New Numerical Solution Methods For Two Kinds Of Equations Base On POD

The proper orthogonal decomposition (POD) is a model reduction technique for the simulation of physical processes governed by partial differential equations, e.g., fluid flows. It has been successfully used in the reduced-order modeling of complex systems. The second part of this paper, the applications of the POD method is extended to a classical finite difference (FD) scheme for the non-stationary Stokes equation with real practical applied background. A reduced FD scheme with lower dimensions and sufficiently high accuracy is established, and the error estimates between the reduced and the classical FD solutions is provided. Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced FD scheme based on the POD method is feasible and efficient for solving the FD scheme of the non-stationary Stokes equation. The third part of this paper is to apply POD method to a classical finite element (FE) formulation for second-order hyperbolic equations with real practical applied background, establish a reduced FE formulation with lower dimensions and sufficiently high accuracy, and also provide the error estimates between the reduced FE solutions and the classical FE solutions and the implementation of algorithm for solving reduced FE formulation. Thus, we provide scientific theoretic basis for service applications. Some numerical examples also illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced FE formulation based on POD method is feasible and efficient for solving FE formulation of second-order hyperbolic equations. Last, this paper gets some conclusion.