波方程中一些新的能量守恒有限体积元方法

发布时间：2018-06-01 18:57

本文选题：两网格 + 有限体积元方法　；参考：《南京师范大学》2016年博士论文

【摘要】：在过去的几十年里,对于偏微分方程数值解的逼近,人们已经提出了各种各样的数值求解方法,如两网格方法,保结构数值方法等.这篇论文主要讨论了这些方法在某些偏微分方程中的应用.文章首先讨论了两网格有限体积元方法在非线性Sobolev方程中的应用,然后介绍了如何利用保结构方法,如离散变分导数方法及哈密尔顿边界值方法等构造保积分的数值方法.首先,我们针对非线性Sobolev方程提出了一类两网格有限体积元方法.该方法是一种基于一个粗网格空间及一个细网格空间的数值方法.我们首先在粗网格上以步长H进行非线性迭代得到原方程的一个近似解UH,然后在细网格上以步长h进行线性求解得到方程的数值解.我们对两网格有限体元方法作了H1范先验误差估计,当h=O(H3|ln H|)时,两网格有限体积元方法的收敛阶为O(H3|ln H|)理论结果表明两网格有限体积元方法相对标准有限体积元方法更为有效.其次,我们针对哈密尔顿偏微分方程提出了一类守恒有限体积元格式.该方法是一类基于离散变分导数方法及有限体积元方法的数值方法.在这一章,我们首先介绍了保能量格式及保动量格式的构造方法,然后对其守恒性及稳定性作了分析.数值实验表明保能量格式及保动量格式可以保证离散不变量的精确守恒.但是,保能量方法比保动量方法有更高的精度及更好的稳定性.最后,我们针对哈密尔顿偏微分方程提出了一类高精度保能量方法.该方法在空间和时间方向分别采用拟谱方法和哈密尔顿边界值方法.数值实验表明数值格式在空间可以达到谱精度,而在时间方向分别达到二阶和四阶精度.此外,该方法可以使离散质量和能量的守恒性达到机器精度.
[Abstract]:In the past decades, a variety of numerical solutions to partial differential equations (PDEs) have been proposed, such as two-grid method, structure-preserving numerical method and so on. This paper mainly discusses the application of these methods in some partial differential equations. In this paper, the application of two-grid finite volume element method to nonlinear Sobolev equation is discussed, and then the method of conserving structure, such as discrete variational derivative method and Hamiltonian boundary value method, is introduced. Firstly, we propose a two-grid finite volume element method for nonlinear Sobolev equations. This method is based on a coarse mesh space and a fine grid space. We first obtain an approximate solution of the original equation UH by nonlinear iteration of step size H on a coarse mesh, and then obtain the numerical solution of the equation by a linear solution of step h on a fine mesh. We estimate the H1-norm prior error of the two-grid finite volume element method. When h=O(H3 ln H), the convergence order of the two-grid finite volume element method is O(H3 ln H). The results show that the two-grid finite volume element method is more effective than the standard finite volume element method. Secondly, we propose a class of conservative finite volume element schemes for Hamiltonian partial differential equations. This method is a kind of numerical method based on discrete variational derivative method and finite volume element method. In this chapter, we first introduce the construction method of energy preserving scheme and momentum preserving scheme, and then analyze their conservation and stability. Numerical experiments show that the energy-conserving scheme and the momentum preserving scheme can guarantee the exact conservation of discrete invariants. However, the energy preserving method has higher accuracy and better stability than the momentum preserving method. Finally, we propose a high accuracy energy preserving method for Hamiltonian partial differential equations. The pseudospectral method and the Hamiltonian boundary value method are used in the space and time directions respectively. Numerical experiments show that the spectral accuracy of the scheme can be achieved in space and the second order and the fourth order in the time direction, respectively. In addition, the conservation of discrete mass and energy can reach the accuracy of the machine.
【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】：O241.82

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