波动方程和二维粘弹性方程的块中心差分方法
发布时间：2018-07-15 14:15
【摘要】：非线性波动方程和二维粘弹性方程是两大重要的微分方程,被广泛应用在物理学,经济学,自然学等许多领域。因而有很多学者对这两类方程做了大量的研究,但是他们只是得到了方程的解的近似值。在本文中,我借助块中心差分方法对二维粘弹性方程和非线性波动方程在剖分的网格上进行讨论,不仅可以得到它们的近似解和解的一阶导数的近似值,而且还得到了近似解的L2模误差估计,解得一阶导数的近似值具有超收敛性。首先,简要介绍这篇论文的所讨论问题的有关研究背景和基本的理论知识。其次,对有界区域上的二维粘弹性方程,在非等距剖分的网格上用块中心差分方法求得了其近似解和解的一阶导数的近似值。从理论上给出了近似解的L2模误差估计,并且得到了解的一阶导数的近似值具有超收敛性,时间上的精度也有了提高。同理,用块中心差分方法对有界区域上的非线性波动方程进行研究,在等距剖分的网格上,也得到了近似解和解的一阶导数的近似值,在理论上给出了近似解的L2模误差估计,并且得到了解的一阶导数的近似值具有超收敛性。最后,对全文进行小结,并提出以后要进一步研究的问题。
[Abstract]:Nonlinear wave equation and two-dimensional viscoelastic equation are two important differential equations, which are widely used in many fields such as physics, economics, nature and so on. Therefore, many scholars have done a lot of research on these two kinds of equations, but they only get the approximate value of the solutions of the equations. In this paper, I discuss two-dimensional viscoelastic equations and nonlinear wave equations on meshes by means of block central difference method. Moreover, the L2-norm error estimates of the approximate solution are obtained, and the approximate value of the first order derivative is superconvergent. Firstly, the research background and basic theoretical knowledge of the problems discussed in this paper are briefly introduced. Secondly, for the two-dimensional viscoelastic equations in the bounded region, the approximate solutions and the first order derivatives are obtained by using the block central difference method on the non-equidistant meshes. The L2-norm error estimates of the approximate solution are given theoretically, and the approximate value of the first derivative of the solution is superconvergent, and the accuracy of the time is improved. Similarly, the block central difference method is used to study the nonlinear wave equation in the bounded region. The approximation of the approximate solution and the first order derivative of the approximate solution is also obtained on the meshes of equidistant subdivision. In theory, the L2 norm error estimates of the approximate solution are given. The approximate value of the first order derivative of the solution is superconvergent. Finally, the paper summarizes the whole paper and puts forward some problems to be further studied.
【学位授予单位】：河南师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
本文编号：2124345
[Abstract]:Nonlinear wave equation and two-dimensional viscoelastic equation are two important differential equations, which are widely used in many fields such as physics, economics, nature and so on. Therefore, many scholars have done a lot of research on these two kinds of equations, but they only get the approximate value of the solutions of the equations. In this paper, I discuss two-dimensional viscoelastic equations and nonlinear wave equations on meshes by means of block central difference method. Moreover, the L2-norm error estimates of the approximate solution are obtained, and the approximate value of the first order derivative is superconvergent. Firstly, the research background and basic theoretical knowledge of the problems discussed in this paper are briefly introduced. Secondly, for the two-dimensional viscoelastic equations in the bounded region, the approximate solutions and the first order derivatives are obtained by using the block central difference method on the non-equidistant meshes. The L2-norm error estimates of the approximate solution are given theoretically, and the approximate value of the first derivative of the solution is superconvergent, and the accuracy of the time is improved. Similarly, the block central difference method is used to study the nonlinear wave equation in the bounded region. The approximation of the approximate solution and the first order derivative of the approximate solution is also obtained on the meshes of equidistant subdivision. In theory, the L2 norm error estimates of the approximate solution are given. The approximate value of the first order derivative of the solution is superconvergent. Finally, the paper summarizes the whole paper and puts forward some problems to be further studied.
【学位授予单位】：河南师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
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