In Groth, C., Zingg, D.W. (eds.) Computational Fluid Dynamics 2004, pp. 177–182. Springer, Berlin (2006) CrossRefGoogle Scholar
  26. 26.
    Tabata, M., Tagami, D.: Error estimates for finite element approximations of drag and lift in nonstationary Navier-Stokes flows. Jpn J. Ind. Appl. Math. 17, 371–389 (2000) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tezduyar, T.: Stabilized finite element formulations for incompressible flow computations. Adv. Appl. Mech. 28, 1–44 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Thomasset, F.: Implementation of Finite Element Methods for Navier-Stokes Equations. Springer, New York (1981) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan

Personalised recommendations