几类结构偏微分方程的平均向量场方法

发布时间：2018-03-09 00:34

本文选题：平均向量场方法　切入点：保能量方法　出处：《海南大学》2017年硕士论文　论文类型：学位论文

【摘要】：在数学物理的研究中,许多偏微分方程可以表示成哈密尔顿系统的辛结构形式或多辛结构形式,如复修正KdV方程,KGS方程,耦合薛定谔Boussinesq方程,BBM方程等.这些偏微分方程具有能量守恒特性,在数值计算中,尽量构造其保持能量守恒特性的数值算法在数值模拟微分方程的行为中具有重要的意义.哈密尔顿系统的辛几何算法在1984年首次由冯康院士及其研究小组提出,在辛几何算法的基础之上Bridges和Reich等人在1997年构造出了多辛算法.以上两种算法具有长时间精确计算的优点,但不足之处在于这两种算法只能近似保持方程的能量守恒.近年来,许多学者构造了保持这些偏微分方程的保能量算法.在1999年保持哈密尔顿系统能量守恒的平均向量场方法被Quispel和McLachlan等人提出了,王雨顺利用平均向量场方法构造了多辛结构偏微分方程的保能量方法.本文利用平均向量场方法和拟谱方法构造耦合和复偏微分方程的高阶保能量格式和多辛整体保能量格式,利用这些新格式对这些偏微分方程进行数值模拟,并对数值结果进行分析.在第一章,时间上利用四阶平均向量场方法,空间上利用傅里叶拟谱方法对复修正KdV方程进行离散,构造了复修正KdV方程的高阶保能量格式,利用构造的高阶保能量格式数值模拟孤立波的演化行为.数值结果表明复修正KdV方程的高阶保能量格式可以很好地模拟孤立波的演化行为,并且可以精确地保持了方程的离散能量.在第二章,对于求解耦合偏微分方程,我们在时间上利用四阶平均向量场方法,空间上利用傅里叶拟谱方法对KGS方程和CSBE方程构造高阶保能量格式,并模拟孤立波的演化行为.数值结果表明KGS方程和CSBE方程的高阶保能量格式可以很好地达到预期效果.在第三章,利用二阶平均向量场方法,拟谱方法对具有多辛结构的一维偏微分方程:BBM方程和复修正KdV方程构造多辛整体保能量格式,利用构造的多辛整体保能量格式数值模拟孤立波的演化行为,并证明了新格式能保方程离散的整体能量守恒特性.数值结果表明BBM方程和复修正KdV的多辛整体保能量格式可以很好地模拟孤立波的演化行为,并且可以精确地保持BBM方程和复修正KdV的离散整体能量守恒特性.在整体保能量守恒特性方面,BBM方程和复修正KdV方程新构造的格式比已有的经典的多辛格式更加精确,计算时间也比高阶平均向量场方法大大缩短.在第四章,对具有多辛结构的二维偏微分方程:ZK方程构造多辛整体保能量格式,利用构造的多辛整体保能量格式数值模拟孤立波的演化行为.数值结果表明整体保能量方法不仅可以长时间的模拟孤立波的演化行为,而且也可以精确地保持ZK方程的离散能量守恒.
[Abstract]:In the study of mathematical physics, many partial differential equations can be expressed as symplectic structure of Hamiltonian system or multi-symplectic structure, such as complex modified KdV equation and KGS equation. Coupled Schrodinger Boussinesq equation and so on. These partial differential equations have the characteristic of energy conservation. It is of great significance to construct a numerical algorithm for preserving the conservation of energy in numerical simulation of differential equations. In 1984, the symplectic geometry algorithm of Hamilton system was first proposed by academician Feng Kang and his research group. On the basis of symplectic geometry algorithm, Bridges and Reich constructed multi-symplectic algorithm in 1997. These two algorithms have the advantage of long time accurate calculation, but the disadvantage of these two algorithms is that they can only keep the energy conservation of equation approximately in recent years. In 1999, the mean vector field method for preserving the energy conservation of Hamiltonian systems was proposed by Quispel and McLachlan et al. Wang Yu-shun constructed the energy-preserving method for multi-symplectic partial differential equations using the mean vector field method. In this paper, the high-order energy-preserving schemes and the multi-symplectic global energy-preserving schemes for coupled and complex partial differential equations are constructed by means of the mean vector field method and the pseudo-spectral method. These partial differential equations are numerically simulated by these new schemes, and the numerical results are analyzed. In chapter 1, the fourth order average vector field method is used to discretize the complex modified KdV equation in space by Fourier pseudo-spectral method. A high order energy preserving scheme for complex modified KdV equations is constructed. The evolution behavior of solitary waves is numerically simulated by using the high-order energy-conserving scheme. The numerical results show that the high-order energy-conserving scheme of complex modified KdV equation can well simulate the evolution behavior of solitary waves. In chapter 2, we use the fourth-order average vector field method to solve coupled partial differential equations. In this paper, Fourier pseudospectral method is used to construct high-order energy-conserving schemes for KGS equation and CSBE equation. The numerical results show that the high order energy preserving schemes of KGS equation and CSBE equation can achieve the desired results. In chapter 3, the second order mean vector field method is used. The pseudospectral method is used to construct multi-symplectic global energy-conserving schemes for one-dimensional partial differential equations with multi-symplectic structure: BBM equation and complex modified KdV equation. The evolution behavior of solitary waves is numerically simulated by using the constructed multi-symplectic global energy-conserving schemes. It is proved that the new scheme can preserve the global energy conservation property of the discrete equation, and the numerical results show that the BBM equation and the multi-symplectic global energy preserving scheme of complex modified KdV can well simulate the evolution behavior of solitary waves. Moreover, the discrete global energy conservation properties of BBM equation and complex modified KdV can be accurately preserved. The new schemes of BBM equation and complex modified KdV equation are more accurate than the classical multi-symplectic schemes. The computational time is also much shorter than that of the high-order average vector field method. In Chapter 4th, the multi-symplectic global energy-preserving scheme is constructed for the 2-D partial differential equation with multi-symplectic structure: ZK equation. The evolution behavior of solitary waves is numerically simulated by using the multi-symplectic global energy preserving scheme. The numerical results show that the global energy conservation method can not only simulate the evolution behavior of solitary waves for a long time. Moreover, the discrete energy conservation of ZK equation can be kept accurately.
【学位授予单位】：海南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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