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

LUO Zhen-dong; OU Qiu-lan; XIE Zheng-hui

doi:10.3879/j.issn.1000-0887.2011.07.004

Volume 32 Issue 7

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2011 > 32(7): 795-806

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:

PDF( 2059 KB)

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

doi: 10.3879/j.issn.1000-0887.2011.07.004

1.
School of Mathematics and Physics, North China Electric Power University, Beijing 102206, P. R. China;
ICCES/LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, P. R. China

Received Date: 2011-03-24
Rev Recd Date: 2011-04-25
Publish Date: 2011-07-15

Abstract

Abstract

The properorthogonal decom position (POD) was amodel reduction technique for the simulation of physical processes governed by partial differen tial equations, e. g. fluid flows. It was success fully used in the reduced-ordermodeling of complex systems. The applications of POD method were extended, i. e., apply POD method to a classical finited ifference (FD) scheme for the non-stationary Stokes equation with real practical applied background, estab lish a reduced FD scheme with lowerd imensions and sufficiently high accuracy, and provide the errorestmi ates between the reduced FD solutions and the classical FD solutions. Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced FD schemebased on POD method is feasible and efficient for solving FD scheme for the non-stationary Stokes equation.
- finite difference scheme,
- proper orthogonal decomposition,
- error estimate,
- non-stationary Stokes equation

FullText(HTML)

References(30)

References

[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.