非定常Stokes方程一种基于POD方法的简化有限差分格式  

A Reduced Finite Difference Scheme and Error Estimates Based on POD Method for the Non-Stationary Stokes Equation

作　　者：罗振东[1] 欧秋兰[1] 谢正辉[2] 

机构地区：[1]华北电力大学数理学院,北京102206 [2]中国科学院大气物理研究所大气科学和地球流体力学数值模拟国家重点实验室,北京100029

出　　处：《应用数学和力学》2011年第7期795-806,共12页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(10871022;11061009;40821092);国家973资助项目(2010CB428403;2009CB421407;2010CB951001);河北省自然科学基金资助项目(A2010001663)

摘　　要：特征正交分解(proper orthogonal decomposition,简记为POD)方法是一种可对偏微分方程的物理模型(如流体流动)做简化的技术.这种方法已经成功地用于对复杂系统模型降阶.该文推广应用POD方法,将POD方法应用于具有实际应用背景的非定常Stokes方程经典的有限差分格式,建立一种维数较低而精度足够高的简化差分格式,并给出简化差分格式解与经典差分格式解的误差估计.数值例子说明数值计算结果与理论结果相吻合.进一步表明基于POD方法的简化差分格式对求解非定常Stokes方程数值解是可行和有效的.The proper orthogonal decomposition（POD） was a model reduction technique for the simulation of physical processes governed by partial differential equations,e.g.fluid flows.It was successfully used in the reduced-order modeling of complex systems.The applications of POD method were extended,i.e.,apply POD method to a classical finite difference（FD） scheme for the non-stationary Stokes equation with real practical applied background,establish a reduced FD scheme with lower dimensions and sufficiently high accuracy,and provide the error estimates between the reduced FD solutions and the classical FD solutions.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 POD method is feasible and efficient for solving FD scheme for the non-stationary Stokes equation.

关 键 词：有限差分格式 特征正交分解 误差估计 非定常STOKES方程 

分 类 号：O241.82[理学—计算数学]