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1、毕业论文(设计)题 目: 热传导方程初边值问题的差分解法 院(系): 数学与计算机科学学院 _ 专业年级: 2008级数学与应用数学系 姓 名: XXX _ _ 学 号: _ _ 指导教师: XXX_ _ 2012年5月摘 要 文章目的是为了探讨热传导方程初边值问题的差分解法。 本文包括以下两部分主要内容： 第一部分即是对比传统热传导方程初边值问题的变量分离法的差分解法； 第二部分即是热传导方程初边值问题差分解法的具体例子。 其中主要涉及到的方法有热传导方程初边值问题的分离变量法和有限差分法。那么先具体介绍有限差分法。基本思想是把连续的定解区域用有限个离散点构成的网格来代替，这些离散点称作网格
2、的节点；把连续定解区域上的连续变量的函数用在网格上定义的离散变量函数来近似；把原方程和定解条件中的微商用差商来近似，积分用积分和来近似，于是原微分方程和定解条件就近似地代之以代数方程组，即有限差分方程组，解此方程组就可以得到原问题在离散点上的近似解。然后再利用插值方法便可以从离散解得到定解问题在整个区域上的近似解。 在采用数值计算方法求解偏微分方程时，若将每一处导数由有限差分近似公式替代，从而把求解偏微分方程的问题转换成求解代数方程的问题，即所谓的有限差分法。 有限差分法求解偏微分方程的步骤如下： 1.区域离散化，即把所给偏微分方程的求解区域细分成由有限个格点组成的网格； 2.近似替代，即采用
3、有限差分公式替代每一个格点的导数； 3.逼近求解。换而言之，这一过程可以看作是用一个插值多项式及其微分来代替偏微分方程的解的过程。 对比与分离变量法，有限差分法有着其特性，方便且更精确的特性。经过下面的一番比较，我们有理由相信有限差分法是大有用途的。 关键词： 差分格式 步长 网络节点 截断误差 Abstract The article aims to explore the heat conduction equation initial boundary value problem of the finite difference method. This paper includes t
4、he following two parts of the main content: The first part is compared with the traditional heat conduction equation initial boundary value problem of the separation of variables method finite difference method; The second part is the heat conduction equation initial boundary value problem of differ
5、ence methods for specific examples. Which mainly relates to a method for heat conduction equation initial boundary value problem of the separation of variables method and finite difference method. It first introduces the finite difference method. The basic idea is to use a continuous solution region
6、 using finite discrete points constitute a grid to replace, the discrete points are called grid node; the continuous solution of continuous variable function is used in the grid defined on a discrete variables function to approximate; the original equations and boundary conditions of the difference
7、quotient to micro commercial approximation, integral integral and to approximate, and the differential equations and boundary conditions is approximately replaced by algebraic equations, finite difference equation, the solution to this equation can get the original problem in the discrete points on
8、the approximate solution. And then using interpolation methods can be determined from the discrete solution solution of the approximate solution on the entire region. In the use of numerical methods for solving partial differential equations, if every derivative by finite difference approximation fo
9、rmula substitution, the solution of partial differential equations of the problem is transformed into solving algebraic equations, the so-called finite difference method. Finite difference method for solving partial differential equations:1discrete regions, which are for solving partial differential
10、 equations by the finite region is subdivided into a lattice grid consisting of;2approximate substitution, i.e. finite difference formula one substitution per lattice points of the derivative;3approximation solution. In other words, this process can be viewed as a polynomial interpolation and its di
11、fferential instead of partial differential equation solution process.In contrast with the method of separation of variables, the finite difference method has the characteristics of convenient, and more precise characteristics. After following a comparison, we have reason to believe that the finite d
12、ifference method is of great use. Key words: differential format step network node truncation error. 目 录绪论.11热传导初边值问题分离变量法的介绍.2 1.1热传导初边值问题分离变量法的具体应用.32热传导初边值问题有限差分法的介绍.5 2.1 对于显式与隐式有限元的理解.72.1.1 两种算法的比较.7 2.1.1.1 显式算法.8 2.1.1.2 隐式算法.8 2.1.2 求解时间.82.1.3 两种方法的应用范围.82.1.4 总结.9 2.2有限差分法求解此热传导方程初边值问题.9 2.3 初边值问题差分法的实例.10致谢.11参考文献.
