Abstract
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.
[1] G. K. Ananth Grama, Anshul Gupta, V. Kumar, Introduction to Parallel Computing. 2nd Ed., Addison-Wesley (2003). Search in Google Scholar
[2] 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/10.1145/1562764.156278310.1145/1562764.1562783Search in Google Scholar
[3] 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/10.1137/11084700710.1137/110847007Search in Google Scholar
[4] 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. 10.1145/1390156.1390170Search in Google Scholar
[5] 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. Search in Google Scholar
[6] 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. Search in Google Scholar
[7] 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/S144618110800033310.1017/S1446181108000333Search in Google Scholar
[8] 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.2012.07.00910.1016/j.jcp.2012.07.009Search in Google Scholar
[9] 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. Search in Google Scholar
[10] 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. Search in Google Scholar
[11] 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. Search in Google Scholar
[12] 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.2011.12.02810.1016/j.jcp.2011.12.028Search in Google Scholar
[13] 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.2011.04.01010.1016/j.jcp.2011.04.010Search in Google Scholar
[14] 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.2011.12.00210.1016/j.cpc.2011.12.002Search in Google Scholar
[15] R. Gorenflo, F. Mainardi, Approximation of lévy-feller diffusion by random walk. J. for Analysis and its Applications 18 (1999), 231–246. Search in Google Scholar
[16] 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. 10.2478/s11534-013-0317-ySearch in Google Scholar
[17] 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.2011.04.00110.1016/j.camwa.2011.04.001Search in Google Scholar
[18] 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.2011.02.00810.1016/j.parco.2011.02.008Search in Google Scholar
[19] 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-X10.1016/S0006-3495(96)79865-XSearch in Google Scholar
[20] 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. Search in Google Scholar
[21] 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.2011.08.01010.1016/j.sigpro.2011.08.010Search in Google Scholar
[22] 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. 10.1029/2001WR000624Search in Google Scholar
[23] 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.2004.01.03310.1016/j.cam.2004.01.033Search in Google Scholar
[24] 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.2005.02.00810.1016/j.apnum.2005.02.008Search in Google Scholar
[25] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York (1993). Search in Google Scholar
[26] 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.2011.08.02010.1016/j.jcp.2011.08.020Search in Google Scholar
[27] 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.01.2011.002810.11121/ijocta.01.2011.0028Search in Google Scholar
[28] 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.2011.10.00510.1016/j.jcp.2011.10.005Search in Google Scholar
[29] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA (1999). Search in Google Scholar
[30] S. Samko, A. Kilbas, O. Maričev, Fractional Integrals and Derivatives, Gordon and Breach Science Publ., Yverdon (1993). Search in Google Scholar
[31] M. Snir, S. W. Otto, D. W. Walker, J. Dongarra, S. Huss-Lederman, MPI: The Complete Reference, MIT Press, Cambridge, MA — USA (1995). Search in Google Scholar
[32] 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.2005.08.00810.1016/j.jcp.2005.08.008Search in Google Scholar
[33] 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.2011.10.07110.1016/j.camwa.2011.10.071Search in Google Scholar
[34] 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.2012.07.01110.1016/j.cpc.2012.07.011Search in Google Scholar
© 2013 Diogenes Co., Sofia
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.