The Journal of Supercomputing

, Volume 68, Issue 3, pp 1521–1537

An efficient parallel solution for Caputo fractional reaction–diffusion equation

  • Chunye Gong
  • Weimin Bao
  • Guojian Tang
  • Bo Yang
  • Jie Liu


The computational complexity of Caputo fractional reaction–diffusion equation is \(O(MN^2)\) compared with \(O(MN)\) of traditional reaction–diffusion equation, where \(M\), \(N\) are the number of time steps and grid points. A efficient parallel solution for Caputo fractional reaction–diffusion equation with explicit difference method is proposed. The parallel solution, which is implemented with MPI parallel programming model, consists of three procedures: preprocessing, parallel solver and postprocessing. The parallel solver involves the parallel tridiagonal matrix vector multiplication, vector vector addition and constant vector multiplication. The sum of constant vector multiplication is optimized. As to the authors’ knowledge, this is the first parallel solution for Caputo fractional reaction–diffusion equation. The experimental results show that the parallel solution compares well with the analytic solution. The parallel solution on single Intel Xeon X5540 CPU runs more than three times faster than the serial solution on single X5540 CPU core, and scales quite well on a distributed memory cluster system.


Fractional differential equation Reaction-diffusion equation  High performance computing Parallel computing Finite difference method 


