【摘要】：許多偏微分方程能被寫成一個(gè)多辛哈密頓系統(tǒng),例如：sine-Gordon方程、非線性薛定諤方程、KdV方程、Camassa-Holm方程、麥克斯韋方程、非線性波動(dòng)方程等.多辛哈密頓系統(tǒng)有三個(gè)局部守恒律,即多辛守恒律,局部能量守恒律和局部動(dòng)量守恒律.如何構(gòu)造保其中一個(gè)或多個(gè)守恒律的數(shù)值算法是非常有意義的.多辛守恒律是多辛哈密頓系統(tǒng)的一個(gè)重要的幾何性質(zhì).在過去的一、二十年里,人們發(fā)展了大量的保離散多辛守恒律的數(shù)值方法.在本文中,我們進(jìn)一步研究了Kawahara方程的多辛Fourier以譜方法,并建立了譜微分矩陣與離散Fourier變換的關(guān)系,從而將快速Fourier算法引入到保結(jié)構(gòu)算法的計(jì)算中.能量守恒是力學(xué)系統(tǒng)中的一個(gè)關(guān)鍵的性質(zhì),它在解的性質(zhì)的研究中扮演著重要的角色.在一些例子中,能量守恒性質(zhì)被直接用來證明數(shù)值方法的穩(wěn)定性.能量是很多發(fā)展方程的最重要的不變量,因此保能量方法引起了很多科研工作者的興趣,并得到了快速的發(fā)展.在本文中,我們?cè)诳臻g上用小波配置方法離散,在時(shí)間上用平均向量場(chǎng)方法離散,從而為一般多辛形式的哈密頓系統(tǒng)構(gòu)造了一個(gè)保全局能量的方法.我們還提出了一個(gè)保局部能量的方法.除了能量守恒律以外,多辛哈密頓系統(tǒng)還擁有動(dòng)量守恒律.動(dòng)量守恒律也是物理中的一個(gè)重要的不變量,但是在文獻(xiàn)中很少有這方面的研究.在本文中,我們給出了一個(gè)保一般多辛形式的哈密頓系統(tǒng)的局部動(dòng)量的方法.值得注意的是,局部保能量方法和局部保動(dòng)量方法與邊界條件無關(guān),它們能被應(yīng)用于一大類守恒型的偏微分方程.在本文中,我們還特別為耦合薛定諤方程構(gòu)造了一個(gè)守恒的Fourier擬譜算法.我們證明了一個(gè)重要的結(jié)果,即由Fourier以譜方法誘導(dǎo)的半范等價(jià)于由有限差分方法誘導(dǎo)的半范.由于這個(gè)結(jié)果以及數(shù)值方法保離散的質(zhì)量和能量守恒的性質(zhì),我們證明Fourier擬譜解在最大模意義下是有界的.從而,我們證明這個(gè)格式是唯一可解的,并且是無條件穩(wěn)定的.僅在原方程的解滿足一定的正則性的條件下,我們分析了算法在L2模意義下的誤差估計(jì),這是保結(jié)構(gòu)擬譜方法的第一個(gè)收斂性證明.數(shù)值實(shí)驗(yàn)印證了理論分析.
[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.
【學(xué)位授予單位】：南京師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類號(hào)】：O241.82

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