计算数学硕士论文——有关定常对流扩散方程的研究《定常对流扩散方程的一种新型差分格式》
摘要 5-6
Abstract 6-7
第一章绪论 10-19
1.1对流扩散方程的背景介绍 10-11
1.2中心和迎风差分格式求解对流扩散方程存在的问题 11-13
1.2.1预备知识 11-12
1.2.2中心差分格式求解对流扩散问题的条件限制 12
1.2.3迎风差分格式求解对流扩散问题的条件限制 12-13
1.3几种非传统的求解对流扩散方程数值方法的简介 13-17
1.3.1特征有限差分法 13-14
1.3.2特征有限元法 14-16
1.3.3有限体积法 16
1.3.4流线扩散法 16-17
1.4本文差分格式构造的主要思想及优势 17-18
1.5本文的主要工作 18-19
第二章一维定常对流扩散方程 19-36
2.1一维定常常系数对流扩散方程 19-28
2.1.1差分格式的构造 19-20
2.1.2差分解的误差估计 20-24
2.1.3数值实验 24-28
2.2一维定常变系数对流扩散方程 28-36
2.2.1差分格式的构造 29-30
2.2.2差分解的误差估计 30
2.2.3数值实验 30-36
第三章二维定常对流扩散方程 36-54
3.1二维定常常系数对流扩散方程 36-50
3.1.1新型差分格式的建立 36-38
3.1.2差分解的误差估计 38-45
3.1.3数值实验 45-50
3.2二维定常变系数对流扩散方程 50-54
3.2.1新型差分格式的建立 50-51
3.2.2数值实验 51-54
第四章总结与展望 54-56
参考文献 56-58
致谢 58-60
本人在研究生期间发表论文情况 60
【摘要】 物质输运与分子扩散的物理过程和黏性流体流动的数学模型通常为对流扩散方程的定解问题,它可以用来描述河流污染、大气污染、核废物污染中污染物质的分布,流体的流动和流体中热传导等众多物理现象。因此,对流扩散方程数值解方法的研究具有十分重要的理论和实际应用意义。求解对流扩散方程的数值方法有多种,如有限差分法、有限元法、有限体积法等,但当对流扩散方程中的对流项占优时,方程具有双曲方程的特点,故源于对流扩散方程中的非对称的对流项所引起的迎风效应使对流扩散方程的数值求解变得困难。用传统的中心差分方法和标准的Galerkin有限元法求解往往会产生数值的振荡,尽管迎风差分格式能够消除对流项非对称效应引起的振荡现象,但它却“过量”的反应了解的情况,导致数值解的扩散。近年来人们关于这类方程数值方法的研究,大都倾向各种非标准的有限元法及差分法,如迎风有限元方法,特征有限元方法,流线扩散有限元法和特征差分法,广义差分法和有限体积法等。考虑到“迎风效应”是由对流扩散方程中的不对称的对流项所引起的,本文直接从原定常对流占优扩散方程出发,通过指数变换,将其等价变换为守恒型扩散方程,然后利用有限体积法对扩散方程进行离散,从而建立了一种新型的差分格式。本文利用能量法证得一维对流扩散方程差分解的误差估计；对于二维对流扩散方程的新型差分格式,本文采用了极值原理、广义差分法及能量估计法三种方法来进行差分解的误差分析,并给出了数值实验结果。数值实验表明,本文差分格式的精度和收敛性是令人满意的,并且差分解没有出现数值振荡和扩散现象。
硕士论文代写【Abstract】 The physical process going with the matter transmiting and molecule diffusing and the flow of glutinous liquid, their mathematical models are ususlly the problems of the convection-diffusion equations with the stable solution that can be described the distribution of the polluted matter in the rivers pollution, air pollution, and nucleus rubbish pollution, are also described the physical phenomenon of the flow and the heated methods in the liquid and so on, so the research on the numerical methods of convection-diffusion equations have theoretical and pratical significance.There are some numerical methods of solving convection-diffusion equations, such as finit difference method, finit element method, and finit volume method and so on. However, when the convection item is dominated in the convection-diffusion equations, the equations take on the characteristics of hyperbola equations, in this case, the upwind domino offect that derived from the anisomerous convection item in the convection-diffusion equations makes it more difficult to solve convection-diffusion equations, and the traditional difference method and standard Galerkin finit element method will generate the numerial oscillation, although upwind difference scheme can eliminate the oscillative phenomenon caused by anisomerous convection item, but it reflects the solution excessively, and leads to the diffusion of the solution. Recent years, most of people are apted to some nonstandard finit element and finit difference methods, such as upwind finit element method, characteristic finit element method, streamline-diffusion finit element method, characteristic difference method, generalized difference method and finit volume method.Considering ’the upwind domino effect’ caused by the anisomerous convection item in the convection-diffusion equations, in this paper, starting frome the original convection-diffusion equation, it is changed into a conservative diffusion equation through the exponent transform, then the conservative diffusion equation is discreted by using the finit volume method, finally a new difference scheme is established. In this paper, the error estimate for one dimensional difference solution is given using the energy method, while maximum principle method, generalized difference method and energy estimate method are adapted to analyze the error for two dimensional difference solution, then numerical solutions for some instances are presented. The results show that the precision and astringency of difference schemes are satisfying, meanwhile the difference solutions haven’t appeared the vibration and diffusion.
【关键词】 对流扩散方程； 指数变换； 有限体积法； 误差估计； 数值实验；
【Key words】 convection-diffusion equation； exponent transform； finit volume method； error estimate； numerical instances；