Reduced Order Modeling Based On Pod For Fluid Dynamic Equations

The proper orthogonal decomposition (POD) method, in combination with the Galerkin projection procedure, has been applied as an efficient way to gen-erate reduced order models of physical processes governed by partial differential equations. This technique essentially provides a low-dimensional description of the time-dependent dynamic system in the optimizing sense that it can capture the greatest possible energy of the system, in order to alleviate the computational load and memory requirements.In this article, an optimizing reduced finite difference scheme (FDS) based on singular value decomposition (SVD) and POD for the chemical vapor deposit (CVD) equations is presented. And the error estimates between the usual finite difference solution and the reduced POD solution of optimizing FDS are derived. Some examples of numerical simulation are given to demonstrate the consistency of the numerical and theoretical results. It is shown that the optimizing reduced FDS based on POD method is of great feasibility and efficiency.A POD reduced order finite element model using unstructured mesh for the Imperial College Ocean Model (ICOM) is also presented. The finite element system matrix is projected to the POD reduced order space to derive a reduced order model. The error estimates between the usual finite element solution and the reduced order POD solution are derived. Finally, some numerical results are presented to demonstrate the consistency of the numerical and theoretical results and the feasibility of the POD method.Furthermore, a POD reduced order model of the parabolized Navier-Stokes (PNS) equations is derived in this article. A space-marching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to con-struct the reduced order model. In order to improve the accuracy and stability of the reduced order model in the presence of high Reynolds number, a Sobolev H1 norm calibration is applied to the POD construction process. Finally, some numerical tests with the high fidelity model as well as the POD reduced order model were carried out to demonstrate the efficiency and accuracy of the reduced order model for solving the PNS equations compared with the full PNS model. The efficiency of the H1 norm POD calibration is illustrated, along with the optimal dissipation coefficient derivation, yielding the best RMSE and correla-tion coefficient when comparing the PNS reduced order model with the full PNS model.A reduced order model based on POD 4-D VAR data assimilation for the parabolized Navier-Stokes (PNS) equations is then derived. Various approaches of POD implementation of the reduced order inverse problem are studied and compared including an ad-hoc POD adaptivity along with a trust region POD adaptivity. The numerical results obtained show that the trust region POD 4-D VAR provides the best results amongst all the POD adaptive methods tested in all error metrics for the reduced order inverse problem of the PNS equations.