G. K. Ananth Grama, Anshul Gupta, V. Kumar, Introduction to Parallel Computing. 2nd Ed., Addison-Wesley (2003).
 K. Asanovic, R. Bodik, J. Demmel, T. Keaveny, K. Keutzer, J. Kubiatowicz, N. Morgan, D. Patterson, K. Sen, J. Wawrzynek, et al., A view of the parallel computing landscape. Communications of the ACM 52 (2009), 56–67. http://dx.doi.org/10.1145/1562764.1562783 [CrossRef] [Web of Science]
 K. Burrage, N. Hale, D. Kay, An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. on Scientific Computing 34 (2012), 2145–2172. http://dx.doi.org/10.1137/110847007 [CrossRef] [Web of Science]
 B. Catanzaro, N. Sundaram, K. Keutzer, Fast support vector machine training and classification on graphics processors. In: Proc. 25th Internat. Conf. on Machine Learning ACM (2008), 104–111.
 J. Chen, F. Liu, Analysis of stability and convergence of numerical approximation for the Riesz fractional reaction-dispersion equation (in Chinese). J. of Xiamen University (Natural Science) 45 (2006), 466–469.
 J. Chen, F. Liu, Stability and convergence of an implicit difference approximation for the space Riesz fractional reaction-dispersion equation. Numerical Mathematics, A Journal of Chinese Universities (EN Ser.) 16 (2007), 253.
 J. Chen, F. Liu, I. Turner, V. Anh, The fundamental and numerical solutions of the Riesz space fractional reaction-dispersion equation. ANZIAM J. 50 (2008), 45–57. http://dx.doi.org/10.1017/S1446181108000333 [CrossRef] [Web of Science]
 G. Colomer, R. Borrell, F. Trias, I. Rodrguez, Parallel algorithms for sn transport sweeps. J. of Computational Physics 232 (2012), 118–135. http://dx.doi.org/10.1016/j.jcp.2012.07.009 [Web of Science] [CrossRef]
 C. Dhaigude, V. Nikam, Solution of fractional partial differential equations using iterative method. Fract. Calc. Appl. Anal. 15, No 4 (2012), 684–699; DOI: 10.2478/s13540-012-0046-8; at http://link.springer.com/journal/13540. [CrossRef]
 K. Diethelm, An efficient parallel algorithm for the numerical solution of fractional differential equations. Fract. Calc. Appl. Anal. 14, No 3 (2011), 475–490; DOI: 10.2478/s13540-011-0029-1; at http://link.springer.com/journal/13540. [CrossRef]
 N. Ford, J. Xiao, Y. Yan, A finite element method for time fractional partial differential equations. Fract. Calc. Appl. Anal. 14, No 3 (2011), 454–474; DOI: 10.2478/s13540-011-0028-2; at http://link.springer.com/journal/13540. [CrossRef]
 G. hua Gao, Z. zhong Sun, Y. nan Zhang, A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J. of Computational Physics 231 (2012), 2865–2879. http://dx.doi.org/10.1016/j.jcp.2011.12.028 [CrossRef] [Web of Science]
 C. Gong, J. Liu, L. Chi, H. Huang, J. Fang, Z. Gong, GPU accelerated simulations of 3D deterministic particle transport using discrete ordinates method. J. of Computational Physics 230 (2011), 6010–6022. http://dx.doi.org/10.1016/j.jcp.2011.04.010 [Web of Science]
 C. Gong, J. Liu, H. Huang, Z. Gong, Particle transport with unstructured grid on GPU. Computer Physics Communications 183 (2012), 588–593. http://dx.doi.org/10.1016/j.cpc.2011.12.002 [CrossRef] [Web of Science]
 R. Gorenflo, F. Mainardi, Approximation of lévy-feller diffusion by random walk. J. for Analysis and its Applications 18 (1999), 231–246.
 H. Hejazi, T. Moroney, F. Liu, A finite volume method for solving the two-sided time-space fractional advection-dispersion equation. In: Proc. FDA’12 — 5th Symposium on Fractional Differentiation and Its Applications, Hohai University, 2012.
