解两类发展方程的新型差分格式研究 Alternative Title Research on New Difference Schemes for Solving Two Kinds of 丁恒飞 Thesis Advisor 张国凤 2008-05-24 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 差分格式 对流扩散方程 双曲方程 截断误差 高精度 Abstract 发展方程是包含时间变数的许多重要的数学物理偏微分方程的统称. 在物理，力学或其他自然科学中，这类方程用来描述随时间而变化的状态或过程.诸如热传导方程、声波与弹性波方程、反应扩散与对流扩散方程、流体与气体力学方程组等，皆属于发展方程的范畴.用有限差分法求解此类问题，需构造出精度高、稳定性好、存储量与计算量都小的差分格式.本文对常见的一维对流扩散方程及一、 二维双曲方程进行了数值方法的研究.通过不同的方法，构造了几个求解上述方程的高精度新型差分格式，并分析了各自的稳定性.每章后的数值实验表明了 理论分析的正确性及有限差分格式的有效性. 本文可分五章：在第一章中，我们介绍了所研究问题的实际背景，回顾了有限差分方法的发展历史及发展现状，概述了有关差分方法的一些基本知识.在第二章中，列举了在以后应用中所需要的一些定义及定理等.在第三章，对一维对流扩散方程，我们分别用Taylor级数展开和积分的方法，再利用\$pad\grave{e}\$逼近的方法，构造了两个高精度，恒稳定的新型差分格式.在第四章，我们首先对一维二阶双曲方程用分步离散的方法，构造出一个紧致差分格式，然后将此方法推广到二维情形，且分析了各自的稳定性.在第五章中，我们回顾了本文的主要内容，并对本文中的方法作了展望. Other Abstract Evolution equation is a collection of many important mathematical physics partial differential equations with time variable. In the physical, mechanical or other natural sciences, such equations are used to describe changes over time in the state or process. Such as thermal conductivity equation, acoustic and elastic wave equation, reaction-diffusion and convection-diffusion equation, equations of fluid mechanics and gas group, and so on, all belong to the scope of the evolution equation. For solving such problems by using difference method, the difference scheme of high-order accuracy, better stability, less storage and calculate amounts need to be constructed. In this paper, the common one-dimensional convection-diffusion equation and the one, two-dimensional hyperbolic equations are studied numerically. Through different methods, we construct several new and high-order accuracy difference schemes for solving above equations and analyze their stability respectively. The numerical experiment at the end of each chapter shows the effectiveness of the theoretical analysis and the correctness of the giving difference scheme. This paper is divided into five chapters： In the first chapter, the actual background of this paper studied is mainly introduced, and the development history and the present situation is recollected, then the basic knowledge about difference method constructed in this paper is simply introduced and some elementary knowledge about finite difference method is outlined. In the second chapter, we list a number of definitions and theorems needed in future application. In the third chapter, by using Taylor series expansion, integral approach and then pade approximation methods, two new high-order accuracy, uncondition stability difference schemes for the one-dimensional convection-diffusion equation are constructed. In the fourth chapter, we firstl construct a compact difference scheme for solving the one-dimensional second-order hyperbolic equations by using step by step discrete methods, then we extend this method to two-dimensional second-order case and analyze their stability respectively. In the fifth chapter, the main content of this paper is reviewed, and the forecast for the development of method is made. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225011 Collection 数学与统计学院 Recommended CitationGB/T 7714 丁恒飞. 解两类发展方程的新型差分格式研究[D]. 兰州. 兰州大学,2008.
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