求解抛物型方程高精度差分格式的并行迭代法

发布时间：2018-04-27 06:05

本文选题：高精度 + 抛物型方程　；参考：《山东大学》2009年硕士论文

【摘要】： 伴随着科学和技术的发展,人们研究问题的深度和广度也在不断发展。而在自然科学和现代工程技术的领域中,很多现象都是用抛物方程或方程组来描述的。因此,用有限差分方法来数值求解抛物方程问题具有重要的理论意义和应用价值。在求解抛物型方程的问题时,需要构造出精度高,稳定性好,存储量和计算量都要小的差分格式。本文从理论与实际应用的角度出发,针对一维抛物型方程的初边值问题,采用组合差商法和参数的应用,构造和研究了高精度差分格式和其并行迭代算法,全文共分为两大部分: 第一部分首先,在空间节点宽度为3,时间层宽度为3的三层局部节点集上设计构造了新的含参数的差分方程,并用待定系数法给出了一类高精度的三层九点含参数的隐式差分格式,使其截断误差达到O(τ~3+h~6),随后用稳定性分析的Fourier方法给出了所得格式的稳定性条件,即该格式无条件绝对稳定。第二部分其次,针对本文构造的隐式差分格式,研究设计了求解抛物型方程三层隐式差分格式的并行迭代算法,其基本思想是根据隐式差分方程组系数矩阵的特点,把差分方程组划分为若干个子方程组来分别同时进行迭代求解。文中给出了构造此算法的过程,并用矩阵的理论推导论证了它的迭代收敛条件和收敛方向。它具有O(τ~3+h~6)的精度阶且绝对稳定,同时也推证了网格加密时的渐进收敛性质,即对任意网格比和任意阶子方程组,迭代过程均收敛,且迭代收敛速度在每段中随网格点数的增加而增加。随后针对具体例子给出了数值试验结果,数值算例验证了理论分析的正确性,表明了算法的可行性与有效性。
[Abstract]:With the development of science and technology, the depth and breadth of people's research problems are also developing. In the field of natural science and modern engineering, many phenomena are described by parabolic equations or equations. Therefore, it is of great theoretical significance and practical value to solve the parabolic equation numerically by the finite difference method. When solving the parabolic equation, we need to construct a difference scheme with high precision, good stability and small storage and computation. In this paper, from the point of view of theory and practical application, for the initial-boundary value problem of one-dimensional parabolic equation, the combined difference quotient method and the application of parameters are used to construct and study the high-precision difference scheme and its parallel iterative algorithm. The thesis is divided into two parts: In the first part, a new difference equation with parameters is designed and constructed on a three-layer local node set with a spatial node width of 3 and a time layer width of 3. By using the undetermined coefficient method, a class of high precision implicit difference schemes with three layers and nine points with parameters are given. The truncation error of the scheme reaches O (蟿 ~ (3) h ~ (6). Then, the stability conditions of the scheme are given by using the Fourier method of stability analysis. That is, the format is unconditionally and absolutely stable. In the second part, for the implicit difference scheme constructed in this paper, we study and design a parallel iterative algorithm for solving the parabolic equation with three levels implicit difference scheme. The basic idea is based on the characteristics of the coefficient matrix of the implicit difference equations. The difference equations are divided into several subequations to be solved iteratively at the same time. In this paper, the process of constructing this algorithm is given, and the iterative convergence conditions and the convergence direction of the algorithm are proved by the theoretical derivation of matrix. It has the order of accuracy and absolute stability of O (蟿 ~ (3) h ~ (6)). It also proves the asymptotic convergence property of mesh encryption, that is, the iterative process is convergent for any mesh ratio and subequations of arbitrary order. The iterative convergence rate increases with the increase of grid points in each segment. Then the numerical test results are given for specific examples. Numerical examples verify the correctness of the theoretical analysis and show the feasibility and effectiveness of the algorithm.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2009
【分类号】：O241.82

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