Article

Science in China Series A: Mathematics

, Volume 50, Issue 8, pp 1186-1196

Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

  • Zhen-dong LuoAffiliated withSchool of Science, Beijing Jiaotong University Email author 
  • , Rui-wen WangAffiliated withInstitute of Atmospheric Physics, Chinese Academy of Sciences
  • , Jiang ZhuAffiliated withInstitute of Atmospheric Physics, Chinese Academy of Sciences

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Abstract

The proper orthogonal decomposition (POD) and the singular value decomposition (SVD) are used to study the finite difference scheme (FDS) for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations. The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD. Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations. The errors between POD approximate solutions and FDS solutions are analyzed. It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results. Moreover, it is also shown that this validates the feasibility and efficiency of POD method.

Keywords

proper orthogonal decomposition singular value decomposition finite difference scheme the nonstationary Navier-Stokes equations

MSC(2000)

65N30 35Q10