哈密顿系统一些保结构算法的构造和分析

发布时间：2018-06-20 21:36

本文选题：哈密顿系统 + 保能量方法　；参考：《南京师范大学》2016年博士论文

【摘要】：一切真实的,耗散可忽略不计的物理过程都可以用哈密顿系统进行描述.哈密顿系统有两个最重要的性质,一个是辛结构,另一个就是能量守恒.正确计算哈密顿系统非常重要.近年来,能够保持哈密顿系统辛结构或能量的保结构方法已经得到了很大的发展.本文讨论哈密顿系统一些保结构算法的构造和分析,主要研究成果如下：I.近几年,人们构造了等离子物理中洛伦兹力系统的保结构格式,比如保体积格式和保辛格式.然而这些格式都不能保持系统能量.我们把洛伦兹力系统写为一个非典则的哈密顿系统,然后利用Boole离散线积分方法进行求解,得到洛伦兹力系统的一个新的格式.该方法可以保持系统哈密顿能量达到机器精度.II.我们研究如何利用二,三和四阶AVF方法求解哈密顿偏微分方程.对非线性薛定谔方程,空间用Fourier拟谱方法半离散,时间用三个AVF方法进行离散,得到该方程三个不同精度的AVF格式.我们用数值实验验证了这三个格式的精度和保能量守恒特性.III.基于根树和B-级数理论,我们给出了5阶树的带入规则的具体公式.利用新得到的带入规则,我们把二阶AVF方法提高到高阶精度,给出了一个新的AVF方法.我们证明了,新方法具有6阶精度,并且可以保持哈密顿系统能量.我们利用六阶AVF方法求解非线性哈密顿系统,并测试了其精度和能量守恒特性.IV.在哈密顿偏微分方程保结构算法框架下,我们研究了基于系统弱形式的空间离散方法.首先,空间用有限元法或谱元法对偏微分方程进行半离散,把得到的常微分方程组写成一个哈密顿系统.然后,我们用一个保结构方法对这个常微分哈密顿系统进行求解,得到一个全离散保结构格式.我们用这个方法对一维非线性薛定谔(NLS)方程进行求解,其中空间用Legendre谱元法,时间用AVF方法,得到一个新的保能量方法.同样对一维NLS方程,我们在空间用Galerkin有限元方法,时间用Crank-Nicolson格式离散,则得到一个同时保能量和质量的格式.对二维NLS方程,空间用Galerkin谱元法,时间用Crank-Nicolson格式离散,得到一个同时保能量和质量的格式.而对Klein-Gordon-Schrodinger方程空间用Galerkin方法,时间用辛Stomer-Verlet方法离散,得到一个显式辛格式.对自旋为1的Bose-Einstein凝聚态(BEC)中耦合Gross-Pitaevskii(GP)方程,空间用Galerkin方法,时间用隐中点辛格式离散,则得到一个新的同时保系统辛结构,质量和磁场强度的格式.对自旋轨道耦合的BEC中耦合GP方程离散,空间用Galerkin方法,时间用Crank-Nicolson格式,得到的新格式可以同时保能量和质量.我们做了数值实验验证理论结果.
[Abstract]:All real, dissipative and negligible physical processes can be described as Hamiltonian systems. Hamiltonian system has two most important properties, one is symplectic structure, the other is energy conservation. It is very important to calculate the Hamiltonian system correctly. In recent years, the conserved structure method which can maintain the symplectic structure or energy of Hamiltonian system has been greatly developed. In this paper, we discuss the construction and analysis of some structure-preserving algorithms for Hamiltonian systems. The main research results are as follows: I. In recent years, the conformal schemes of Lorentz force system in plasma physics, such as volume preserving scheme and symplectic scheme, have been constructed. However, none of these formats can maintain system energy. We write the Lorentz force system as a Hamiltonian system of SARS, then solve it by Boole discrete line integral method, and obtain a new scheme of Lorentz force system. This method can keep the Hamiltonian energy of the system to the accuracy of the machine. II. We study how to solve Hamiltonian partial differential equations by using the second, third and fourth order AVF methods. For the nonlinear Schrodinger equation, the Fourier pseudospectral method is used in space and the time is discretized by three AVF methods, and three AVF schemes with different accuracy are obtained. The accuracy and energy conservation characteristics of the three schemes are verified by numerical experiments. Based on the theory of root tree and B-series, we give the formula of introducing rule of order 5 tree. By using the new bring rule, we improve the second order AVF method to higher order precision, and give a new AVF method. It is proved that the new method has 6 order accuracy and can maintain the energy of Hamiltonian system. We use the sixth order AVF method to solve the nonlinear Hamiltonian system and test its accuracy and energy conservation. In the framework of Hamiltonian partial differential equation preserving structure algorithm, we study the spatial discretization method based on the weak form of the system. Firstly, the partial differential equations are semi-discretized by finite element method or spectral element method, and the resulting ordinary differential equations are written as a Hamiltonian system. Then we solve the ordinary differential Hamiltonian system by a structure-preserving method and obtain a fully discrete-time preserving structure scheme. We use this method to solve the one-dimensional nonlinear Schrodinger NLSs equation. A new energy preserving method is obtained by using Legendre spectral element method in space and AVF method in time. Similarly, for one-dimensional NLS equations, we use Galerkin finite element method in space, and discretize time by Crank-Nicolson scheme, then we obtain a scheme that preserves both energy and mass. For two-dimensional NLS equations, the Galerkin spectral element method is used in space and the time is discretized by Crank-Nicolson scheme. The space of Klein-Gordon-Schrodinger equation is discretized by Galerkin method and the time is discretized by symplectic Stomer-Verlet method, and an explicit symplectic scheme is obtained. For the coupled Gross-Pitaevskii GPP equation in Bose-Einstein condensed matter (BECs) with spin 1, the Galerkin method is used in space and the time is discretized by the implicit midpoint symplectic scheme. A new scheme for simultaneous symplectic structure, mass and magnetic field intensity is obtained. The coupled GP equations in spin-orbit coupled bec are discretized. The Galerkin method is used in space and Crank-Nicolson scheme is used in time. The new scheme can preserve both energy and mass. We have done numerical experiments to verify the theoretical results.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.8

【相似文献】