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Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations. (English) Zbl 1134.35387
Summary: 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 optimized 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 the POD method.
MSC:
 35Q30 Stokes and Navier-Stokes equations 76M20 Finite difference methods (fluid mechanics) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
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