本文選題：降維模型 + 向后歐拉有限元格式　； 參考：《延邊大學(xué)》2017年碩士論文

【摘要】：在物理、工程、醫(yī)學(xué)、經(jīng)濟(jì)等科學(xué)研究中,遇到的很多問(wèn)題都是用偏微分方程來(lái)表示,為了得到有用的數(shù)據(jù)和預(yù)測(cè)結(jié)果,需要對(duì)其進(jìn)行求解.但是絕大多數(shù)偏微分方程的解很難以實(shí)用的解析形式來(lái)表示,于是偏微分方程的數(shù)值解法就成了求解偏微分方程的重要手段,在一定程度上彌補(bǔ)了這一問(wèn)題的不足.然而數(shù)值方法也有其局限性,在求解復(fù)雜的偏微分方程問(wèn)題的時(shí)候,無(wú)論是多么好的離散化格式,都需要很多的自由度,從而在內(nèi)存和計(jì)算上付出很高的代價(jià).因此,在保證方程的數(shù)值解具有足夠高精度的情況下,簡(jiǎn)化計(jì)算量、截?cái)嗾`差的控制、節(jié)省運(yùn)算時(shí)間和降低內(nèi)存要求就成為了很有必要的研究問(wèn)題.降維方法就是解決這一問(wèn)題的有效方法之一,其中特征正交分解(Proper Orthogonal Decomposition)方法是大家比較熟悉的一種降維方法,已成功的用于對(duì)復(fù)雜系統(tǒng)模型的降維.特征正交分解方法的實(shí)質(zhì)就是對(duì)物理過(guò)程進(jìn)行低維近似描述,最優(yōu)的逼近已知數(shù)據(jù),從而達(dá)到減化計(jì)算、節(jié)省計(jì)算時(shí)間和降低內(nèi)存的目的.在本文主要研究了如下兩個(gè)方面的內(nèi)容:首先,主要把特征正交分解方法應(yīng)用BBM-Burgers方程通常的歐拉有限元格式,為了克服BBM-Burgers方程通常的歐拉有限元格式計(jì)算量大的缺點(diǎn),我們?cè)谟邢拊庵谐槿×怂蚕窦?然后用POD基張成的子空間,取代了有限元格式的有限元空間,將維數(shù)較高的歐拉有限元格式簡(jiǎn)化為維數(shù)較低且具有足夠高精度的POD向后歐拉有限元格式.并給出了降維后的歐拉有限元誤差估計(jì).其次,闡述了如何構(gòu)造基于特征正交分解方法的Rosenau-RLW方程通常的歐拉有限元格式,簡(jiǎn)化其為一個(gè)計(jì)算量很少但具有足夠高精度的POD向后歐拉有限元格式,并給出了簡(jiǎn)化后的有限元誤差估計(jì).POD向后歐拉有限元格式比通常的歐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.
【學(xué)位授予單位】：延邊大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.82

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本文編號(hào)：1952779