| Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations |
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| 作者姓名： | Zhen-dong LUO~ |
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| 作者单位： | Zhen-dong LUO（School of Science, Beijing Jiaotong University, Beijing 100044, China） ;
Rui-wen WANG（Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China） ;
Jiang ZHU（Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China） ; |
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| 摘 要： | 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.
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| 收稿时间： | 18 July 2006 |
| 修稿时间： | 14 March 2007 |
Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations |
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| Zhen-dong LUO.Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations[J].Science in China(Mathematics),2007,50(8):1186-1196. |
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| Authors: | Zhen-dong Luo Rui-wen Wang Jiang Zhu |
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| Institution: | 1. School of Science, Beijing Jiaotong University, Beijing 100044, China
2. Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China |
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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. |
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| Keywords: | proper orthogonal decomposition singular value decomposition finite difference scheme the nonstationary Navier-Stokes equations |
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