偏微分方程保结构算法构造及分析

发布时间：2019-01-17 12:02

【摘要】：随着科学的快速发展,越来越多的物理、化学和生物过程可以用非线性发展方程或者电磁场方程来描述.在许多情况下这些系统是保守的,因此如何为这些系统设计高效且能保持系统守恒量的算法一直是计算科学的研究热点.本博士学位论文致力于几类非线性发展方程的局部保结构算法(多辛算法,局部能量、动量守恒算法)和三维Maxwell方程高效保结构算法的研究及其数值分析.主要研究成果包括：1)辛算法以及全局保能量、动量等传统保结构算法在处理偏微分方程时,除了要考察方程是否是保守系统外,还必须考虑边界条件是否适当,只有在适当的边界条件下才可以使用这些保结构算法.为了增加保结构算法的适用范围,我们以耦合非线性Schrodinger系统、Boussinesq系统和Klein-Gordon-Schrodinger系统为研究对象,为其构造了一系列的局部保结构算法,包括多辛算法,局部能量、动量守恒算法.这些局部保结构算法能在任何时、空区域上保持离散的局部守恒律.当边界条件适当时,这些局部保结构算法自然就是全局保结构算法,反之不然.除此之外,我们还分析了其中一些局部保结构算法的非线性稳定性和收敛性.数值实验表明局部保结构算法不但能得到较好的数值解,而且确实能保持系统的局部守恒律和全局守恒律.通过与文献中已有的数值算法比较,展现了本文算法的优点.2)三维Maxwell方程具有双哈密尔顿结构.应用谱方法离散哈密尔顿函数和哈密尔顿算子,再对所得到的有限维哈密尔顿系统用平均向量场方法求积,从而我们得到求解三维Maxwell方程的两个格式(下称“AVF(2)”和"AVF(4)"). AVF(2)和AVF(4)自动保持系统的两个哈密尔顿.我们证明了AVF(2)和AVF(4)保持离散的能量、动量和散度.数值色散分析表明它们是无条件稳定的且无耗散的.严格的误差分析表明AVF(2)和AVF(4)分别在时问方向具有二阶和四阶收敛性,在空间方向都具有谱精度.数值实验很好地证实了理论分析结果.3)AVF(2)和AVF(4)是通过直接离散Maxwell方程得到的,计算时编程实现比较复杂.为了设计更高效的保能量算法,我们利用指数算子分裂和组合的技巧分别得到逼近Maxwell方程的时间二阶和四阶分裂方法.这些分裂模型的每个子问题都是哈密尔顿系统且和原问题具有相同的哈密尔顿函数.对每个子问题,分别应用谱方法离散哈密尔顿函数及哈密尔顿算子,进而应用平均向量场方法求积所得的有限维哈密尔顿系统,从而得到时间二阶和四阶分裂格式(下称"S-AVF(2)"和"S-AVF(4)").我们证明了S-AVF(2)和S-AVF(4)能同时保持4个离散能量且是无条件稳定的.此外,利用离散的傅里叶变换,我们还可以将所得格式写成显式形式.借助于能量分析方法,我们得到了S-AVF(2)和S-AVF(4)的误差估计.数值实验证实了理论分析结果.
[Abstract]:With the rapid development of science, more and more physical, chemical and biological processes can be described by nonlinear evolution equations or electromagnetic field equations. In many cases, these systems are conservative, so how to design efficient and conservative algorithms for these systems has been a hot topic in computational science. This dissertation is devoted to the study and numerical analysis of locally conserved algorithms for nonlinear evolution equations (multi-symplectic algorithm, local energy, momentum conservation algorithm) and three-dimensional Maxwell equation efficient structure-conserving algorithm. The main research results are as follows: 1) in dealing with partial differential equations, the symplectic algorithm and the traditional conserved structure algorithms such as global conserving energy and momentum must consider not only whether the equation is a conservative system, but also whether the boundary conditions are appropriate. These algorithms can only be used under proper boundary conditions. In order to increase the applicability of structure-preserving algorithm, we construct a series of local structure-preserving algorithms for coupled nonlinear Schrodinger system, Boussinesq system and Klein-Gordon-Schrodinger system, including multi-symplectic algorithm, local energy, and local energy. Momentum conservation algorithm. These locally structure-preserving algorithms can maintain discrete local conservation laws in the space domain at any time. When the boundary conditions are appropriate, these local structure preserving algorithms are naturally global preserving algorithms, otherwise they are not. In addition, we also analyze the nonlinear stability and convergence of some locally conserved algorithms. Numerical experiments show that the locally conserved structure algorithm can not only obtain a good numerical solution, but also preserve the local conservation law and global conservation law of the system. The advantages of the proposed algorithm are demonstrated by comparing with the existing numerical algorithms in the literature. 2) the three-dimensional Maxwell equation has a double Hamiltonian structure. The Hamiltonian function and Hamiltonian operator are discretized by spectral method, and then the finite dimensional Hamiltonian system is quadrature by the mean vector field method. Thus, we obtain two schemes for solving three dimensional Maxwell equations (AVF (2) and AVF (4). AVF (2) and AVF (4) automatically hold the two Hamiltonian systems. We prove that AVF (2) and AVF (4) maintain discrete energy, momentum and divergence. Numerical dispersion analysis shows that they are unconditionally stable and nondissipative. The strict error analysis shows that AVF (2) and AVF (4) have the second order and fourth order convergence respectively in the time direction and the spectral accuracy in the space direction. The results of theoretical analysis are well confirmed by numerical experiments. 3) AVF (2) and AVF (4) are obtained by direct discrete Maxwell equations. In order to design a more efficient energy preserving algorithm, we obtain time second order and fourth order splitting methods for approximating Maxwell equations by using the techniques of exponential operator splitting and combination, respectively. Each subproblem of these splitting models is a Hamiltonian system and has the same Hamiltonian function as the original problem. For each subproblem, the Hamiltonian function and Hamiltonian operator are discretized by spectral method, and then the finite dimensional Hamiltonian system is obtained by means of the average vector field method. The second and fourth order splitting schemes are obtained (S-AVF (2) and S-AVF (4). We prove that S-AVF (2) and S-AVF (4) can maintain four discrete energies at the same time and are unconditionally stable. In addition, using discrete Fourier transform, we can write the obtained scheme into explicit form. By means of energy analysis, we obtain the error estimates of S-AVF (2) and S-AVF (4). The results of theoretical analysis are confirmed by numerical experiments.
【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】：O175.2

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