KGS方程保能量算法的设计与实现

发布时间：2018-03-13 10:12

本文选题：哈密顿系统　切入点：KGS方程　出处：《南京师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：z1合Klein-Gordon-Schrodinger(KGS)方程是一类重要偏微分方程,在量子场论中有非常重要的应用.能量守恒是该方程本质特征之一.本文致力于发展KGS方程的保能量数值算法.近年来,常微分方程的能量守恒算法非常热门,提出了许多如平均向量场方法和哈密顿边界值方法之类的新方法.本文将利用这些常微分方程的新的能量守恒算法来构造偏微分方程的数值算法.我们首先给出KGS方程的无穷维哈密顿系统的形式,并得到了相应的性质.空间离散采用离散奇异卷积方法,得到了半离散的常微分方程系统以及对应的Hamilton形式.在时间离散上,我们分别利用平均向量场方法,有限差分方法和哈密顿边界值方法离散得到的Hamilton系统,给出了 KGS方程的多个全离散的数值格式.理论上我们严格证明这些新格式满足能量守恒定律.最后,数值实验结果验证了理论分析,并说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.
【学位授予单位】：南京师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.2

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