基于MPI的并行有限差分法对几类偏微分方程的数值求解

发布时间：2018-03-06 12:09

本文选题：偏微分方程　切入点：差分方法　出处：《青岛科技大学》2017年硕士论文　论文类型：学位论文

【摘要】：数值计算求解偏微分广泛应用于数学与工程领域。求解偏微分的方法主要包括有限元法和有限差分法。随着分布式计算平台的快速发展,其中可并行的有限差分格式在并行机上进行快速有效的执行,正受到越来越多的重视。在本文中,主要探究了运用分组显式方法对若干偏微分方程的数值求解,以及在MPI(Message Passing Interface)并行运算环境下对上述方程构造了多种并行模式。在本文绪论中,首先分析了并行差分格式的研究意义,研究现状以及国内外的发展趋势,之后介绍了MPI并行技术在当前的发展趋势以及研究意义。在本文第一章中,简单介绍了并行计算原理以及MPI的配置过程。在第二章中,研究了抛物方程的并行数值算法。首先,对Saul’yev非对称格式进行合适的组合,针对二阶抛物型偏微分方程,构造了分组显式方法,并简单扼要分析了该格式的稳定性。之后本章着重介绍了如何在MPI并行环境下对该格式进行数值计算,构建了两种不同的并行算法并与非并行状态下的有限差分格式做出比较,即阻塞通信(等待模式)和非阻塞通信(非等待模式)模式。相对于单个进程求解偏微分方程,两种模式都表现出较好的效果,其中非阻塞通信相较于阻塞通信模式亦表现出较好的并行效率。第三章探讨了高阶抛物型方程的MPI并行算法。首先,利用Saul’yev非对称格式建立了求解高阶抛物方程的四点格式。四点格式是显式求解的,因此可以将求解空间区域分为若干子区域,每个子区域独立计算。验证分析表明,该格式是绝对稳定的。随后针对四点格式,构造了两种不同的MPI并行算法,相对于串行算法运用四点格式求解四阶抛物方程,两种MPI并行模式都表现出极好的效果,而且,非阻塞通信模式下的计算由于相对减少了一部分数据的通信等待时间,使得相对于阻塞通信,非阻塞通信表现出较好的并行效率。为了进一步提升MPI并行模型的效率,分别给出了在不同进程数目下,两种消息传递模型的运算时间。在第四章中,探究了非线性偏微分方程的MPI并行算法,以Burgers方程为例,首先将其线性化处理,然后构建有限并行差分格式,然后构造了与之相适应的MPI并行算法,并运用于大规模的数值模拟运算,得到并行计算相对于串行计算的效率分析结果及加速比。
[Abstract]:Partial differential is widely used in mathematics and engineering. The methods of solving partial differential mainly include finite element method and finite difference method. With the rapid development of distributed computing platform, The parallel finite-difference scheme, which can be implemented quickly and efficiently on parallel machines, is being paid more and more attention. In this paper, the numerical solution of some partial differential equations by grouping explicit method is mainly discussed. In the introduction of this paper, the significance of parallel difference scheme, the research status and the development trend of parallel difference scheme at home and abroad are analyzed. In the first chapter, the principle of parallel computing and the configuration process of MPI are introduced. In the second chapter, parallel numerical algorithms for parabolic equations are studied. For the second order parabolic partial differential equation, an explicit grouping method is constructed for the Saul'yev asymmetric scheme. The stability of the scheme is briefly analyzed. After that, this chapter focuses on how to numerically calculate the scheme in the MPI parallel environment, and constructs two different parallel algorithms and compares them with the finite difference scheme in the non-parallel state. That is, blocking communication (waiting mode) and non-blocking communication (non-waiting mode) mode. Non-blocking communication also shows better parallel efficiency than blocking communication mode. Chapter three discusses the MPI parallel algorithm for high-order parabolic equations. A four-point scheme for solving high-order parabolic equations is established by using Saul'yev 's asymmetric scheme. The four-point scheme is explicitly solved, so the spatial region of the solution can be divided into several subregions, each of which can be independently calculated. This scheme is absolutely stable. Then, two different MPI parallel algorithms are constructed for the four-point scheme. Compared with the serial algorithm using the four-point scheme to solve the fourth-order parabolic equations, the two MPI parallel schemes show excellent results. In the non-blocking communication mode, due to the relative reduction of the waiting time of some data, the non-blocking communication shows a better parallel efficiency than the blocking communication, in order to further improve the efficiency of the MPI parallel model. In Chapter 4th, the MPI parallel algorithm for nonlinear partial differential equations is studied. Taking Burgers equation as an example, the linearization of the two message passing models is given. Then the finite parallel difference scheme is constructed and the corresponding MPI parallel algorithm is constructed and applied to large-scale numerical simulation. The efficiency analysis results and speedup ratio of parallel computing compared with serial computation are obtained.
【学位授予单位】：青岛科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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