本文選題：哈密頓系統(tǒng) + 保能量方法��； 參考：《南京師范大學(xué)》2016年博士論文

【摘要】：一切真實的,耗散可忽略不計的物理過程都可以用哈密頓系統(tǒng)進行描述.哈密頓系統(tǒng)有兩個最重要的性質(zhì),一個是辛結(jié)構(gòu),另一個就是能量守恒.正確計算哈密頓系統(tǒng)非常重要.近年來,能夠保持哈密頓系統(tǒng)辛結(jié)構(gòu)或能量的保結(jié)構(gòu)方法已經(jīng)得到了很大的發(fā)展.本文討論哈密頓系統(tǒng)一些保結(jié)構(gòu)算法的構(gòu)造和分析,主要研究成果如下：I.近幾年,人們構(gòu)造了等離子物理中洛倫茲力系統(tǒng)的保結(jié)構(gòu)格式,比如保體積格式和保辛格式.然而這些格式都不能保持系統(tǒng)能量.我們把洛倫茲力系統(tǒng)寫為一個非典則的哈密頓系統(tǒng),然后利用Boole離散線積分方法進行求解,得到洛倫茲力系統(tǒng)的一個新的格式.該方法可以保持系統(tǒng)哈密頓能量達到機器精度.II.我們研究如何利用二,三和四階AVF方法求解哈密頓偏微分方程.對非線性薛定諤方程,空間用Fourier擬譜方法半離散,時間用三個AVF方法進行離散,得到該方程三個不同精度的AVF格式.我們用數(shù)值實驗驗證了這三個格式的精度和保能量守恒特性.III.基于根樹和B-級數(shù)理論,我們給出了5階樹的帶入規(guī)則的具體公式.利用新得到的帶入規(guī)則,我們把二階AVF方法提高到高階精度,給出了一個新的AVF方法.我們證明了,新方法具有6階精度,并且可以保持哈密頓系統(tǒng)能量.我們利用六階AVF方法求解非線性哈密頓系統(tǒng),并測試了其精度和能量守恒特性.IV.在哈密頓偏微分方程保結(jié)構(gòu)算法框架下,我們研究了基于系統(tǒng)弱形式的空間離散方法.首先,空間用有限元法或譜元法對偏微分方程進行半離散,把得到的常微分方程組寫成一個哈密頓系統(tǒng).然后,我們用一個保結(jié)構(gòu)方法對這個常微分哈密頓系統(tǒng)進行求解,得到一個全離散保結(jié)構(gòu)格式.我們用這個方法對一維非線性薛定諤(NLS)方程進行求解,其中空間用Legendre譜元法,時間用AVF方法,得到一個新的保能量方法.同樣對一維NLS方程,我們在空間用Galerkin有限元方法,時間用Crank-Nicolson格式離散,則得到一個同時保能量和質(zhì)量的格式.對二維NLS方程,空間用Galerkin譜元法,時間用Crank-Nicolson格式離散,得到一個同時保能量和質(zhì)量的格式.而對Klein-Gordon-Schrodinger方程空間用Galerkin方法,時間用辛Stomer-Verlet方法離散,得到一個顯式辛格式.對自旋為1的Bose-Einstein凝聚態(tài)(BEC)中耦合Gross-Pitaevskii(GP)方程,空間用Galerkin方法,時間用隱中點辛格式離散,則得到一個新的同時保系統(tǒng)辛結(jié)構(gòu),質(zhì)量和磁場強度的格式.對自旋軌道耦合的BEC中耦合GP方程離散,空間用Galerkin方法,時間用Crank-Nicolson格式,得到的新格式可以同時保能量和質(zhì)量.我們做了數(shù)值實驗驗證理論結(jié)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.
【學(xué)位授予單位】：南京師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O241.8

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