本文關(guān)鍵詞： Navier-Stokes方程 鞍點(diǎn)問題 迭代方法 矩陣分裂 預(yù)處理子　出處：《蘭州大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：當(dāng)今,很多工程和物理應(yīng)用問題,如計(jì)算流體動力學(xué),計(jì)算電磁學(xué),約束優(yōu)化問題等,最后都會歸為線性方程組的求解.一些微分方程,例如Navier-Stokes方程,求它們的解析解是非常困難的,此時(shí)研究它們的數(shù)值解就變得尤為重要.一般采用有限差分或有限元等方法去離散這些微分方程為大型稀疏的線性方程組,這樣就把微分方程的數(shù)值解問題最后都轉(zhuǎn)化為對應(yīng)的線性方程組求解問題.因此,研究這些線性方程組有效的解法具有非常重要的理論意義和應(yīng)用價(jià)值.本學(xué)位論文主要研究了鞍點(diǎn)線性系統(tǒng)的矩陣分裂迭代方法和預(yù)處理技術(shù);還研究了基于雙分裂的并行多分裂迭代方法.首先,關(guān)于非對稱鞍點(diǎn)問題,提出了一類修正的位移分裂(MSS)預(yù)處理子,同時(shí)MSS預(yù)處理子對應(yīng)的MSS迭代方法的收斂性質(zhì)會被討論;另外,進(jìn)一步提出了局部的MSS(LMSS)預(yù)處理子,也討論了LMSS預(yù)處理子對應(yīng)的LMSS迭代方法的收斂性質(zhì);接著,討論了MSS和LMSS預(yù)處理子的最優(yōu)參數(shù)的選取方法;數(shù)值實(shí)驗(yàn)驗(yàn)證了MSS預(yù)處理子和LMSS預(yù)處理子的有效性.其次,提出了廣義鞍點(diǎn)問題的正則的埃爾米特和反埃爾米特分裂(RHSS)迭代法和RHSS預(yù)處理子,且研究了RHSS迭代方法的收斂性質(zhì);接著,推出了修正的RHSS(MRHSS)預(yù)處理子,并分析了MRHSS預(yù)處理的廣義鞍點(diǎn)矩陣的譜性質(zhì);此外,分別研究了RHSS和MRHSS預(yù)處理子最優(yōu)參數(shù)的選取方法;數(shù)值實(shí)驗(yàn)驗(yàn)證了RHSS迭代方法的優(yōu)勢,以及RHSS預(yù)處理子和MRHSS預(yù)處理子的預(yù)處理效果.再次,為了克服修正的維數(shù)分裂(MDS)預(yù)處理子和廣義的松弛分裂(GRS)預(yù)處理子的不足,給出了松弛的塊三角分裂(RBTS)預(yù)處理子.因?yàn)镽BTS預(yù)處理子有更簡單的塊結(jié)構(gòu),所以這個(gè)新的預(yù)處理子比MDS和GRS預(yù)處理子更容易實(shí)施;接著,推導(dǎo)了RBTS預(yù)處理的鞍點(diǎn)矩陣的譜分布和最小多項(xiàng)式次數(shù)的上界;另外,提出了RBTS預(yù)處理子最優(yōu)參數(shù)的選取方法.數(shù)值實(shí)驗(yàn)證實(shí)了RBTS預(yù)處理子的有效性.然后,關(guān)于廣義鞍點(diǎn)問題,給出一類修正的GRS(MGRS)預(yù)處理子和一類修正的塊三角分裂(MBTS)預(yù)處理子;接著,分別研究了MGRS和MBTS預(yù)處理的鞍點(diǎn)矩陣的譜性質(zhì)及它們的最小多項(xiàng)式次數(shù)的上界;進(jìn)而,分別討論了MGRS和MBTS預(yù)處理子最優(yōu)參數(shù)的選取方法;另外,應(yīng)用這兩類新的預(yù)處理子到三維線性化的Navier-Stokes方程,并分別討論了對應(yīng)的MGRS和MBTS預(yù)處理子最優(yōu)參數(shù)的選取;最后,通過數(shù)值實(shí)驗(yàn)來驗(yàn)證了兩類新預(yù)處理子的有效性.最后,基于系數(shù)矩陣的雙分裂提出了并行多分裂迭代法和并行多分裂兩階段迭代法.當(dāng)系數(shù)矩陣為單調(diào)矩陣或H矩陣時(shí),研究了新方法的收斂性,也進(jìn)一步討論了新方法的比較結(jié)果.此外,提出了鞍點(diǎn)線性系統(tǒng)的基于雙分裂的多分裂迭代法.
[Abstract]:Nowadays, many engineering and physical application problems, such as computational fluid dynamics, computational electromagnetics, constrained optimization problems and so on, will be classified as the solution of linear equations, some differential equations, such as Navier-Stokes equations, It is very difficult to find their analytical solutions, so it is very important to study their numerical solutions. The finite difference method or finite element method is usually used to discretize these differential equations for large sparse linear equations. In this way, the numerical solutions of the differential equations are transformed into the corresponding linear equations. It is of great theoretical significance and practical value to study the effective solutions of these linear equations. In this dissertation, the matrix splitting iterative method and preprocessing technique for saddle point linear systems are studied. The parallel multi-splitting iterative method based on double splitting is also studied. Firstly, for the asymmetric saddle point problem, a modified displacement-splitting MSS) preprocessor is proposed, and the convergence property of the MSS iterative method corresponding to the MSS preprocessor is discussed. In addition, the local MSSS-LMSS) preprocessor is proposed, and the convergence property of the LMSS iteration method corresponding to the LMSS preprocessor is also discussed, and then the methods of selecting the optimal parameters of the MSS and LMSS preconditioners are discussed. Numerical experiments show the validity of MSS preprocessor and LMSS preprocessor. Secondly, regular Hermitian and anti-Hermitian splitting RHSS iterative method and RHSS preprocessor for generalized saddle point problem are proposed, and the convergence properties of RHSS iterative method are studied. Then, the modified RHSS-MRHSS preprocessor is proposed, and the spectral properties of the generalized saddle point matrix of MRHSS preprocessing are analyzed. In addition, the methods of selecting optimal parameters of RHSS and MRHSS preconditioners are studied, and the advantages of RHSS iterative method are verified by numerical experiments. Thirdly, in order to overcome the shortcomings of modified dimension splitters and generalized relaxation splitters, the effects of RHSS preconditioners and MRHSS preconditioners are also discussed. This new preprocessor is easier to implement than the MDS and GRS preprocessor because the RBTS preprocessor has a simpler block structure. The spectral distribution and upper bound of minimum polynomial degree of saddle point matrix of RBTS preprocessing are derived. In addition, a method for selecting optimal parameters of RBTS preprocessor is proposed. Numerical experiments show the validity of RBTS preprocessor. For the generalized saddle point problem, a class of modified GRS MGRS preconditioners and a class of modified block triangulation splitters (MBTS) preconditioners are given, and then the spectral properties of the saddle point matrices of MGRS and MBTS preprocessing and the upper bounds of their minimum polynomial degree are studied, respectively. Furthermore, the methods of selecting the optimal parameters of MGRS and MBTS preprocessor are discussed respectively. In addition, the two new preconditioners are applied to the three-dimensional linearized Navier-Stokes equation, and the optimal parameters of the corresponding MGRS and MBTS preconditioners are discussed respectively. The validity of two kinds of new preprocessor is verified by numerical experiments. Finally, the parallel multi-splitting iterative method and the parallel multi-splitting two-stage iterative method are proposed based on the coefficient matrix. When the coefficient matrix is monotone matrix or H matrix, In this paper, the convergence of the new method is studied, and the comparison results of the new method are also discussed. In addition, a double splitting iterative method based on double splitting is proposed for saddle point linear systems.
【學(xué)位授予單位】：蘭州大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O241.6

