THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Hou-Ping Dai

,

Zhou-Shun Zheng

,

Wei Tan

LATTICE BOLTZMANN MODEL FOR THE RIESZ SPACE FRACTIONAL REACTION-DIFFUSION

ABSTRACT
In this paper, a Riesz space fractional reaction-diffusion equation with non-linear source term is considered on a finite domain. This equation is commonly used to describe anomalous diffusion in thermal science. To solve the diffusion equation, a new fractional lattice Boltzmann method is proposed. Firstly, a difference approximation for the global spatial correlation of Riesz fractional derivative is derived by applying the numerical discretization technique, and a brief convergence analysis is presented. Then the global spatial correlation process is inserted into the evolution process of the standard lattice Boltzmann method. With combining Taylor expansion, Chapman-Enskog expansion and the multi-scales expansion, the governing evolution equation is recovered from the continuous Boltzmann equation. Three numerical examples are provided to confirm our theoretical analysis and illustrate the effectiveness of our method at last.
KEYWORDS
Lattice Boltzmann Method, non-linear source term, Taylor expansion, Riesz space fractional reaction-diffusion equation, multi-scales expansion
PAPER SUBMITTED: 2017-09-05
PAPER REVISED: 2017-09-28
PAPER ACCEPTED: 2017-09-28
PUBLISHED ONLINE: 2018-09-10
DOI REFERENCE: https://doi.org/10.2298/TSCI1804831D
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2018, VOLUME 22, ISSUE Issue 4, PAGES [1831 - 1843]
REFERENCES
  1. Chaves, A. S., A Fractional Diffusion Equation to Describe Levy Flights, Physics Letters A, 239 (1998), 1-2, pp. 13-16
  2. Benson, D. A., et al., Application of a Fractional Advection-Dispersion Equation, Water Resources Re-search, 36 (2000), 5, pp. 1403-1412
  3. Benson, D. A., et al., The Fractional‐Order Governing Equation of Levy Motion, Water Resources Re-search, 36 (2000), 6, pp. 1413-1424
  4. Tadjerana, C., A Second-Order Accurate Numerical Approximation for the Fractional Diffusion Equa-tion, Journal of Computational Physics, 213 (2006), 1, pp. 205-213
  5. He, J.-H., Li, Z.-B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334
  6. Zhang, H., et al., Galerkin Finite Element Approximation of Symmetric Space-Fractional Partial Differ-ential Equations, Applied Mathematics & Computation, 217 (2010), 6, pp. 2534-2545
  7. Sousa, E., A Second Order Explicit Finite Difference Method for the Fractional Advection Diffusion Equation, Computers & Mathematics with Applications, 64 (2012), 10, pp. 3141-3152
  8. Xu, Q., et al. Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations, Society for Industrial and Applied Mathematics, 52 (2014), 1, pp. 405-423
  9. Chen, S., Doolen, G. D., Lattice Boltzmann Method for Fluid Flows, Annual Review of Fluid Mechanics, 30 (2012), 1, pp. 329-364
  10. Wang, C. C., et al., Application of Lattice Boltzmann Method and Field Synergy Principle to the Heat Transfer Analysis of Channel Flow with Obstacles Inside, Thermal Science, 15 (2011), 5, pp. 75-80
  11. Walther, E., et al., Lattice Boltzmann Method and Diffusion in Materials with Large Diffusivity Ratios, Thermal Science, 21 (2017), 3, pp. 1173-1182
  12. Jourabian, M., et al., Lattice Boltzmann Simulation of Melting Phenomenon with Natural Convection from an Eccentric Annulus, Thermal Science, 17 (2013), 3, pp. 877-890
  13. Dawson, S. P., et al., Lattice Boltzmann Computations for Reaction-Diffusion Equations, Journal of Chemical Physics, 98 (1993), 2, pp. 1514-1523
  14. Chai, Z., Zhao, T. S., Lattice Boltzmann Model for the Convection-Diffusion Equation, Physical Review E Statistical Nonlinear & Soft Matter Physics, 87 (2013), 6, pp. 1-15
  15. Yoshida, H., Nagaoka, M., Multiple-Relaxation-Time Lattice Boltzmann Model for the Convection and Anisotropic Diffusion Equation, Journal of Computational Physics, 229 (2010), 20, pp. 7774-7795
  16. Xiang, X., et al., Modified Lattice Boltzmann Scheme for Nonlinear Convection Diffusion Equations, Communications in Nonlinear Science & Numerical Simulation, 17 (2012), 6, pp. 2415-2425
  17. Li, Q., et al., Numerical Solution of Nonlinear Klein-Gordon Equation Using Lattice Boltzmann Meth-od, Applied Mathematics, 02 (2011), 12, pp. 1479-1485
  18. Lai, H., Ma, C., Numerical Study of the Nonlinear Combined Sine-Cosine-Gordon Equation with the Lattice Boltzmann Method, Journal of Scientific Computing, 53 (2012), 3, pp. 569-585
  19. Duan, Y. L., Liu, R. X., Lattice Boltzmann Model for Two-Dimensional Unsteady Burgers' Equation, Journal of Computational & Applied Mathematics, 206 (2007), 1, pp. 432-439
  20. Yan, G., Zhang, J., A Higher-Order Moment Method of the Lattice Boltzmann Model for the Korteweg-de Vries Equation, Mathematics & Computers in Simulation, 79 (2009), 5, pp. 1554-1565
  21. Zhang, J., Yan, G., Lattice Boltzmann Method for the Fractional Sub‐Diffusion Equation, International Journal for Numerical Methods in Fluids, 80 (2016), 8, pp. 490-507
  22. Zhou, J. G., et al., Lattice Boltzmann Method for the Fractional Advection-Diffusion Equation, Physical Review E, 93 (2016), 4, pp. 043310
  23. Gorenflo, R., Mainardi, F., Random Walk Models for Space-Fractional Diffusion Processes, Lecture Notes in Physics, 2 (1998), 2, pp. 148-166
  24. Lin, Y., Xu, C., Finite Difference/Spectral Approximations for the Time-Fractional Diffusion Equation, Journal of Computational Physics, 225 (2007), 2, pp. 1533-1552
  25. Buick, J. M., Greated, C. A., Gravity in a Lattice Boltzmann Model, Physical Review E Statistical Phys-ics Plasmas Fluids & Related Interdisciplinary Topics, 61 (2000), 5A, pp. 5307-5320
  26. Liu, F., Zhang, H. M., Numerical Simulation of the Riesz Fractional Diffusion Equation with a Nonline-ar Source Term, Journal of Applied Mathematics & Informatics, 26 (2008),1-2, pp. 1-14
  27. Liu, F., et al., Numerical Solution of the Space Fractional Fokker-Planck Equation, Journal of Computa-tional and Applied Mathematics. 166 (2004), 1, pp. 209-219
  28. Baeumer, B., et al., Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels, Bulletin of Mathematical Biology, 69 (2007), 7, pp. 2281-2297
PDF VERSION [DOWNLOAD]
Lattice Boltzmann model for the Riesz space fractional reaction-diffusion

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence