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Reduced finite difference scheme and error estimates based on POD method for non-stationary Stokes equation

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Abstract

The proper orthogonal decomposition (POD) is a model reduction technique for the simulation of physical processes governed by partial differential equations (e.g., fluid flows). It has been successfully used in the reduced-order modeling of complex systems. In this paper, the applications of the POD method are extended, i.e., the POD method is applied to a classical finite difference (FD) scheme for the non-stationary Stokes equation with a real practical applied background. A reduced FD scheme is established with lower dimensions and sufficiently high accuracy, and the error estimates are provided between the reduced and the classical FD solutions. Some numerical examples illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced FD scheme based on the POD method is feasible and efficient in solving the FD scheme for the non-stationary Stokes equation.

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References

  1. Brezzi, F. and Douglas, Jr. J. Stabilized mixed method for the Stokes problem. Numer. Math., 53(1–2), 225–235 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  2. Douglas, Jr. J. and Wang, J. P. An absolutely stabilized finite element method for the Stokes problem. Math. Comp., 52, 495–508 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chung, T. Computational Fluid Dynamics, Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  4. Holmes, P., Lumley, J. L., and Berkooz, G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK (1996)

    Book  MATH  Google Scholar 

  5. Fukunaga, K. Introduction to Statistical Pattern Recognition, Academic Press, Boston (1990)

    MATH  Google Scholar 

  6. Jolliffe, I. T. Principal Component Analysis, Springer-Verlag, Berlin (2002)

    MATH  Google Scholar 

  7. Crommelin, D. T. and Majda, A. J. Strategies for model reduction: comparing different optimal bases. J. Atmos. Sci., 61, 2306–2317 (2004)

    MathSciNet  Google Scholar 

  8. Majda, A. J., Timofeyev, I., and Vanden-Eijnden, E. Systematic strategies for stochastic mode reduction in climate. J. Atmos. Sci., 60, 1705–1723 (2003)

    Article  MathSciNet  Google Scholar 

  9. Selten, F. Baroclinic empirical orthogonal functions as basis functions in an atmospheric model. J. Atmos. Sci., 54, 2100–2114 (1997)

    Google Scholar 

  10. Lumley, J. L. Coherent Structures in Turbulence, In Transition and Turbulence (ed. Meyer, R. E.), Academic Press, New York, 215–242 (1981)

    Google Scholar 

  11. Aubry, Y. N., Holmes, P., Lumley, J. L., and Stone, E. The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech., 192, 115–173 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Sirovich, L. Turbulence and the dynamics of coherent structures: part I–III. Quart. Appl. Math., 45(3), 561–590 (1987)

    MATH  MathSciNet  Google Scholar 

  13. Joslin, R. D., Gunzburger, M. D., Nicolaides, R., Erlebacher, G., and Hussaini, M. Y. A self-contained automated methodology for optimal flow control validated for transition delay. AIAA Journal, 35, 816–824 (1997)

    Article  MATH  Google Scholar 

  14. Ly, H. V. and Tran, H. T. Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Quart. Appl. Math., 60, 631–656 (2002)

    MATH  MathSciNet  Google Scholar 

  15. Moin, P. and Moser, R. D. Characteristic-eddy decomposition of turbulence in channel. J. Fluid Mech., 200, 471–509 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Rajaee, M., Karlsson, S. K. F., and Sirovich, L. Low dimensional description of free shear flow coherent structures and their dynamical behavior. J. Fluid Mech., 258, 1–29 (1994)

    Article  MATH  Google Scholar 

  17. Kunisch, K. and Volkwein, S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math., 90, 117–148 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kunisch, K. and Volkwein, S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal., 40, 492–515 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kunisch, K. and Volkwein, S. Control of Burgers’ equation by a reduced order approach using proper orthogonal decomposition. J. Optim. Theory Appl., 102, 345–371 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ahlman, D., Södelund, F., Jackson, J., Kurdila, A., and Shyy, W. Proper orthogonal decomposition for time-dependent lid-driven cavity flows. Numer. Heat Transfer Part B, 42(4), 285–306 (2002)

    Article  Google Scholar 

  21. Luo, Z. D., Wang, R. W., Zhu, J. Finite difference scheme based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations. Sci. China Ser. A: Math., 50(8), 1186–1196 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Luo, Z. D., Chen, J., Zhu, J., Wang, R. W., and Navon, I. M. An optimizing reduced order FDS for the tropical pacific ocean reduced gravity model. Int. J. Numer. Methods Fluids, 55, 143–161 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Luo, Z. D., Chen, J., Navon, I. M., and Yang, X. Z. Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations. SIAM J. Numer. Anal., 47(1), 1–19 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Luo, Z. D., Chen, J., Sun, P., and Yang, X. Z. Finite element formulation based on proper orthogonal decomposition for parabolic equations. Sci. China Ser. A: Math., 52(3), 585–596 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sun, P., Luo, Z. D., and Zhou, Y. J. Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations. Appl. Numer. Math., 60, 154–164 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Volkwein, S. Optimal control of a phase-field model using the proper orthogonal decomposition. ZFA Math. Mech., 81, 83–97 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Antoulas, A. Approximation of large-scale dynamical systems. Soc. Indust. Appl. Math., 10, 12–60 (2005)

    Google Scholar 

  28. Stewart, G. W. Introduction to Matrix Computations, Academic Press, New York (1973)

    MATH  Google Scholar 

  29. Noble, B. Applied Linear Algebra, Prentice-Hall, New Jersey (1969)

    MATH  Google Scholar 

  30. Girault, V. and Raviart, P. A. Finite Element Approximations of the Navier-Stokes Equations, Theorem and Algorithms, Springer-Verlag, New York (1986)

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, P. R. China

    Zhen-dong Luo  (罗振东) & Qiu-lan Ou  (欧秋兰)

  2. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, 100029, P. R. China

    Zheng-hui Xie  (谢正辉)

Authors
  1. Zhen-dong Luo  (罗振东)
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  2. Qiu-lan Ou  (欧秋兰)
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  3. Zheng-hui Xie  (谢正辉)
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Corresponding author

Correspondence to Zhen-dong Luo  (罗振东).

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 10871022, 11061009, and 40821092), the National Basic Research Program of China (973 Program) (Nos. 2010CB428403, 2009CB421407, and 2010CB951001), and the Natural Science Foundation of Hebei Province of China (No.A2010001663)

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Luo, Zd., Ou, Ql. & Xie, Zh. Reduced finite difference scheme and error estimates based on POD method for non-stationary Stokes equation. Appl. Math. Mech.-Engl. Ed. 32, 847–858 (2011). https://doi.org/10.1007/s10483-011-1464-9

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  • DOI: https://doi.org/10.1007/s10483-011-1464-9

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