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Applied Mathematics and Mechanics

, Volume 32, Issue 7, pp 847–858 | Cite as

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

  • Zhen-dong Luo (罗振东)Email author
  • Qiu-lan Ou (欧秋兰)
  • Zheng-hui Xie (谢正辉)
Article

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.

Key words

finite difference scheme proper orthogonal decomposition error estimate non-stationary Stokes equation 

Chinese Library Classification

O241.82 

2010 Mathematics Subject Classification

65N30 65N55 35Q10 

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Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Zhen-dong Luo (罗振东)
    • 1
    Email author
  • Qiu-lan Ou (欧秋兰)
    • 1
  • Zheng-hui Xie (谢正辉)
    • 2
  1. 1.School of Mathematics and PhysicsNorth China Electric Power UniversityBeijingP. R. China
  2. 2.State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric PhysicsChinese Academy of SciencesBeijingP. R. China

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