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Fractional Calculus and Applied Analysis

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Volume 19, Issue 1

Solving 3D Time-Fractional Diffusion Equations by High-Performance Parallel Computing

Wei Zhang / Xing Cai
  • Department of Informatics, University of Oslo, P.O. Box 1080, Blindern, N-0316 Oslo, NORWAY
  • Simula Research Laboratory, P.O. Box 134, N-1325 Lysaker, NORWAY
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Published Online: 2016-03-09 | DOI: https://doi.org/10.1515/fca-2016-0008


Numerically solving time-fractional diffusion equations, especially in three space dimensions, is a daunting computational task. This is due to the huge requirements of both computation time and memory storage. Compared with solving integer-ordered diffusion equations, the costs for time and storage both increase by a factor that equals the number of time steps involved. Aiming to overcome these two obstacles, we study in this paper three programming techniques: loop unrolling, vectorization and parallelization. For a representative numerical scheme that adopts finite differencing and explicit time integration, the performance-enhancing techniques are indeed shown to dramatically reduce the computation time, while allowing the use of many CPU cores and thereby a large amount of memory storage. Moreover, we have developed simple-to-use performance models that support our empirical findings, which are based on using up to 8192 CPU cores and 12.2 terabytes.

MSC: Primary 65Y05; Secondary 65L05; 65R20

Key Words and Phrases: fractional differential equations; loop unrolling; vectorization; parallel computing


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About the article

Received: 2014-12-26

Revised: 2015-07-19

Published Online: 2016-03-09

Published in Print: 2016-02-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 19, Issue 1, Pages 140–160, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2016-0008.

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