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非定常Stokes方程一种基于POD方法的简化有限差分格式

罗振东 欧秋兰 谢正辉

罗振东, 欧秋兰, 谢正辉. 非定常Stokes方程一种基于POD方法的简化有限差分格式[J]. 应用数学和力学, 2011, 32(7): 795-806. doi: 10.3879/j.issn.1000-0887.2011.07.004
引用本文: 罗振东, 欧秋兰, 谢正辉. 非定常Stokes方程一种基于POD方法的简化有限差分格式[J]. 应用数学和力学, 2011, 32(7): 795-806. doi: 10.3879/j.issn.1000-0887.2011.07.004
LUO Zhen-dong, OU Qiu-lan, XIE Zheng-hui. A Reduced Finite Difference Scheme and Error Estimates Based on POD Method for the Non-Stationary Stokes Equation[J]. Applied Mathematics and Mechanics, 2011, 32(7): 795-806. doi: 10.3879/j.issn.1000-0887.2011.07.004
Citation: LUO Zhen-dong, OU Qiu-lan, XIE Zheng-hui. A Reduced Finite Difference Scheme and Error Estimates Based on POD Method for the Non-Stationary Stokes Equation[J]. Applied Mathematics and Mechanics, 2011, 32(7): 795-806. doi: 10.3879/j.issn.1000-0887.2011.07.004

非定常Stokes方程一种基于POD方法的简化有限差分格式

doi: 10.3879/j.issn.1000-0887.2011.07.004
基金项目: 国家自然科学基金资助项目(10871022;11061009;40821092);国家973资助项目(2010CB428403;2009CB421407;2010CB951001);河北省自然科学基金资助项目(A2010001663)
详细信息
    作者简介:

    罗振东(1958- ),男,广西人,教授,博士,博士生导师(联系人.E-mai:lzhdluo@163.com)

  • 中图分类号: O241.82

A Reduced Finite Difference Scheme and Error Estimates Based on POD Method for the Non-Stationary Stokes Equation

