| 求解抛物及双曲方程若干差分格式的加速迭代并行算法 |
| Alternative Title | Accelerating iterative and parallel algorithms of some difference schemes for solving parabolic and hyperbolic equations
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| 郭瑜超 |
| Thesis Advisor | 伍渝江
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| 2010-05-29
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 硕士
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| Keyword | 有限差分
紧格式
交替方向隐格式
加速并行迭代算法
渐进收敛性质
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| Abstract | 随着电子计算机的发展，偏微分方程的数值解法也得到了巨大的发展。差分方法是一种求解偏微分方程的主要方法。众所周知，显式差分格式有理想的并行性，适合于并行计算，但是它多为条件稳定，尤其是在处理高维问题时经常受到限制。一般隐式差分格式是绝对稳定的，但每个时间层上需求解线性方程组。本文的第一部分首先针对抛物型差分方程的紧格式，构造了加速并行迭代算法，这种算法是对紧差分格式的线性方程组的系数矩阵进行分裂，然后对每个子方程组进行分别迭代求解，本文证明了算法的收敛性以及在网格加密时的收敛性质。接下来对于二维抛物型方程的紧交替方向隐格式，构造了加速并行迭代算法。本文的第二部分主要是针对双曲型偏微分方程，本文以波动方程的初边值问题为例，构造了古典隐式差分格式和紧差分格式的加速并行迭代算法。对于二维双曲型偏微分方程，本文以隐式交替方向差分格式为基础，构造了加速并行迭代算法。本文最后进行了数值试验，数值试验的结果与理论分析的结果一致，证明了算法的有效性。 |
| Other Abstract | With the development of computer, numerical solution for partial differential equations has also been a huge development. Difference method is a primary method for solving partial differential equations. As well known, the explicit difference scheme is suitable for parallel computation, but it has the limitation of stability, and it is often restricted especially in the treatment of high-dimensional problems. The implicit difference scheme is absolutely stable generally, but it is necessary to solve different linear systems at each level of time. In chapter one of the paper, we construct a parallel algorithm of accelerative iteration for solving compact scheme of one-dimensional parabolic differential equations at first. We split the coefficient matrix of the linear systems with respect to a compact difference scheme, and then we use iterative methods to solve the subsystems one by one. The convergence of the algorithm and the property of asymptotic convergence are proved. For the two-dimensional parabolic partial differential equation, we discuss the compact and alternating direction implicit scheme. The parallel algorithm of accelerative iteration is constructed. In the chapter two, we mainly study the hyperbolic partial differential equation. The initial and boundary value problem of the wave equation is used as an example to construct the parallel algorithm of accelerative iteration with respect to the classical implicit difference scheme and the compact difference scheme. Then the alternating direction implicit scheme and its parallel algorithm of accelerative iteration for two-dimensional hyperbolic partial differential equation are also discussed. The numerical examples are carried out at last. The results of numerical examples show that the analysis is correct and the algorithm is feasible and efficient. |
| URL | 查看原文
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/224792
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
郭瑜超. 求解抛物及双曲方程若干差分格式的加速迭代并行算法[D]. 兰州. 兰州大学,2010.
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