本文選題：擴(kuò)散方程 + 橢圓界面問題��； 參考：《中國(guó)科學(xué)技術(shù)大學(xué)》2016年博士論文

【摘要】：本文主要研究?jī)?nèi)容分為兩個(gè)方面：(1)橢圓界面問題保持單調(diào)性的有限體積格式；(2)適用于結(jié)構(gòu)及非結(jié)構(gòu)網(wǎng)格的守恒強(qiáng)制保正修復(fù)算法.首先,考慮橢圓界面問題,采用貼體四邊形網(wǎng)格對(duì)計(jì)算區(qū)域進(jìn)行剖分,用兩種方法構(gòu)造物質(zhì)界面上的離散法向通量,并給出了間斷界面兩側(cè)物理量的計(jì)算公式.可以證明兩種方法均具有單調(diào)性.數(shù)值結(jié)果表明,方法二能夠有效捕捉界面的間斷信息,在矩形網(wǎng)格和隨機(jī)網(wǎng)格上未知量的L2范數(shù)和L∞范數(shù)具有二階精度,法向通量的L2范數(shù)具有一階精度.其次,針對(duì)常用擴(kuò)散格式(如Kershaw格式,九點(diǎn)格式等)在扭曲結(jié)構(gòu)網(wǎng)格上數(shù)值解出負(fù)的現(xiàn)象,利用結(jié)構(gòu)網(wǎng)格的優(yōu)勢(shì),采用維數(shù)分裂的思想構(gòu)造了守恒的強(qiáng)制遇負(fù)置零算法(CENZ),該方法是對(duì)傳統(tǒng)遇負(fù)置零(ENZ)方法的改進(jìn).它不僅能夠使修復(fù)后的數(shù)值解非負(fù),而且保證總能量和局部通量守恒.此外,該算法可直接用于修復(fù)任何不具有保極值原理特性的有限體積格式的數(shù)值解.數(shù)值實(shí)驗(yàn)表明,該算法在數(shù)值解的精度和守恒性方面優(yōu)于其它修復(fù)算法.最后,針對(duì)非結(jié)構(gòu)網(wǎng)格上擴(kuò)散格式計(jì)算出負(fù)的問題,構(gòu)造了適用于一般網(wǎng)格的守恒強(qiáng)制遇負(fù)置零算法(GCENZ)該算法使得修復(fù)后的數(shù)值解非負(fù),且滿足總能量守恒及局部通量守恒.分析及數(shù)值結(jié)果表明,所提出的GCENZ算法可以對(duì)數(shù)值解出負(fù)現(xiàn)象進(jìn)行守恒的非負(fù)修正,守恒誤差明顯低于ENZ算法.
[Abstract]:The main contents of this paper are divided into two parts: 1) the finite volume scheme for elliptic interface problems with monotonicity is suitable for the conservation and forced orthodontic restoration of structural and unstructured meshes. Firstly, considering the elliptic interface problem, a body-fitted quadrilateral mesh is used to divide the calculation area, and two methods are used to construct the discrete normal flux on the material interface, and the formulas for calculating the physical quantities on both sides of the discontinuous interface are given. It can be proved that both methods have monotonicity. The numerical results show that the second method can effectively capture the discontinuous information of the interface. The L _ 2 norm and L _ 鈭，

本文編號(hào)：1968588