代写数学硕士论文《有关变系数椭圆型方程的紧差分格式探究》
目录
摘要5-6
Abstract6
目录7-9
第一章绪论9-13
1.1微分方程数值解法介绍9-10
1.2求解偏微分方程的有限差分法10-11
1.3紧差分格式11
1.4椭圆型偏微分方程差分逼近11-12
1.5本文主要工作12-13
第二章椭圆型方程的传统差分格式13-17
2.1一维椭圆型方程的传统差分格式13-14
2.2二维椭圆型方程的传统差分格式14-17
第三章一维椭圆型方程的紧差分格式17-27
3.1引言17
3.2紧差分格式建立与截断误差估计17-21
3.3差分解的先验估计21-24
3.4差分格式解的存在性、稳定性和收敛性24-25
3.5数值实验25-27
第四章二维椭圆型方程的紧差分格式27-47
4.1二维变系数椭圆型方程的紧差分格式27-37
4.1.1引言27-28
4.1.2紧差分格式建立与截断误差估计28-33
4.1.3差分解的先验估计33-35
4.1.4差分解的存在性、稳定性和收敛性35-36
4.1.5数值实验36-37
4.2一般二维椭圆型方程的紧差分格式37-47
4.2.1引言37-38
4.2.2紧差分格式建立与截断误差估计38-40
4.2.3差分解的先验估计40-46
4.2.4差分解的存在性、稳定性和收敛性46-47
第五章总结47-49
参考文献49-51
致谢51
【摘要】 许多物理和工程实际问题的数学模型都可以用椭圆型偏微分方程来描述,例如扩散问题、导体中电流分布问题和静电学问题。但是椭圆型方程边值问题的精确解只有在特殊情况下才能得到,因此必须采用数值方法求解这些问题。有限差分法是数值求解椭圆型方程的通用而有效的方法,目前已经有很多建立椭圆型方程差分逼近的方法。求解一维椭圆型方程的差分方法有直接差分法、积分插值法、变分差分法等,但是这些差分方法只能达到二阶精度,要提高精度需要加密网格,然而这将大大增加计算量。求解二维变系数椭圆型方程差分方法的精度一般比较低。因此构造求解变系数椭圆型方程的高精度差分格式具有理论和实际意义。本文主要内容如下：(1)简要介绍了有限差分法的基本思想和求解变系数椭圆型方程的一些传统有限差分格式。(2)对一维变系数椭圆型方程建立了一种具有四阶精度的紧差分格式,给出了差分解的先验估计,进而证明了差分格式解的存在唯一性、稳定性和收敛性,并通过数值实验验证了该差分格式的高精度。(3)对二维变系数椭圆型方程建立了具有四阶精度的紧差分格式,证明了差分解的存在唯一性、稳定性和收敛性,并且利用数值实验验证理论结果。数值实验结果表明本文建立的紧差分格式是一种高精度的、稳定且收敛的差分格式。
代写硕士论文-【Abstract】 Many mathematic models of physics and engineering practical problem can be described by elliptic partial differential equations, such as the diffusion problem, the current distribution problems in conductor and electrostatic problem. But the exact solutions of elliptic equation boundary value problem can be obtained only in exceptional condition. It is essential to solve these problems by numerical method. The finite difference method is a popular and effective numerical method for solving elliptic equation, a number of elliptic equation difference approximation schemes have been constructed up to now. The difference methods for solving one-dimensional elliptic equation are direct difference method, integral interpolation method, variation difference method and so on, but these methods can only reach second order accuracy, in order to improve the accuracy, we need to use fine grid, but this will increase the computation quantity greatly. The accuracy order of many difference schemes for solving two-dimensional elliptic equation with variable coefficients is relatively low. Therefore, constructing a high-accuracy difference scheme for solving elliptic equation with variable coefficients has theoretical and practical significance. The main contents are as follows.Mainly introduce the basic idea of finite difference method and some traditional finite difference schemes for solving elliptic equation with variable coefficients.Construct a fourth-order accuracy compact difference scheme for one-dimensional elliptic equation with variable coefficients, give the prior estimates of the difference solution, and prove the existence, uniqueness, stability and convergence of the difference solution. Then verify the high-accuracy of the difference scheme by numerical experiment.Construct fourth-order accuracy compact difference scheme for two-dimensional elliptic equation with variable coefficients and prove the existence, uniqueness, stability and convergence of the difference solution. Then verify the theoretical result by numerical experiment. Numerical results show that the compact difference scheme in this paper is a high-accuracy, stable and convergence difference scheme.
【关键词】 椭圆型方程； 紧差分格式； 稳定性； 收敛性；
【Key words】 elliptic equation； compact difference scheme； stability； convergence；