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.
-
关键词:
- 有限差分格式 /
- 特征正交分解 /
- 误差估计 /
- 非定常Stokes方程
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. -
[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, Sdelund 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.
点击查看大图
计量
- 文章访问数: 1613
- HTML全文浏览量: 126
- PDF下载量: 926
- 被引次数: 0