The Study Of Fluid Mechanics And Ocean Numerical Simulation Dimensionality Reduction Method Based On The Pod Technology

POD (Proper Orthogonal Decomposition) is a model reduction technique for the simulation of physical processes governed by partial differential equations. This is an evidently effective method of data analyse, and the aim of the method is to describe a multidimensional physical processes in a low dimensional way approximately (series expansion). It provides a way to find optimal lower dimensional approximations of the given data, and It enormously reduce the storage of the data which is needed when the physical processes is recurred. The physical process can be described in series expansion function that only is related to the time and space. It is optimal approximation of the given data in a certain least squares optimal sense. The most obvious advantage of the method is that it reduces the computation cost and computation time markedly.In this paper, POD method and SVD are used to study the finite difference scheme for the nonstationary Navier-Stokes equations. Combing the POD technique with a Galerkin projection approach yields a new optimizing finite difference scheme with lower dimensions and high accuracy for the nonstationary Navier-Stokes equations. The errors between POD approximation solutions and the solutions of direct numerical simulation are analyzed. It is shown by the results of the numerical simulation that the errors are consistent with theoretical results.We also combined the POD technique with mixed finite element (MFE) method to study the tropical Pacific Ocean reduced gravity model, the lower dimension MFE scheme and an error estimate of POD approximate solution based on MFE method are derived. It is shown by numerical examples that the errors between POD approximate solution and reference solution are consistent with theoretical results, thus validating the feasibility and efficiency of POD method in decreasing dimension of the problems. The "Equation-Free" (E-F) method is often used in complex, multi-scale problems. In such cases it is necessary to know the closed form of the required evolution equations about macroscopic variables within some applied fields. Conceptually such equations exist, however, they are not available in closed form. The EF method can bypass this difficulty.In this paper we apply the E-F POD-assisted method to the reduced modeling of a large-scale upper ocean circulation in the tropical Pacific domain. We discuss the convergence and accuracy of this method along with a series of numerical experiments that were carried in order to discuss some factors that affect the results. These factors include the number of snapshots, basis functions based on proper orthogonal decomposition (POD) mode and the ratio between large- and short-scale time steps. An observed reduction in the computation work in comparison to the full model was observed, for instance, the computational cost of the E-F POD-assisted method was about 6% of that of the full model. Compared with the POD method, the method provided more accurate results and it did not require the availability of any explicit equations or the right-hand-side (RHS) of the evolution equation.