基于POD方法的两类波动方程向后欧拉有限元降维格式

发布时间：2018-05-29 22:46

本文选题：降维模型 + 向后欧拉有限元格式　；参考：《延边大学》2017年硕士论文

【摘要】：在物理、工程、医学、经济等科学研究中,遇到的很多问题都是用偏微分方程来表示,为了得到有用的数据和预测结果,需要对其进行求解.但是绝大多数偏微分方程的解很难以实用的解析形式来表示,于是偏微分方程的数值解法就成了求解偏微分方程的重要手段,在一定程度上弥补了这一问题的不足.然而数值方法也有其局限性,在求解复杂的偏微分方程问题的时候,无论是多么好的离散化格式,都需要很多的自由度,从而在内存和计算上付出很高的代价.因此,在保证方程的数值解具有足够高精度的情况下,简化计算量、截断误差的控制、节省运算时间和降低内存要求就成为了很有必要的研究问题.降维方法就是解决这一问题的有效方法之一,其中特征正交分解(Proper Orthogonal Decomposition)方法是大家比较熟悉的一种降维方法,已成功的用于对复杂系统模型的降维.特征正交分解方法的实质就是对物理过程进行低维近似描述,最优的逼近已知数据,从而达到减化计算、节省计算时间和降低内存的目的.在本文主要研究了如下两个方面的内容:首先,主要把特征正交分解方法应用BBM-Burgers方程通常的欧拉有限元格式,为了克服BBM-Burgers方程通常的欧拉有限元格式计算量大的缺点,我们在有限元解中抽取了瞬像集合,然后用POD基张成的子空间,取代了有限元格式的有限元空间,将维数较高的欧拉有限元格式简化为维数较低且具有足够高精度的POD向后欧拉有限元格式.并给出了降维后的欧拉有限元误差估计.其次,阐述了如何构造基于特征正交分解方法的Rosenau-RLW方程通常的欧拉有限元格式,简化其为一个计算量很少但具有足够高精度的POD向后欧拉有限元格式,并给出了简化后的有限元误差估计.POD向后欧拉有限元格式比通常的欧拉有限元格式更有效.
[Abstract]:In physics, engineering, medicine, economics and other scientific research, many problems are expressed by partial differential equations, in order to obtain useful data and prediction results, it needs to be solved. However, most of the solutions of partial differential equations are difficult to express in practical analytical form, so the numerical solution of partial differential equations becomes an important means of solving partial differential equations, which to some extent makes up for the deficiency of this problem. However, numerical methods also have their limitations. No matter how good the discretization scheme is, it requires a lot of degrees of freedom when solving complex partial differential equations, thus paying a high cost in memory and computation. Therefore, under the condition that the numerical solution of the equation is sufficiently accurate, it is necessary to simplify the calculation, control the truncation error, save the operation time and reduce the memory requirement. The dimensionality reduction method is one of the effective methods to solve this problem. The characteristic orthogonal decomposition (Proper Orthogonal Decomposition) method is a familiar dimensionality reduction method, which has been successfully used to reduce the dimension of complex system models. The essence of the feature orthogonal decomposition method is to describe the physical process in a low-dimensional approximation and to approach the known data optimally so as to reduce the computation time and memory. In this paper, the following two aspects are mainly studied: firstly, the characteristic orthogonal decomposition method is mainly applied to the Euler finite element scheme of the BBM-Burgers equation, in order to overcome the disadvantages of the Euler finite element scheme of the BBM-Burgers equation. We extract the instantaneous image set from the finite element solution and replace the finite element space with the subspace of POD basis Zhang Cheng. The high dimensional Euler finite element scheme is simplified to a POD backward Euler finite element scheme with low dimension and sufficient precision. The error estimation of Euler finite element after dimensionality reduction is given. Secondly, how to construct the usual Euler finite element scheme of Rosenau-RLW equation based on the method of characteristic orthogonal decomposition is introduced, which is simplified to a POD backward Euler finite element scheme with less computation and enough precision. The simplified finite element error estimation. POD backward Euler finite element scheme is more effective than the conventional Euler finite element method.
【学位授予单位】：延边大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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