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Applied Mathematics and Mechanics

, Volume 24, Issue 3, pp 326–337 | Cite as

IMD based nonlinear Galerkin method

  • Hou Yan-ren
  • Li Kai-tai
Article
  • 28 Downloads

Abstract

By taking example of the 2D Navier-Stokes equations, a kind of improved version of the nonlinear Galerkin method of Marion-Temam type based on the new concept of the inertial manifold with delay (IMD) is presented, which is focused on overcoming the defect that the feasibility of the M-T type nonlinear Galerkin method heavily depended on the least solving scale. It is shown that the improved version can greatly reduce the feasible conditions as well as preserve the superiority of the former version. Therefore, the version obtained here is an applicable, high performance and stable algorithm.

Key words

nonlinear Galerkin method inertial manifold with delay Navier-Stokes equation 

Chinese Library Classification

O241.82 

2000 MR Subject Classification

65M12 65M70 76D05 

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References

  1. [1]
    Foias C, Sell G R, Temam R. Inertial manifolds for nonlinear evolutionary equations[J].J Diff Eqs, 1998,73(2):309–353.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Foias C, Manley O, Temam R. On the interaction of small eddies in two-dimensional turbulence flows[J].Math Modeling and Numerical Analysis, 1998,22:93–114.MathSciNetGoogle Scholar
  3. [3]
    Marion M, Temam R. Nonlinear Galerkin methods[J].SIAM J Numer Anal, 1989,21(5): 1139–1157.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Marion M, Temam R. Nonlinear Galerkin methods: the finite elements case[J].Numer Math, 1990,57(3):205–226.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Marion M, XU Jin-chao. Error estimates on a new nonlinear Galerkin method based on two-grid finite elements[J].SIAM J Numer Anal, 1995,32(4):1170–1184.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    SHEN Jie, Temam R. Nonlinear Galerkin method using Chebyshev and Legendre polynomials I: the one-dimensional case[J].SIAM J Numer Anal, 1995,32(1):215–234.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    LI Kai-tai, HOU Yan-ren. Fourier nonlinear Galerkin method for the Navier-Stokes equations[J].Discrete and Continuous Dynamical Systems, 1996,2(4):497–524.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Dubois T, Jauberteau F, Marion M,et al. Subgrid modelling and the interaction of small and large wavelengths in turbulent flows[J].Computer Physics Communications, 1991,65(1–3):100–106.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    Debussche A, Temam R, Inertial manifolds with delay[J].Applied Math Letters, 1995,8(1): 21–24.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    LI Kai-tai, HOU Yan-ren. Inertial manifold with delay and the family of approximate inertial manifold with delay[J].Acta Mathematica Sinica, 2000,43(3):435–444. (in Chinese)MathSciNetGoogle Scholar
  11. [11]
    Temam R,Navier-Stokes Equations[M]. 3rd Edition. Amsterdam: North-Holland, 1984.Google Scholar
  12. [12]
    Canuto C, Hussaini M Y, Quarteroni A,et al.Spectral Methods in Fluid Dynamics[M]. New York-Heidelberg-Berlin: Springer-Verlag, 1987.Google Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 2003

Authors and Affiliations

  • Hou Yan-ren
    • 1
  • Li Kai-tai
    • 1
  1. 1.College of ScienceXi'an Jiaotong UniversityXi'anPR China

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