 J. T. Katsikadelis, The BEM for numerical solution of partial fractional differential equations. Comput. Math. Appl. 62 (2011), 891–901. http://dx.doi.org/10.1016/j.camwa.2011.04.001 [CrossRef]
 D. J. Kerbyson, M. Lang, S. Pakin, Adapting wave-front algorithms to efficiently utilize systems with deep communication hierarchies. Parallel Computing 37 (2011), 550–561. http://dx.doi.org/10.1016/j.parco.2011.02.008 [CrossRef] [Web of Science]
 M. Köpf, C. Corinth, O. Haferkamp, T. Nonnenmacher, Anomalous diffusion of water in biological tissues. Biophysical Journal 70 (1996), 2950–2958. http://dx.doi.org/10.1016/S0006-3495(96)79865-X [CrossRef]
 C. Li, F. Zeng, F. Liu, Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, No 3 (2012), 383–406; DOI: 10.2478/s13540-012-0028-x; at http://link.springer.com/journal/13540. [CrossRef]
 J. Lima, R. C. de Souza, The fractional Fourier transform over finite fields. Signal Processing 92 (2012), 465–476. http://dx.doi.org/10.1016/j.sigpro.2011.08.010 [CrossRef]
 S. Lu, F. Molz, G. Fix, Possible problems of scale dependency in applications of the three-dimensional fractional advection-dispersion equation to natural porous media. Water Resour. Res. 38 (2002), 1165.
 M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations. J. of Computational and Applied Mathematics 172 (2004), 65–77. http://dx.doi.org/10.1016/j.cam.2004.01.033 [CrossRef]
 M. Meerschaert, C. Tadjeran, Finite difference approximations for twosided space-fractional partial differential equations. Applied Numerical Mathematics 56 (2006), 80–90. http://dx.doi.org/10.1016/j.apnum.2005.02.008 [Web of Science] [CrossRef]
 K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York (1993).
 Y. nan Zhang, Z. zhong Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation. J. of Computational Physics 230 (2011), 8713–8728. http://dx.doi.org/10.1016/j.jcp.2011.08.020 [CrossRef] [Web of Science]
 N. Ozdemir, D. Avci, B. Iskender, The numerical solutions of a two-dimensional space-time Riesz-Caputo fractional diffusion equation. Intern. J. of Optimization and Control: Theories & Applications (IJOCTA) 1, (2011), 17–26. http://dx.doi.org/10.11121/ijocta.01.2011.0028
 H.-K. Pang, H.-W. Sun, Multigrid method for fractional diffusion equations. J. of Computational Physics 231 (2012), 693–703. http://dx.doi.org/10.1016/j.jcp.2011.10.005 [CrossRef]
 I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA (1999).
 S. Samko, A. Kilbas, O. Maričev, Fractional Integrals and Derivatives, Gordon and Breach Science Publ., Yverdon (1993).
 M. Snir, S. W. Otto, D. W. Walker, J. Dongarra, S. Huss-Lederman, MPI: The Complete Reference, MIT Press, Cambridge, MA — USA (1995).
 C. Tadjeran, M. M. Meerschaert, H.-P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation. J. of Computational Physics 213 (2006), 205–213. http://dx.doi.org/10.1016/j.jcp.2005.08.008 [CrossRef]
 Y. Xu, Z. He, The short memory principle for solving abel differential equation of fractional order. Computers & Mathematics with Applications 62 (2011), 4796–4805. http://dx.doi.org/10.1016/j.camwa.2011.10.071 [CrossRef] [Web of Science]
 S. B. Yuste, J. Quintana-Murillo, A finite difference method with nonuniform timesteps for fractional diffusion equations. Computer Physics Communications 183 (2012), 2594–2600. http://dx.doi.org/10.1016/j.cpc.2012.07.011 [CrossRef] [Web of Science]
Fractional Calculus and Applied Analysis
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A parallel algorithm for the Riesz fractional reaction-diffusion equation with explicit finite difference method
1College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, 410073, China
2Science and Technology on Space Physics Libratory, Beijing, 10076, China
3School of Computer Science, National University of Defense Technology, Changsha, 410073, China
© 2013 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)
Citation Information: Fractional Calculus and Applied Analysis. Volume 16, Issue 3, Pages 654–669, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: 10.2478/s13540-013-0041-8, June 2013
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The fractional reaction-diffusion equations play an important role in dynamical systems. Indeed, it is time consuming to numerically solve differential fractional diffusion equations. In this paper, we present a parallel algorithm for the Riesz space fractional diffusion equation. The parallel algorithm, which is implemented with MPI parallel programming model, consists of three procedures: preprocessing, parallel solver and postprocessing. The parallel solver involves the parallel matrix vector multiplication and vector vector addition. As to the authors’ knowledge, this is the first parallel algorithm for the Riesz space fractional reaction-diffusion equation. The experimental results show that the parallel algorithm is as accurate as the serial algorithm. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.3-3.4 times faster than the serial algorithm on single CPU core. The parallel efficiency of 64 processes is up to 79.39% compared with 8 processes on a distributed memory cluster system.
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