Preconditioned iterative methods for solving a fractional advection-diffusion equation
DOI:
https://doi.org/10.5269/bspm.64142Abstract
In this paper, we consider numerical solutions of a fractional advection-diffusion equation. We first, propose an implicit method based on Grunwald formulae and then discuss its stability and consistency. To improve the implicit method, we use a preconditioned generalized minimal residual (PGMRES) method and preconditioned conjugate gradient normal residual (PCGNR) method. Numerical experiments are given to illustrate efficiency of the method.
References
[1] Liu, F., Zhuang, P., Anh, V., Turner, I., & Burrage, K., Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation, Applied Mathematics and Computation. 191(1), 12-20, (2007).
[2] Bai, J., & Feng, X. C., Fractional-order anisotropic diffusion for image denoising, IEEE transactions on image processing. 16(10), 2492-2502, (2007).
[3] Van Beinum, W., Meeussen, J. C., Edwards, A. C., & Van Riemsdijk, W. H., Transport of ions in physically heterogeneous systems; convection and diffusion in a column filled with alginate gel beads, predicted by a two-region model, Water Research. 34(7), 2043-2050, (2000).
[4] Sokolov, I. M., Klafter, J., & Blumen, A., Fractional kinetics, Physics Today, 55(11), 48-54, (2002).
[5] Savovic, S., & Djordjevich, A., ´ Finite difference solution of the one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media, International Journal of Heat and Mass Transfer. 55(15-16), 4291-4294 (2012).
[6] Dhawan, S., Kapoor, S., & Kumar, S., Numerical method for advection diffusion equation using FEM and B-splines, Journal of Computational Science, 3(5), 429-437, (2012).
[7] Xu, Y., & He, Z., The short memory principle for solving Abel differential equation of fractional order, Computers & Mathematics with Applications, 62(12), 4796-4805, (2011).
[8] Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional integrals and derivatives, Gordon and Breach Science Publishers, theory and applications. (1993).
[9] Zheng, Y., Li, C., & Zhao, Z., A note on the finite element method for the space-fractional advection diffusion equation, Computers & Mathematics with Applications. 59(5), 1718-1726, (2010).
[10] Lax, P. D., & Richtmyer, R. D., Survey of the stability of linear finite difference equations, In Selected Papers Volume I, Springer, New York, NY. 125-151, (2005).
[11] Magin, R. L., Fractional calculus in bioengineering begell house publishers, Inc., Connecticut. (2006).
[12] Tadjeran, C., Meerschaert, M. M., & Scheffler, H. P., A second-order accurate numerical approximation for the fractional diffusion equation, Journal of computational physics. 213(1), 205-213, (2006).
[13] Meerschaert, M. M., & Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Applied numerical mathematics. 56(1), 80-90, (2006).
[14] Meerschaert, M. M., & Tadjeran, C., Finite difference approximations for fractional advection–dispersion flow equations, Journal of computational and applied mathematics. 172(1), 65-77, (2004).
[15] Podlubny, I., Fractional differential equations, mathematics in science and engineering, (1999).
[16] Pang, H. K., & Sun, H. W., Multigrid method for fractional diffusion equations, Journal of Computational Physics. 231(2), 693-703, (2012).
[17] Saad, Y., & Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on scientific and statistical computing. 7(3), 856-869, (1986).
[18] Saad, Y., Iterative methods for sparse linear systems, Society for Industrial and Applied Mathematics. (2003)
[2] Bai, J., & Feng, X. C., Fractional-order anisotropic diffusion for image denoising, IEEE transactions on image processing. 16(10), 2492-2502, (2007).
[3] Van Beinum, W., Meeussen, J. C., Edwards, A. C., & Van Riemsdijk, W. H., Transport of ions in physically heterogeneous systems; convection and diffusion in a column filled with alginate gel beads, predicted by a two-region model, Water Research. 34(7), 2043-2050, (2000).
[4] Sokolov, I. M., Klafter, J., & Blumen, A., Fractional kinetics, Physics Today, 55(11), 48-54, (2002).
[5] Savovic, S., & Djordjevich, A., ´ Finite difference solution of the one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media, International Journal of Heat and Mass Transfer. 55(15-16), 4291-4294 (2012).
[6] Dhawan, S., Kapoor, S., & Kumar, S., Numerical method for advection diffusion equation using FEM and B-splines, Journal of Computational Science, 3(5), 429-437, (2012).
[7] Xu, Y., & He, Z., The short memory principle for solving Abel differential equation of fractional order, Computers & Mathematics with Applications, 62(12), 4796-4805, (2011).
[8] Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional integrals and derivatives, Gordon and Breach Science Publishers, theory and applications. (1993).
[9] Zheng, Y., Li, C., & Zhao, Z., A note on the finite element method for the space-fractional advection diffusion equation, Computers & Mathematics with Applications. 59(5), 1718-1726, (2010).
[10] Lax, P. D., & