多辛哈密顿系统中的一些新的保结构算法
发布时间：2018-12-13 19:12
【摘要】：许多偏微分方程能被写成一个多辛哈密顿系统,例如：sine-Gordon方程、非线性薛定谔方程、KdV方程、Camassa-Holm方程、麦克斯韦方程、非线性波动方程等.多辛哈密顿系统有三个局部守恒律,即多辛守恒律,局部能量守恒律和局部动量守恒律.如何构造保其中一个或多个守恒律的数值算法是非常有意义的.多辛守恒律是多辛哈密顿系统的一个重要的几何性质.在过去的一、二十年里,人们发展了大量的保离散多辛守恒律的数值方法.在本文中,我们进一步研究了Kawahara方程的多辛Fourier以谱方法,并建立了谱微分矩阵与离散Fourier变换的关系,从而将快速Fourier算法引入到保结构算法的计算中.能量守恒是力学系统中的一个关键的性质,它在解的性质的研究中扮演着重要的角色.在一些例子中,能量守恒性质被直接用来证明数值方法的稳定性.能量是很多发展方程的最重要的不变量,因此保能量方法引起了很多科研工作者的兴趣,并得到了快速的发展.在本文中,我们在空间上用小波配置方法离散,在时间上用平均向量场方法离散,从而为一般多辛形式的哈密顿系统构造了一个保全局能量的方法.我们还提出了一个保局部能量的方法.除了能量守恒律以外,多辛哈密顿系统还拥有动量守恒律.动量守恒律也是物理中的一个重要的不变量,但是在文献中很少有这方面的研究.在本文中,我们给出了一个保一般多辛形式的哈密顿系统的局部动量的方法.值得注意的是,局部保能量方法和局部保动量方法与边界条件无关,它们能被应用于一大类守恒型的偏微分方程.在本文中,我们还特别为耦合薛定谔方程构造了一个守恒的Fourier拟谱算法.我们证明了一个重要的结果,即由Fourier以谱方法诱导的半范等价于由有限差分方法诱导的半范.由于这个结果以及数值方法保离散的质量和能量守恒的性质,我们证明Fourier拟谱解在最大模意义下是有界的.从而,我们证明这个格式是唯一可解的,并且是无条件稳定的.仅在原方程的解满足一定的正则性的条件下,我们分析了算法在L2模意义下的误差估计,这是保结构拟谱方法的第一个收敛性证明.数值实验印证了理论分析.
[Abstract]:Many partial differential equations can be written into a multi-symplectic Hamiltonian system, such as sine-Gordon equation, nonlinear Schrodinger equation, KdV equation, Camassa-Holm equation, Maxwell equation, nonlinear wave equation and so on. There are three local conservation laws for multi-symplectic Hamiltonian systems, namely, multi-symplectic conservation laws, local energy conservation laws and local momentum conservation laws. It is very meaningful to construct a numerical algorithm that preserves one or more of the conservation laws. Multi-symplectic conservation law is an important geometric property of multi-symplectic Hamiltonian system. In the past ten or twenty years, a large number of numerical methods for preserving discrete multiple symplectic conservation laws have been developed. In this paper, we further study the multi-symplectic Fourier spectral method for Kawahara equation, and establish the relationship between spectral differential matrix and discrete Fourier transformation, so that the fast Fourier algorithm is introduced into the computation of the conserved structure algorithm. Energy conservation is a key property in mechanical systems, which plays an important role in the study of the properties of solutions. In some examples, the conservation of energy is directly used to prove the stability of numerical methods. Energy is the most important invariant of many evolution equations, so energy conservation method has attracted the interest of many researchers and has been developed rapidly. In this paper, we use wavelet collocation method in space and average vector field method in time to construct a global energy preserving method for general multi-symplectic Hamiltonian systems. We also propose a method for preserving local energy. In addition to energy conservation laws, the multi-symplectic Hamiltonian system also has momentum conservation laws. Momentum conservation law is also an important invariant in physics, but it is seldom studied in the literature. In this paper, we give a method for preserving the local momentum of a general multi-symplectic form Hamiltonian system. It is worth noting that the local energy preserving method and the local momentum preserving method are independent of boundary conditions and can be applied to a large class of conservative partial differential equations. In this paper, we also construct a conserved Fourier pseudospectral algorithm for coupled Schrodinger equation. We prove an important result that the semi-norm induced by Fourier by spectral method is equivalent to that induced by finite-difference method. Due to this result and the conservation of discrete mass and energy in numerical methods, we prove that the Fourier pseudospectral solution is bounded in the sense of maximum modulus. Thus, we prove that the scheme is solvable and unconditionally stable. Only if the solution of the original equation satisfies some regularity, we analyze the error estimate of the algorithm in the sense of L2-norm, which is the first proof of convergence of the structure-preserving pseudospectral method. Numerical experiments confirm the theoretical analysis.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.82
