基于MPI的一种有限并行差分格式求解四阶抛物方程

A Finite Difference Parallel Scheme Based on MPI Implementation for Fourth Order Parabolic Equations

摘要: 在大规模的科学与工程计算问题中，并行计算能够节省大量的时间，本文针对一维四阶抛物方程给出了一类并行差分格式。利用Saul’yev非对称格式进行恰当的组合，形成求解抛物方程的四点格式。四点格式是显式求解的，因此可以将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。随后本文着重介绍了在MPI并行环境下对该格式进行数值计算，构建了两种不同的MPI并行算法并与串行状态下的有限差分格式做出比较，即阻塞通信(等待通信)和非阻塞通信(非等待通信)模式。相对于串行算法运用四点格式求解四阶抛物方程，两种并行通信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模式，非阻塞通信模式表现出较好的并行效率。

Abstract: Parallel computing can save a lot of time in the field of large-scale scientific computing. In this paper, the main idea is that a finite difference parallel scheme for fourth order parabolic equations. The scheme is constructed by Saul’yev asymmetric difference schemes which called the four-point scheme. It’s one explicit difference scheme, the computational domain can be divided into a number of large areas; each sub-region computes themselves, and the parallel scheme is unconditionally stable. Then, the paper focuses on the numerical calculation of the four-point scheme in MPI parallel environment. Two different MPI parallel algorithms are constructed, one is blocking com- munication (wait communication) mode, and the other is non-blocking communication (non-wait communication) mode. These two parallel algorithms both better than serial algorithm to calculate numerical solutions use four-point scheme, and the non-blocking communication mode is higher computational than the other, because the wait time in non-blocking communication mode is less than blocking communication mode.