Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

NUMERICAL PROPERTIES OF GAUGE METHOD FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

  • Pyo, Jae-Hong (DEPARTMENT OF MATHEMATICS, KANGWON NATIONAL UNIVERSITY)
  • Received : 2010.03.06
  • Accepted : 2010.03.09
  • Published : 2010.03.25
Download PDF
⟨ Previous Next ⟩

Abstract

The representative numerical algorithms to solve the time dependent Navier-Stokes equations are projection type methods. Lots of projection schemes have been developed to find more accurate solutions. But most of projection methods [4, 11] suffer from inconsistency and requesting unknown datum. E and Liu in [5] constructed the gauge method which splits the velocity $u=a+{\nabla}{\phi}$ to make consistent and to replace requesting of the unknown values to known datum of non-physical variables a and ${\phi}$. The errors are evaluated in [9]. But gauge method is not still obvious to find out suitable combination of discrete finite element spaces and to compute boundary derivative of the gauge variable ${\phi}$. In this paper, we define 4 gauge algorithms via combining both 2 decomposition operators and 2 boundary conditions. And we derive variational derivative on boundary and analyze numerical results of 4 gauge algorithms in various discrete spaces combinations to search right discrete space relation.

Keywords

References

  1. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, (1991).
  2. D.L. Brown, R. Cortez, and M.L. Minion Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. phys., 168 (2001), 464-499. https://doi.org/10.1006/jcph.2001.6715
  3. G.F. Carey, S.S. Chow and M.K. Seager, Approximate boundary-flux calculations Comput. Meth. Appl. Mech. Eng., 50 (1985), 107-120. https://doi.org/10.1016/0045-7825(85)90085-4
  4. A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22, (1968), 745-762. https://doi.org/10.1090/S0025-5718-1968-0242392-2
  5. W. E and J.-G. Liu, Gauge method for viscous incompressible flows, Comm. Math. Sci., 1 (2003), 317-332. https://doi.org/10.4310/CMS.2003.v1.n2.a6
  6. Weinan E and J.-G. Liu, Projection method I: Convergence and numerical boundary layers, SIAM J. Numer. Anal., 32 (1995), 1017-1057. https://doi.org/10.1137/0732047
  7. Vivette Girault, and Pierre-Arnaud Raviart, Finite Element Methods for Navier-stokes Equations, Springer-Verlag, (1986).
  8. R.H. Nochetto and J.-H. Pyo, Optimal relaxation parameter for the Uzawa method Numer. Math., 98 (2004), 695-702. https://doi.org/10.1007/s00211-004-0522-0
  9. R. Nochetto and J.-H. Pyo, Error Estimates for Semi-discrete Gauge Methods for the Navier-Stokes Equations, Math. Comp., 74 (2005), 521-542.
  10. A.I. Pehlivanov, R.D. Lazarov, G.F. Carey, and S.S. Chow, Superconvergence analysis of approximate boundary-flux calculations, Numer. Math., 63 (1992), 483-501. https://doi.org/10.1007/BF01385871
  11. Andreas Prohl Projection and Quasi-Compressiblity Methods for Solving the Incompressible Navier-Stokes Equations, B.G.Teubner Stuttgart, (1997).
  12. R. Rannacher, On Chorin's projection method for the incompressible Navier-Stokes equations, Lecture Notes in Mathematics, 1530 (1992), 167-183. https://doi.org/10.1007/BFb0090341
  13. A. Schmidt and K.G. Siebert, ALBERT: An adaptive hierarchical finite element toolbox , Manual, 244 p., Preprint 06/2000 Freiburg.
  14. J. Shen, On error estimates of projection methods for Navier-Stokes equation: first order schemes, SIAM J. Numer. Anal., 29 (1992), 57-77. https://doi.org/10.1137/0729004
  15. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Siam CBMS 66, (1995).
  16. R. Temam, Sur l'approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires. II. (French) Arch. Rational Mech. Anal., 33 (1969), 377-385.