流体力学及海洋数值模拟基于POD技术的降维方法研究

【作者】 王瑞文；

【导师】 罗振东；

【作者基本信息】 首都师范大学 ， 应用数学， 2007， 博士

【摘要】 在模拟由偏微分方程控制的物理过程中，POD(Proper Orthogonal Decomposition)方法是一种高效降维方法。这是一种功能强大，效果显著的数据分析方法，其目标是把多维的物理过程进行低维的近似描述(即用最佳级数展开)，进而大大减小用于重现物理过程时所需存储的数据量。它提供了一种描绘物理过程的有效方法，该物理过程可以写成一个只与时间相关的函数和一个只与空间相关的函数的展开式序列，且它们在均方意义上是最优的，在展开式中只需要少量的项数就可较准确地描述该物理过程，因此降低计算维数、减少计算量和节省计算时间是这个方法最明显的优点。本文将POD方法和SVD方法结合起来研究非定常的Navier-Stokes方程的有限差分格式，结合Galerkin投影方法导出了Navier-Stokes方程具有较高精确度的低维模型，给出了Navier-Stokes方程最优化的有限差分格式，并导出了这种最优化的有限差分格式的逼近解和通常的全阶有限差分格式的逼近解的误差估计。数值例子表明基于POD方法的降维格式解和全阶有限差分格式解的误差与理论分析结果是一致的。然后我们将POD技术结合混合有限元方法应用在热带太平洋约化重力模式上，得到了降维模式，导出了混合有限元方法的降维格式，并给出了用混合有限元法得到的全阶模拟逼近解和基于POD技术的降维模拟逼近解理论上的误差估计，通过数值试验得到了全阶模拟的数值解和POD降维模拟数值解之间的误差，数值结果和理论结果一致。从而验证POD方法在降维问题上的有效性和可行性。Equation-Free方法是常用来解决复杂、多尺度问题。在实际问题中，我们通常需要知道描述宏观活动的物理系统的具体方程形式，而这些具体形式理论上存在实际中又非常难于知道。Equation-Free方法就能解决这个问题，即不需要知道方程的具体形式而达到能够求解的目的。本文我们将基于POD方法的Equation-Free POD方法应用到热带太平洋区域上的大尺度大洋环流的约化重力模式。讨论了该方法的收敛性和精度，并且通过数值试验研究了影响该方法效果的几个因素，主要是大小时间步长的比例是多少，基函数的个数应该取多少，还有snapshots(瞬像)点个数等。数值结果表明，Equation-Free方法的计算量是全阶模式计算量的6％，并且通过均方根误差和相关系数说明和POD方法相比都有大的改进，尤其是我们在应用POD方法的过程中不再需要知道计算时间系数的方程的具体形式。更多还原

【Abstract】 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.更多还原