本文编号：2377072
[Abstract]:Many partial differential equations can be written into a multi-symplectic Hamiltonian system, such as sine-Gordon equation, nonlinear Schrodinger equation, KdV equation, Camassa-Holm equation, Maxwell equation, nonlinear wave equation and so on. There are three local conservation laws for multi-symplectic Hamiltonian systems, namely, multi-symplectic conservation laws, local energy conservation laws and local momentum conservation laws. It is very meaningful to construct a numerical algorithm that preserves one or more of the conservation laws. Multi-symplectic conservation law is an important geometric property of multi-symplectic Hamiltonian system. In the past ten or twenty years, a large number of numerical methods for preserving discrete multiple symplectic conservation laws have been developed. In this paper, we further study the multi-symplectic Fourier spectral method for Kawahara equation, and establish the relationship between spectral differential matrix and discrete Fourier transformation, so that the fast Fourier algorithm is introduced into the computation of the conserved structure algorithm. Energy conservation is a key property in mechanical systems, which plays an important role in the study of the properties of solutions. In some examples, the conservation of energy is directly used to prove the stability of numerical methods. Energy is the most important invariant of many evolution equations, so energy conservation method has attracted the interest of many researchers and has been developed rapidly. In this paper, we use wavelet collocation method in space and average vector field method in time to construct a global energy preserving method for general multi-symplectic Hamiltonian systems. We also propose a method for preserving local energy. In addition to energy conservation laws, the multi-symplectic Hamiltonian system also has momentum conservation laws. Momentum conservation law is also an important invariant in physics, but it is seldom studied in the literature. In this paper, we give a method for preserving the local momentum of a general multi-symplectic form Hamiltonian system. It is worth noting that the local energy preserving method and the local momentum preserving method are independent of boundary conditions and can be applied to a large class of conservative partial differential equations. In this paper, we also construct a conserved Fourier pseudospectral algorithm for coupled Schrodinger equation. We prove an important result that the semi-norm induced by Fourier by spectral method is equivalent to that induced by finite-difference method. Due to this result and the conservation of discrete mass and energy in numerical methods, we prove that the Fourier pseudospectral solution is bounded in the sense of maximum modulus. Thus, we prove that the scheme is solvable and unconditionally stable. Only if the solution of the original equation satisfies some regularity, we analyze the error estimate of the algorithm in the sense of L2-norm, which is the first proof of convergence of the structure-preserving pseudospectral method. Numerical experiments confirm the theoretical analysis.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.82
【相似文献】
相关期刊论文 前2条
1 方金英;印度新总理 帕·文·纳拉辛哈·拉奥[J];世界知识;1991年14期
2 ;[J];;年期
相关重要报纸文章 前10条
1 江亚平、王建刚;印中关系已相当成熟[N];人民日报;2003年
2 记者 熊昌义;印称已经制定出印巴会谈“路线图”[N];新华每日电讯;2003年
3 ;“强奸”言论不当印度高官引众怒[N];新华每日电讯;2013年
4 记者章亚东;斯总理希望与中国建立强有力经济关系[N];人民日报;2003年
5 本报驻印度记者 钱峰;印度也称要“先发制人”[N];人民日报;2003年
6 记者熊昌义;印外长对印中关系表示乐观[N];人民日报;2003年
7 记者吴迎春、张静宇、钱峰;印愿加强与我全面合作[N];人民日报;2003年
8 记者 缪毅容;加强合作促进共同发展[N];解放日报;2012年
9 记者 章亚东;猛虎组织再次拒绝和平建议[N];新华每日电讯;2003年
10 本报驻印度记者 张静宇;印向巴发出积极信息[N];人民日报;2003年
相关博士学位论文 前1条
1 龚跃政;多辛哈密顿系统中的一些新的保结构算法[D];南京师范大学;2015年
，本文编号：2377072
本文链接：https://www.wllwen.com/kejilunwen/yysx/2377072.html