  • 摘要: 特征正交分解(proper orthogonal decomposition,简记为POD)方法是一种可对偏微分方程的物理模型(如流体流动)做简化的技术.这种方法已经成功地用于对复杂系统模型降阶.推广应用POD方法,将POD方法应用于具有实际应用背景的非定常Stokes方程经典的有限差分格式,建立一种维数较低而精度足够高的简化差分格式,并给出简化差分格式解与经典差分格式解的误差估计.数值例子说明数值计算结果与理论结果相吻合.进一步表明基于POD方法的简化差分格式对求解非定常Stokes方程数值解是可行和有效的.
  • [1] Brezzi F, Douglas J. Stabilized mixed method for the Stokes problem[J]. Numer Math,1988, 53(1/2): 225-235.
    [2] Douglas J,Wang J P. An absolutely stabilized finite element method for the Stokes problem[J]. Math Comp, 1989, 52(186): 495-508. doi: 10.1090/S0025-5718-1989-0958871-X
    [3] Chung T. Computational Fluid Dynamics[M]. Cambridge: Cambridge University Press, 2002.
    [4] Holmes P, Lumley J L, Berkooz G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry[M]. Cambridge: Cambridge University Press, 1996.
    [5] Fukunaga K. Introduction to Statistical Pattern Recognition[M]. Bostn: Academic Press, 1990.
    [6] Jolliffe I T. Principal Component Analysis[M]. Berlin: Springer-Verlag, 2002.
    [7] Crommelin D T, Majda A J. Strategies for model reduction: comparing different optimal bases[J]. J Atmos Sci, 2004, 61(17): 2206-2217. doi: 10.1175/1520-0469(2004)061<2206:SFMRCD>2.0.CO;2
    [8] Majda A J, Timofeyev I, Vanden-Eijnden E. Systematic strategies for stochastic mode reduction in climate[J]. J Atmos Sci, 2003, 60(14): 1705-1722. doi: 10.1175/1520-0469(2003)060<1705:SSFSMR>2.0.CO;2
    [9] Selten F. Baroclinic empirical orthogonal functions as basis functions in an atmospheric model[J]. J Atmos Sci, 1997, 54(16): 2099-2114. doi: 10.1175/1520-0469(1997)054<2099:BEOFAB>2.0.CO;2
    [10] Lumley J L. Coherent structures in turbulence[C]Meyer R E ed.Transition and Turbulence. New York: Academic Press, 1981: 215-242.
    [11] Aubry Y N, Holmes P,Lumley J L, Stone E. The dynamics of coherent structures in the wall region of a turbulent boundary layer[J]. J Fluid Mech, 1988, 192: 115-173. doi: 10.1017/S0022112088001818
    [12] Sirovich L. Turbulence and the dynamics of coherent structures: part Ⅰ-Ⅲ[J]. Quart Appl Math, 1987, 45(3): 561-590.
    [13] Joslin R D, Gunzburger M D, Nicolaides R A, Erlebacher G, Hussaini M Y. A self-contained automated methodology for optimal flow control validated for transition delay[J]. AIAA J, 1997, 35(5): 816-824. doi: 10.2514/2.7452
    [14] Ly H V, Tran H T. Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor[J]. Quart Appl Math, 2002, 60(4): 631-656.
    [15] Moin P,Moser R D. Characteristic-eddy decomposition of turbulence in channel[J]. J Fluid Mech, 1989, 200: 417-509.
    [16] Rajaee M, Karlsson S K F, Sirovich L. Low dimensional description of free shear flow coherent structures and their dynamical behavior[J]. J Fluid Mech, 1994, 258: 1401-1402.
    [17] Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for parabolic problems[J]. Numer Math, 2001, 90(1): 117-148. doi: 10.1007/s002110100282
    [18] Kunisch K,Volkwein S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics[J].SIAM J Numer Anal, 2002, 40(2): 492-515. doi: 10.1137/S0036142900382612
    [19] Kunisch K,Volkwein S. Control of Burgers’ equation by a reduced order approach using proper orthogonal decomposition[J].J Optim Theory Appl,1999, 102(2): 345-371. doi: 10.1023/A:1021732508059
    [20] Ahlman D, Sdelund F, Jackson J, Kurdila A, Shyy W. Proper orthogonal decomposition for time-dependent lid-driven cavity flows[J]. Numer Heat Transfer Part B: Fund, 2002, 42(4): 285-306.
    [21] Luo Z D,Wang R W, Zhu J. Finite difference scheme based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations[J]. Sci China, Ser A: Math, 2007, 50(8): 1186-1196.
    [22] Luo Z D,Chen J, Zhu J, Wang R, Navan I M. An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model[J]. I J Numer Methods Fluids, 2007, 55(2): 143-161. doi: 10.1002/fld.1452
    [23] Luo Z D,Chen J,Navon I M, Yang X Z. Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the non- stationary Navier-Stokes equations[J].SIAM J Numer Anal, 2008, 47(1): 1-19.
    [24] Luo Z D,Chen J,Sun P, Yang X Z. Finite element formulation based on proper orthogonal decomposition for parabolic equations[J].Sci China, Ser A: Math, 2009, 52(3): 585-596. doi: 10.1007/s11425-008-0125-9
    [25] Sun P, Luo Z D, Zhou Y J. Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations[J]. Appl Numer Math, 2010, 60(1/2): 154-164. doi: 10.1016/j.apnum.2009.10.008
    [26] Volkwein S. Optimal control of a phase-field model using the proper orthogonal decomposition[J]. ZFA Math Mech, 2001, 81(2): 83-97. doi: 10.1002/1521-4001(200102)81:2<83::AID-ZAMM83>3.0.CO;2-R
    [27] Antoulas A. Approximation of Large-Scale Dynamical Systems[M]. Philadelphia: Society for Industrial and Applied Mathematics, 2005.
    [28] Stewart G W. Introduction to Matrix Computations[M].New York: Academic Press,1973.
    [29] Noble B. Applied Linear Algebra[M]. Englewood Clis: Prentice-Hall, 1969.
    [30] Girault V,Raviart P A. Finite Element Approximations of the Navier-Stokes Equations, Theorem and Algorithms[M]. New York: Springer-Verlag, 1986.
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出版历程
  • 收稿日期:  2011-03-24
  • 修回日期:  2011-04-25
  • 刊出日期:  2011-07-15

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