本文選題：偏微分方程　切入點(diǎn)：差分方法　出處：《青島科技大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：數(shù)值計(jì)算求解偏微分廣泛應(yīng)用于數(shù)學(xué)與工程領(lǐng)域。求解偏微分的方法主要包括有限元法和有限差分法。隨著分布式計(jì)算平臺的快速發(fā)展,其中可并行的有限差分格式在并行機(jī)上進(jìn)行快速有效的執(zhí)行,正受到越來越多的重視。在本文中,主要探究了運(yùn)用分組顯式方法對若干偏微分方程的數(shù)值求解,以及在MPI(Message Passing Interface)并行運(yùn)算環(huán)境下對上述方程構(gòu)造了多種并行模式。在本文緒論中,首先分析了并行差分格式的研究意義,研究現(xiàn)狀以及國內(nèi)外的發(fā)展趨勢,之后介紹了MPI并行技術(shù)在當(dāng)前的發(fā)展趨勢以及研究意義。在本文第一章中,簡單介紹了并行計(jì)算原理以及MPI的配置過程。在第二章中,研究了拋物方程的并行數(shù)值算法。首先,對Saul’yev非對稱格式進(jìn)行合適的組合,針對二階拋物型偏微分方程,構(gòu)造了分組顯式方法,并簡單扼要分析了該格式的穩(wěn)定性。之后本章著重介紹了如何在MPI并行環(huán)境下對該格式進(jìn)行數(shù)值計(jì)算,構(gòu)建了兩種不同的并行算法并與非并行狀態(tài)下的有限差分格式做出比較,即阻塞通信(等待模式)和非阻塞通信(非等待模式)模式。相對于單個進(jìn)程求解偏微分方程,兩種模式都表現(xiàn)出較好的效果,其中非阻塞通信相較于阻塞通信模式亦表現(xiàn)出較好的并行效率。第三章探討了高階拋物型方程的MPI并行算法。首先,利用Saul’yev非對稱格式建立了求解高階拋物方程的四點(diǎn)格式。四點(diǎn)格式是顯式求解的,因此可以將求解空間區(qū)域分為若干子區(qū)域,每個子區(qū)域獨(dú)立計(jì)算。驗(yàn)證分析表明,該格式是絕對穩(wěn)定的。隨后針對四點(diǎn)格式,構(gòu)造了兩種不同的MPI并行算法,相對于串行算法運(yùn)用四點(diǎn)格式求解四階拋物方程,兩種MPI并行模式都表現(xiàn)出極好的效果,而且,非阻塞通信模式下的計(jì)算由于相對減少了一部分?jǐn)?shù)據(jù)的通信等待時間,使得相對于阻塞通信,非阻塞通信表現(xiàn)出較好的并行效率。為了進(jìn)一步提升MPI并行模型的效率,分別給出了在不同進(jìn)程數(shù)目下,兩種消息傳遞模型的運(yùn)算時間。在第四章中,探究了非線性偏微分方程的MPI并行算法,以Burgers方程為例,首先將其線性化處理,然后構(gòu)建有限并行差分格式,然后構(gòu)造了與之相適應(yīng)的MPI并行算法,并運(yùn)用于大規(guī)模的數(shù)值模擬運(yùn)算,得到并行計(jì)算相對于串行計(jì)算的效率分析結(jié)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.
【學(xué)位授予單位】：青島科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.82

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