  1. 1.
    Campos R, Rico-Melgoza J, Chvez E (2012) A new formulation of the fast fractional fourier transform. SIAM J Sci Comput 34(2):A1110–A1125. doi:10.1137/100812677 Google Scholar
  2. 2.
    Cao X, Mo Z, Liu X, Xu X, Zhang A (2011) Parallel implementation of fast multipole method based on jasmin. Sci China Inform Sci 54:757–766CrossRefMathSciNetGoogle Scholar
  3. 3.
    Cecilia J, Abellán J, Fernández J, Acacio M, Garca J, Ujaldn M (2012) Stencil computations on heterogeneous platforms for the jacobi method: Gpus versus cell be. J Supercomput 62:787–803. doi:10.1007/s11227-012-0749-y CrossRefGoogle Scholar
  4. 4.
    Chen J (2007) An implicit approximation for the caputo fractional reaction-dispersion equation (in chinese). J Xiamen Univ (Nat Sci) 46(5):616–619Google Scholar
  5. 5.
    Chen J, Liu F, Turner I, Anh V (2008) The fundamental and numerical solutions of the riesz space fractional reaction-dispersion equation. ANZIAM J 50:45–57CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chen S, Jiang X (2012) Analytical solutions to time-fractional partial differential equations in a two-dimensional multilayer annulus. Phys A Stat Mech Appl 391(15):3865–3874. doi:10.1016/j.physa.2012.03.014 CrossRefMathSciNetGoogle Scholar
  7. 7.
    Diethelm K (2011) An efficient parallel algorithm for the numerical solution of fractional differential equations. Fract Calc Appl Anal 14:475–490. doi:10.2478/s13540-011-0029-1 CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dursun H, Kunaseth M, Nomura KI, Chame J, Lucas R, Chen C, Hall M, Kalia R, Nakano A, Vashishta P (2012) Hierarchical parallelization and optimization of high-order stencil computations on multicore clusters. J Supercomput 62:946–966. doi:10.1007/s11227-012-0764-z CrossRefGoogle Scholar
  9. 9.
    Fatone L, Giacinti M, Mariani F, Recchioni M, Zirilli F (2012) Parallel option pricing on gpu: barrier options and realized variance options. J Supercomput 62:1480–1501. doi:10.1007/s11227-012-0813-7 CrossRefGoogle Scholar
  10. 10.
    Gafiychuk V, Datsko B, Meleshko V (2008) Mathematical modeling of time fractional reaction-diffusion systems. J Comput Appl Math 220(1):215–225. doi:10.1016/ CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Gong C, Bao W, Tang G (2013) A parallel algorithm for the riesz fractional reaction-diffusion equation with explicit finite difference method. Fract Calc Appl Anal 16(3):654–669CrossRefMathSciNetGoogle Scholar
  12. 12.
    Gong C, Liu J, Chi L, Huang H, Fang J, Gong Z (2011) GPU accelerated simulations of 3D deterministic particle transport using discrete ordinates method. J Comput Phys 230(15):6010–6022. doi:10.1016/ CrossRefMATHGoogle Scholar
  13. 13.
    Gong C, Liu J, Huang H, Gong Z (2012) Particle transport with unstructured grid on gpu. Comput Phys Commun 183(3):588–593. doi:10.1016/j.cpc.2011.12.002 CrossRefGoogle Scholar
  14. 14.
    Goude A, Engblom S (2012) Adaptive fast multipole methods on the gpu. J Supercomput 1–22. doi:10.1007/s11227-012-0836-0
  15. 15.
    Haubold H, Mathai A, Saxena R (2011) Further solutions of fractional reactiondiffusion equations in terms of the h-function. J Comput Appl Math 235(5):1311–1316. doi:10.1016/ CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Hennessy JL, Patterson DA (2012) Computer architecture: a quantitative approach. Elsevier, AmsterdamGoogle Scholar
  17. 17.
    Henry B, Wearne S (2000) Fractional reaction-diffusion. Phys A Stat Mech Appl 276(3):448–455. doi:10.1016/S0378-4371(99)00469-0 Google Scholar
  18. 18.
    Huang F, Liu F (2005) The time fractional diffusion equation and the advection-dispersion equation. ANZIAM J 46(3):317–330CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Keshavarz-Kohjerdi F, Bagheri A (2013) An efficient parallel algorithm for the longest path problem in meshes. J Supercomput 1–19. doi:10.1007/s11227-012-0852-0.
  20. 20.
    Klages R, Radons G, Sokolov I (2008) Anomalous transport: foundations and applications. Wiley, WeinheimCrossRefGoogle Scholar
  21. 21.
    Li C, Zeng F, Liu F (2012) Spectral approximations to the fractional integral and derivative. Fract Calc Appl Anal 15:383–406. doi:10.2478/s13540-012-0028-x
  22. 22.
    Li R, Saad Y (2012) Gpu-accelerated preconditioned iterative linear solvers. J Supercomput 1–24. doi:10.1007/s11227-012-0825-3
  23. 23.
    Li X, Xu C (2009) A space-time spectral method for the time fractional diffusion equation. SIAM J Numer Anal 47(3):2108–2131. doi:10.1137/080718942 CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Liu Q, Liu F, Turner I, Anh V (2009) Numerical simulation for the 3d seepage flow with fractional derivatives in porous media. IMA J Appl Math 74(2):201–229. doi:10.1093/imamat/hxn044 CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Podlubny I (1999) Fractional differential equations. Academic Press, San DiegoMATHGoogle Scholar
  26. 26.
    Qi H, Jiang X (2011) Solutions of the space-time fractional cattaneo diffusion equation. Phys A Stat Mech Appl 390(11):1876–1883. doi:10.1016/j.physa.2011.02.010 CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Rida S, El-Sayed A, Arafa A (2010) On the solutions of time-fractional reactiondiffusion equations. Commun Nonlinear Sci Numer Simul 15(12):3847–3854. doi:10.1016/j.cnsns.2010.02.007 CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Salvadore F, Bernardini M, Botti M (2013) Gpu accelerated flow solver for direct numerical simulation of turbulent flows. J Comput Phys 235:129–142. doi:10.1016/ CrossRefMathSciNetGoogle Scholar
  29. 29.
    Saxena R, Mathai A, Haubold H (2006) Fractional reaction-diffusion equations. Astrophys Space Sci 305:289–296. doi:10.1007/s10509-006-9189-6 CrossRefMATHGoogle Scholar
  30. 30.
    Saxena R, Mathai A, Haubold H (2006) Solution of generalized fractional reaction-diffusion equations. Astrophys Space Sci 305:305–313. doi:10.1007/s10509-006-9191-z CrossRefMATHGoogle Scholar
  31. 31.
    Shen S, Liu F, Anh V, Turner I (2008) The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation. IMA J Appl Math 73(6):850–872. doi:10.1093/imamat/hxn033 CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Teijeiro C, Sutmann G, Taboada G, Tourio J (2012) Parallel simulation of brownian dynamics on shared memory systems with openmp and unified parallel c. J Supercomput 1–13. doi:10.1007/s11227-012-0843-1
  33. 33.
    Williams S, Oliker L, Vuduc R, Shalf J, Yelick K, Demmel J (2009) Optimization of sparse matrixvector multiplication on emerging multicore platforms. Parallel Comput 35(3):178–194. doi:10.1016/j.parco.2008.12.006 CrossRefGoogle Scholar
  34. 34.
    Xu Y, He Z (2011) The short memory principle for solving abel differential equation of fractional order. Comput Math Appl 62(12):4796–4805. doi:10.1016/j.camwa.2011.10.071 CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Yu Q, Liu F, Anh V, Turner I (2008) Solving linear and non-linear space-time fractional reaction-diffusion equations by the adomian decomposition method. Int J Numer Methods Eng 74(1):138–158. doi:10.1002/nme.2165 CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Hang X, Liu J, Wei L, Ma C. Finite element method for grwünwaldletnikov time-fractional partial differential equation. Appl Anal 1–12. doi:10.1080/00036811.2012.718332
  37. 37.
    Hang Y, Sun Z, Zhao X (2012) Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J Numer Anal 50(3):1535–1555. doi:10.1137/110840959 CrossRefMathSciNetGoogle Scholar
  38. 38.
    Zhang Z, Wang K, Li Q (2013) Accelerating a three-dimensional moc calculation using gpu with cuda and two-level gcmfd method. Ann Nucl Energy 62:445–451CrossRefMathSciNetGoogle Scholar
  39. 39.
    Huang P, Liu F, Anh V, Turner I (2009) Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process. IMA J Appl Math 74(5):645–667. doi:10.1093/imamat/hxp015 Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Chunye Gong
    • 1
    • 2
    • 3
  • Weimin Bao
    • 1
    • 2
  • Guojian Tang
    • 1
    • 2
  • Bo Yang
    • 3
  • Jie Liu
    • 3
  1. 1.College of Aerospace Science and EngineeringNational University of Defense TechnologyChangshaChina
  2. 2.Science and Technology on Space Physics LibratoryBeijingChina
  3. 3.Department of Computer SciencesNational University of Defense TechnologyChangshaChina

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