一种基于隐式差分法的二维时间分数阶扩散方程的并行算法。

A parallel algorithm for the two-dimensional time fractional diffusion equation with implicit difference method.

作者信息

Gong Chunye, Bao Weimin, Tang Guojian, Jiang Yuewen, Liu Jie

机构信息

College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China ; Science and Technology on Space Physics Laboratory, Beijing 100076, China ; School of Computer Science, National University of Defense Technology, Changsha 410073, China.

College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China ; Science and Technology on Space Physics Laboratory, Beijing 100076, China.

出版信息

ScientificWorldJournal. 2014 Mar 12;2014:219580. doi: 10.1155/2014/219580. eCollection 2014.

DOI:10.1155/2014/219580

PMID:24744680

原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC3972955/

Abstract

It is very time consuming to solve fractional differential equations. The computational complexity of two-dimensional fractional differential equation (2D-TFDE) with iterative implicit finite difference method is O(M(x)M(y)N(2)). In this paper, we present a parallel algorithm for 2D-TFDE and give an in-depth discussion about this algorithm. A task distribution model and data layout with virtual boundary are designed for this parallel algorithm. The experimental results show that the parallel algorithm compares well with the exact solution. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.16-4.17 times faster than the serial algorithm on single CPU core. The parallel efficiency of 81 processes is up to 88.24% compared with 9 processes on a distributed memory cluster system. We do think that the parallel computing technology will become a very basic method for the computational intensive fractional applications in the near future.

摘要

求解分数阶微分方程非常耗时。采用迭代隐式有限差分法求解二维分数阶微分方程（2D-TFDE）的计算复杂度为O(M(x)M(y)N(2))。本文提出了一种用于2D-TFDE的并行算法，并对该算法进行了深入讨论。针对此并行算法设计了一种带有虚拟边界的任务分配模型和数据布局。实验结果表明，该并行算法与精确解相比效果良好。在单个英特尔至强X5540 CPU上运行的并行算法比在单个CPU核心上运行的串行算法快3.16 - 4.17倍。在分布式内存集群系统上，81个进程的并行效率与9个进程相比高达88.24%。我们确实认为，在不久的将来，并行计算技术将成为计算密集型分数阶应用中一种非常基本的方法。