求解抛物方程的MPI并行方法

Parallel Methods for Parabolic Equations Based on MPI Implementation

摘要: 数值求解偏微分方程广泛应用于数学与工程领域。大规模数值计算在当今科学技术运用中得到飞速发展，其中可并行的有限差分格式受到越来越多的重视。在本文中，主要阐述了经典的分组显示方法求解抛物方程，并简单扼要的分析了该格式的建立以及稳定性。随后本文着重介绍了如何在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计算模型，即阻塞通信(等待模式)和非阻塞通信(即非等待模式)模式。并与非并行状态下的差分格式做出比较，结果表明，相对于一个进程求解偏微分方程，两种模式都表现出较好的效果，而且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。

Abstract: Many applications in mathematics and engineering involve numerical solutions of partial diffe-rential equations (PDEs). The demands of large-scale computing are quickly increasing in modern science and technology, and parallel computing has received more and more attention. In this paper, the main idea is that classical Group Explicit method (GEM) for parabolic equations, the group explicit method is established briefly and the stability analysis of the method is indicated simply. Then we focus on how to calculate the format in MPI parallel environment. Two parallel MPI algorithms are established and compared with non-parallel algorithm based on GEM. They are MPI block communication (wait communication) and non-blocking communication (no-wait communication). These two MPI schemas both better than one single process to calculate numerical solutions use group explicit method. Also, the non-blocking communication program has higher computational efficiency than blocking communication program.