本文選題：哈密頓系統(tǒng)　切入點(diǎn)：KGS方程　出處：《南京師范大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：z1合Klein-Gordon-Schrodinger(KGS)方程是一類重要偏微分方程,在量子場(chǎng)論中有非常重要的應(yīng)用.能量守恒是該方程本質(zhì)特征之一.本文致力于發(fā)展KGS方程的保能量數(shù)值算法.近年來,常微分方程的能量守恒算法非常熱門,提出了許多如平均向量場(chǎng)方法和哈密頓邊界值方法之類的新方法.本文將利用這些常微分方程的新的能量守恒算法來構(gòu)造偏微分方程的數(shù)值算法.我們首先給出KGS方程的無窮維哈密頓系統(tǒng)的形式,并得到了相應(yīng)的性質(zhì).空間離散采用離散奇異卷積方法,得到了半離散的常微分方程系統(tǒng)以及對(duì)應(yīng)的Hamilton形式.在時(shí)間離散上,我們分別利用平均向量場(chǎng)方法,有限差分方法和哈密頓邊界值方法離散得到的Hamilton系統(tǒng),給出了 KGS方程的多個(gè)全離散的數(shù)值格式.理論上我們嚴(yán)格證明這些新格式滿足能量守恒定律.最后,數(shù)值實(shí)驗(yàn)結(jié)果驗(yàn)證了理論分析,并說明了本文提出的保能量方法的有效性.
[Abstract]:The Z1 and Klein-Gordon-Schrodingern KGSequation is a kind of important partial differential equation, which has very important applications in quantum field theory. Energy conservation is one of the essential characteristics of the equation. This paper is devoted to the development of energy-preserving numerical algorithm for KGS equation. Energy conservation algorithms for ordinary differential equations are very popular. In this paper, many new methods such as mean vector field method and Hamiltonian boundary value method are proposed. In this paper, the new energy conservation algorithms of these ordinary differential equations are used to construct the numerical algorithms for partial differential equations. The form of infinite dimensional Hamiltonian systems for KGS equations, The corresponding properties are obtained. The discrete singular convolution method is used in space discretization, and the semi-discrete ordinary differential equation system and its corresponding Hamilton form are obtained. In time discretization, we use the mean vector field method, respectively. The finite difference method and Hamiltonian boundary value method are used to discretize the Hamilton system. Several fully discrete numerical schemes of KGS equation are given. In theory, we strictly prove that these new schemes satisfy the law of conservation of energy. The numerical results verify the theoretical analysis and illustrate the effectiveness of the energy conservation method proposed in this paper.
【學(xué)位授予單位】：南京師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175.2

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本文編號(hào)：1605929