本文選題：特征線方法 + 間斷Galerkin方法�。� 參考：《蘭州大學》2016年博士論文

【摘要】：本文我們主要考慮兩類微分方程:時間依賴的不可壓縮Navier-Stokes方程和二維空間Riemann-Liouville分數(shù)階方程.對于Navier-Stokes方程,首先,我們引入一個輔助變量來分離擴散算子使原始的高階方程轉化為一階系統(tǒng),從而降低解決高階方程的難度.其次,通過變分、精心地選擇數(shù)值流、添加罰項,我們設計了一個穩(wěn)定對稱的局部間斷Galerkin(LDG)格式.這樣我們完成了方程的空間離散,但是我們并沒有離散非線性對流項.我們知道求解Navier-Stokes方程最大的困難有兩個:一是如何處理非線性對流項,另一個是如何處理方程中的壓強函數(shù).由于特征線方法在求解對流占優(yōu)問題上有很大的優(yōu)勢,所以我們考慮特征線方法同時離散非線性對流項和時間導數(shù)項.從而,我們成功地解決了非線性問題,并且得到了一個穩(wěn)定的對稱的特征線局部間斷Galerkin格式(CLDG).在穩(wěn)定性推導的過程中,我們發(fā)現(xiàn)特征線方法使得理論推導簡易明了.特別是應用格式的對稱性,使得我們消去了一些比較難處理的算子.由于在NavierStokes方程中速度函數(shù)和壓強函數(shù)沒有緊密的聯(lián)系,所以當我們得到速度的誤差估計時,不是很容易得到壓強的誤差估計.為了解決這個困難,我們應用經典的速度和壓強的連續(xù)上下界條件,建立速度誤差和壓強誤差的聯(lián)系.實際上,在間斷有限元空間我們通常利用速度和壓強的離散上下界條件,但是在本文,我們將連續(xù)上下界條件和離散空間結合起來得到壓強的誤差估計,這也是第三章的一個亮點.最后,我們給出四個不同的數(shù)值例子來驗證理論結果,并且看到數(shù)值結果不僅達到了預期的效果而且比預期的更好.對于二維空間Riemann-Liouville分數(shù)階方程,首先,我們引進兩個輔助變量來分離Riemann-Liouville分數(shù)階導數(shù).由于Riemann-Liouville分數(shù)階導數(shù)本身具有奇異性,Riemann-Liouville分數(shù)階積分沒有奇異性,所以一個輔助變量用來代替函數(shù)的梯度項,另一個輔助變量用來代替Riemann-Liouville分數(shù)階積分.這樣我們成功地分離了Riemann-Liouville分數(shù)階導數(shù)并且把高階導數(shù)方程降低為一階系統(tǒng).然后,通過變分、精確地選擇數(shù)值流、添加罰項,我們設計了一個混合的間斷Galerkin(HDG)格式,從而完成了方程的半離散.最后,我們給出三種時間離散方法,針對Riemann-Liouville分數(shù)階擴散問題,我們應用一般的差分方法進行時間離散,由于方法的簡易性,我們沒有給出相應的離散過程.對于Riemann-Liouville分數(shù)階對流擴散問題,我們應用一階和二階特征線方法同時離散時間導數(shù)項和對流項,并且給出相應的離散格式、穩(wěn)定性分析、誤差估計.在研究二維空間Riemann-Liouville分數(shù)階方程時,我們發(fā)現(xiàn)如何找到一種有效的方法進行理論分析是比較困難的,特別是誤差估計.為了解決這一困難,我們沒有考慮有效的分析工具而是從格式設計出發(fā),設計有效的數(shù)值格式.基于這樣的考慮我們得到了不同的數(shù)值格式,不僅將一維問題推廣到二維而且得到較好的數(shù)值結果.在第四章最后一節(jié),我們分別應用三個數(shù)值例子驗證Riemann-Liouville分數(shù)階方程和一階二階HDG格式.總之,本篇論文我們成功地將間斷Galerkin方法和特征線方法結合起來求解不可壓縮的Navier-Stokes方程和二維空間分數(shù)階微分方程.我們可以看到數(shù)值實驗結果與理論結果是一致的并且是有效的.因此,這些方法可進一步改進且應用到其他問題中.
[Abstract]:In this paper, we mainly consider two kinds of differential equations: time dependent incompressible Navier-Stokes equation and two-dimensional space Riemann-Liouville fractional equation. For the Navier-Stokes equation, first, we introduce a auxiliary variable to separate the diffusion operator and turn the original high order equation into the first order system, thus reducing the high order equation. Secondly, we design a stable symmetric local discontinuous Galerkin (LDG) format by selecting the numerical flow and adding the penalty term by the variation of the variation. We have completed the space discretization of the equation, but we do not have the discrete nonlinear convection term. We know that there are two difficulties in solving the maximum Navier-Stokes equation: one is how The nonlinear convection term is dealt with, and the other is how to deal with the pressure function in the equation. Because the characteristic line method has a great advantage in solving the convection dominated problem, we consider the characteristic line method to discrete the nonlinear convection term and the time derivative. Thus, we successfully solve the nonlinear problem and get a stable problem. In the process of stability derivation, we find that the characteristic line method makes the theoretical derivation simple and clear in the process of stability deduction. In particular, the symmetry of the applied format makes us eliminate some of the more difficult operators. Because the velocity function and pressure function in the NavierStokes equation are not tight, we have not tightened the pressure function in the NavierStokes equation. In order to solve this problem, we apply the continuous upper and lower bounds of the classical velocity and pressure to establish the relation between the velocity error and the pressure error. In fact, we usually use the velocity and pressure in the discontinuous finite element space. In this paper, we combine the continuous upper and lower boundary conditions with the discrete space to get the error estimation of the pressure, which is also a bright spot in the third chapter. Finally, we give four different numerical examples to verify the theoretical results and see that the number results not only achieve the expected effect but also compare with the expected results. Better. For the two-dimensional space Riemann-Liouville fractional equation, first, we introduce two auxiliary variables to separate the Riemann-Liouville fractional derivative. Because the fractional derivative of the Riemann-Liouville is singularity itself, the fractional integral of Riemann-Liouville has no singularity, so a auxiliary variable is used to replace the gradient of the function. The other auxiliary variable is used to replace the Riemann-Liouville fractional integral. So we successfully separate the Riemann-Liouville fractional derivative and reduce the high order derivative equation to the first order system. Then, we choose the numerical flow accurately by the variational, and add the penalty term, and we design a mixed discontinuous Galerkin (HDG) format. And finally, we have completed the semi discrete equation. Finally, we give three time discrete methods. In view of the Riemann-Liouville fractional diffusion problem, we use the general difference method to make time discretization. Because of the simplicity of the method, we do not give the corresponding discrete process. For the Riemann-Liouville fractional convection diffusion problem, we should do it. The first and two order characteristic lines are used to simultaneously discrete time derivatives and convective terms, and the corresponding discrete schemes, stability analysis and error estimation are given. In the study of the two-dimensional space Riemann-Liouville fractional equation, it is found that it is difficult to find an effective method for the theory analysis, especially the error estimation. In order to solve this problem, we do not consider the effective analysis tool but design the effective numerical format from the format design. Based on this, we get different numerical schemes, not only to extend the one-dimensional problem to two-dimensional but also to get better numerical results. In the last section of the fourth chapter, we apply three respectively. The numerical examples verify the Riemann-Liouville fractional equation and the first order two order HDG scheme. In this paper, we successfully combine the discontinuous Galerkin method and the characteristic line method to solve the incompressible Navier-Stokes equation and the two-dimensional fractional order differential equation. We can see that the numerical experiment results are in agreement with the theoretical results. And they are effective. Therefore, these methods can be further improved and applied to other problems.
【學位授予單位】：蘭州大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O241.82

