Burgers方程基于POD方法的降维CN有限元外推算法
A Reduced-Order CN Finite Element Extrapolating Algorithm Based on POD for Burgers Equation

作者: 李 宏 , 黄春霞 :内蒙古大学数学科学学院,呼和浩特; 罗振东 :华北电力大学数理学院,北京;

关键词: 二维Burgers方程特征投影分解方法Crank-Nicolson有限元降维外推算法误差估计 Two-Dimensional Burgers Equation Proper Orthogonal Decomposition Technique Crank-Nicolson Finite Element Reduced-Order Extrapolating Algorithm Error Estimate

摘要:

建立二维Burgers方程基于特征投影分解(POD)方法的时间二阶精度的Crank-Nicolson (CN)有限元降维外推算法,给出这种算法的误差估计,并用误差估计作为算法的POD基数目选取及POD更新的准则。最后用数值实验说明该算法的优越性,这表明了该算法对于求解二维Burgers方程的数值解是有效可行的。

A Crank-Nicolson (CN) finite element reduced-order extrapolating algorithm with second-order accuracy based on proper orthogonal decomposition (POD) technique is established for two-dimensional Burgers equation, its error estimates are provided for criterions of the CN finite element reduced-order extrapolating algorithm to choose the number of POD basis and to renew POD basis. Some numerical experiments are used to show that the advantage of the CN finite element reduced-order extrapolating algorithm. It is shown that the CN finite element reduced-order extrapolating algorithm based on POD technique is feasible and efficient for finding the numerical solutions for two-dimensional Burgers equation.

 

文章引用: 李 宏 , 黄春霞 , 罗振东 (2013) Burgers方程基于POD方法的降维CN有限元外推算法。 流体动力学, 1, 1-9. doi: 10.12677/IJFD.2013.11001

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[38] Holmes P, Lumley J L, Berkooz G.Turbulence, coherent structures, dynamical systems and symmetry[M]. Cambridge: Cambridge University Press, 1996.

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[42] Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for parabolic problems[J]. Numer. Math., 2001, 90: 117−148.

[43] Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics[J]. SIAM Journal on Numerical Analysis, 2002, 40: 492−515.

[44] Luo Z D, Chen J, Xie Z H, et al. A reduced second-order time accurate finite element formulation based on POD for parabolic equations[J]. Sci Sin Math, 2011, 41(5): 447−460.

[45] Luo Z D, Li H, Zhou Y J, et al. A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem[J]. Journal of Mathematical Analysis and Applications, 2012, 385: 371−383.

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[65] Adams R A. Sobolev Spaces[M]. New York: Academic Press, 1975.

[66] Girault V, Raviant P A. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms[M]. Berlin: Spinger-Verlag, 1986.

[67] Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods[M]. New York: Springer-Verlag, 1991.

[68] Rudin W. Functional and Analysis (2nd Ed)[M]. McGraw-Hill Companies, Inc. 1973.

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