【相似文獻(xiàn)】

相關(guān)期刊論文 前10條

1 萬維明,遲曉恒;廣義齊三次系統(tǒng)鞍點(diǎn)量問題[J];大連鐵道學(xué)院學(xué)報(bào);2001年03期

2 徐子珊;嚴(yán)格鞍點(diǎn)的查找算法[J];重慶工商大學(xué)學(xué)報(bào)(自然科學(xué)版);2004年05期

3 桑波;朱思銘;;焦點(diǎn)量與鞍點(diǎn)量的關(guān)系[J];數(shù)學(xué)年刊A輯(中文版);2007年02期

4 徐天博;李偉;;缺參數(shù)a_(23),b_(32)的齊五次系統(tǒng)的前四階鞍點(diǎn)量公式[J];大連交通大學(xué)學(xué)報(bào);2008年02期

5 趙景余;張國鳳;常巖磊;;求解鞍點(diǎn)問題的一種新的結(jié)構(gòu)算法[J];數(shù)值計(jì)算與計(jì)算機(jī)應(yīng)用;2009年02期

6 萬維明;周文;;齊四次系統(tǒng)鞍點(diǎn)量公式[J];大連交通大學(xué)學(xué)報(bào);2010年06期

7 葉惟寅;二次系統(tǒng)鞍點(diǎn)量的計(jì)算[J];南京師大學(xué)報(bào)(自然科學(xué)版);1987年02期

8 李文輝;;鞍點(diǎn)的穩(wěn)定性分析[J];沈陽化工學(xué)院學(xué)報(bào);1992年03期

9 遲曉恒;三次系統(tǒng)第一第二鞍點(diǎn)量計(jì)算公式[J];東北師大學(xué)報(bào)(自然科學(xué)版);1995年01期

10 謝佐恒;動態(tài)系統(tǒng)中鞍點(diǎn)處的熵與分維[J];系統(tǒng)科學(xué)與數(shù)學(xué);1996年01期

相關(guān)會議論文 前2條

1 朱懷念;植t熀，

本文編號：1529997