Abstract
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.
Acknowledgements
Wei Zhang has been funded by a grant from the China Scholarship Council. A part of the computer time used for the paper has been provided by a grant from the NOTUR project of Norway.
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Please cite to this paper as published in:
Fract. Calc. Appl. Anal., Vol. 19, No 1 (2016), pp. 140–160, DOI: 10.1515/j_fca-2016-0008
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