【相似文獻】

相關期刊論文 前10條

1 羅振東;二維空間的Navier-Stokes問題的混合有限元分析[J];高校應用數(shù)學學報A輯(中文版);1991年02期

2 韓慶書,王唯;求解二維 Navier-Stokes 方程的譜元法[J];北京聯(lián)合大學學報;1998年S1期

3 張余;關于平面 Navier-Stokes 流的估計[J];內蒙古工業(yè)大學學報(自然科學版);1998年03期

4 趙紅星;姚正安;;Navier-Stokes流近臨界指標的均勻化[J];廣州大學學報(自然科學版);2009年01期

5 武曉松,王棟,郭錫福;彈丸超聲速繞流的 Navier-Stokes 模擬[J];空氣動力學學報;1998年02期

6 段獻葆;秦新強;;一種改進的水平集方法在Navier-Stokes問題形狀優(yōu)化中的應用[J];工程數(shù)學學報;2010年02期

7 岳寶增,劉延柱;帶自由液面Navier-Stokes流動問題的ALE分步有限元方法[J];水動力學研究與進展(A輯);2003年04期

8 羅振東;Navier-Stokes問題的六面體單元的混合有限元法[J];應用數(shù)學和力學;1992年12期

9 張克偉;二維Navier-Stokes方程的吸引子和二維Navier-Stokes流[J];內蒙古大學學報(自然科學版);1999年03期

10 張克偉;;定常的Navier-Stokes流的Dirichlet積分的增長估計[J];內蒙古大學學報(自然科學版);1986年04期

相關會議論文 前4條

1 汪淳;林志良;樊濤;;原始變量定常Navier-Stokes的廣義邊界元方法[A];第二十一屆全國水動力學研討會暨第八屆全國水動力學學術會議暨兩岸船舶與海洋工程水動力學研討會文集[C];2008年

2 任玉新;孫宇濤;;求解Euler和Navier-Stokes的多維迎風格式[A];近代空氣動力學研討會論文集[C];2005年

3 羅俊;徐昆;;模擬Euler和Navier-Stokes解的高階緊致氣體動力學方法[A];中國力學大會——2013論文摘要集[C];2013年

4 孫丹;楊建剛;;基于局部DQ-Lagrange不可壓縮N-S方程的數(shù)值求解[A];中國計算力學大會'2010（CCCM2010）暨第八屆南方計算力學學術會議（SCCM8）論文集[C];2010年

相關博士學位論文 前2條

1 王淑琴;間斷Galerkin方法求解Navier-Stokes和分數(shù)階方程[D];蘭州大學;2016年

2 蔡小云;幾類Navier-Stokes系統(tǒng)的整體正則性問題[D];南京大學;2014年

相關碩士學位論文 前3條

1 駱艷;黏性不可壓縮流動Stokes（Navier-Stokes）方程的間斷有限元法[D];四川大學;2006年

2 余水源;穩(wěn)態(tài)Navier-Stokes和Darcy流耦合問題的非協(xié)調混合元方法[D];山東大學;2013年

3 杜模云;帶時間平均的Smoluchowski方程耦合Navier-stokes方程的正則性[D];湘潭大學;2009年

，

本文編號：2053604