一类偏微分方程数值求解的交替方向算法研究

发布时间：2018-04-02 03:22

本文选题：偏微分方程　切入点：椭圆型方程　出处：《西安理工大学》2017年硕士论文

【摘要】：偏微分方程在自然科学与工程技术中有着广泛的应用,许多领域中的数学模型都可以用偏微分方程来描述,很多重要的物理、力学等学科的基本方程本身就是偏微分方程。近几十年来,偏微分方程的发展非常迅速,研究成果层出不穷。偏微分方程的数值求解方法一向是研究的热点,其中交替方向隐式(Alternating direction implicit,简称ADI)法是一种离散过程简单、计算速度快、求解精度高,便于编程实现的求解算法。本文系统地研究了一类变系数偏微分方程数值求解的交替方向隐式法,主要包括以下几个方面:1.阐述了偏微分方程的相关背景和研究意义,概述了国内外研究新进展,对本文的研究工作做了详细的安排,给出了相关的预备知识。2.对一类变系数偏微分方程(包括二维抛物型方程、二维和三维椭圆型方程)给出了利用交替方向隐式法(ADI)求解的迭代格式,建立了相应的截断误差分析理论,给出了数值模拟及结果分析,并对在迭代过程中的迭代参数进行讨论。3.对提出的一类变系数偏微分方程的交替方向隐格式分别进行了算法设计,编写了通用的求解程序。
[Abstract]:Partial differential equations are widely used in natural science and engineering technology. Many mathematical models in many fields can be described by partial differential equations, many important physics, The basic equations of mechanics and other disciplines are partial differential equations themselves. In recent decades, the partial differential equations have developed very rapidly, and the research results have been endless. The numerical solution of partial differential equations has always been a hot research topic. Among them, the alternating direction implicit direction implicit (ADI) method is a simple discrete process, fast calculation speed and high accuracy. The alternating direction implicit method for solving a class of partial differential equations with variable coefficients is studied systematically in this paper, including the following aspects: 1. The background and significance of the partial differential equations are expounded. This paper summarizes the new research progress at home and abroad, makes a detailed arrangement of the research work in this paper, and gives the relevant preparatory knowledge .2. for a class of partial differential equations with variable coefficients (including two-dimensional parabolic equations), In this paper, the iterative scheme for solving the elliptic equations by alternating direction implicit method (ADI) is given. The corresponding truncation error analysis theory is established, and the numerical simulation and the result analysis are given. The iterative parameters in the iterative process are discussed. 3. The alternating direction implicit schemes for a class of partial differential equations with variable coefficients are designed, and a general solution program is compiled.
【学位授予单位】：西安理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

【参考文献】