非线性Klein-Gordon-Zakharov方程组的多辛算法

【作者】 张星；

【导师】 单双荣；

【作者基本信息】 华侨大学 ， 基础数学， 2009， 硕士

【摘要】 一切耗散效应可以忽略不计的物理过程都可表示成能够保持辛几何结构不变的哈密尔顿系统的形式,它在自然界中具有普适性,也就是说大多数孤子方程都可以表示成哈密尔顿形式.现代数值计算的基本原则是尽可能保持原问题的本质特征.因此,研究保持哈密尔顿系统的辛几何结构特征的数值方法是必然的.本论文主要讨论无穷维Hamilton系统的多辛几何算法.与辛几何算法的主要差别是,多辛算法不仅在一定的边界条件下保持系统的离散空间上的辛形式之和,而且能够保持局部的辛形式,从而多辛几何算法更多的是体现在系统局部的守恒性质,更能体现系统的本质特征.本文研究的无穷维Hamilton系统中Klein-Gordon-Zakharov (简称KGZ)方程组是一个重要的模型,它是由一个Klein-Gordon方程和一个Zakharov方程耦合而成.我们通过对KGZ方程组作正则变换后,得到了它的一个多辛方程组及其几个相关守恒律.然后用Gauss-Legendre Runge-Kutta方法对此多辛方程组离散,得到了KGZ方程组的多辛格式,证明了该格式具有离散形式的多辛守恒律.对中点格式,通过消去中间变量得到了与多辛格式等价的多辛Preissman格式.我们通过大量数值实验验证了所构造的多辛格式的有效性和长时间的数值稳定性,同时多辛格式还能很好地模拟原孤立波的波形,说明我们的理论分析是正确的.另外,本文还通过对空间和时间方向分别用Fourier拟谱方法和中点方法离散KGZ方程组的多辛方程组,得到了非线性KGZ方程组的多辛Fourier拟谱格式.我们通过数值实验证明了该格式的有效性.数值结果表明多辛Fourier拟谱格式是正确可行的.更多还原

【Abstract】 All the physical courses whose dissipative effects are negligible can be expressed as Hamiltonian systems which preserve energy conservation and symplectic geometric structure. The Hamiltonian system is universal in the nature, in other words, most soliton equations can be written into Hamiltonian formalism. The basic principle of modern numerical computation is to preserve the intrincal character of the original problems. Therefore, it is necessary to study numerical methods which preserve the symplectic structure of the Hamiltonian system. This thesis illustrates the multi-symplectic algorithms for infinite dimensional Hamiltonian systems. The multi-symplectic algorithms conserve not only the sum of symplectic forms on discrete space under appropriate boundary conditions, but also symplectic forms in local representations. Therefore, multi-symplectic algorithms have remarkable local properties, and describe the system more essentially.In the dissertation, Klein-Gordon-Zakharov(KGZ) equations are important model of infinite dimensional Hamiltonian systems in physics, which are coupled by the Klein-Gordon equation and Zakharov equation. Multi-symplectic equations for KGZ equations are presented which posses some conservation laws by canonical transformations. Using the Gauss-Legendre Runge-Kutta method to discrete multi-symplectic equations, we present the multi-symplectic schemes for the KGZ equations. They preserve discretic multi-symplectic conservation law exactly. We eliminate middle variables and amount to a single variable multi-symplectic schemes for mid-point schemes, and call them Preissman schemes. We perform a lot of experiments to show that the schemes preserve the multi-symplectic geometry structure exactly by satisfying the discrete multi-symplectic conservation law, and can simulate the original waves in a long time. Numerical experiments demonstrate the consistency between the theoretical analysis and the numerical results.In other words, in the dissertation we present multi-symplectic Fourier pseudo-spectral schemes for the KGZ equations, using Fourier pseudo-spectral method and mid-point method to discrete multi-symplectic equations in space and time directions, respectively. We also carry out some experiments to illustrate their validity. Numerical results show that multi-symplectic Fourier pseudo-spectral schemes are right.更多还原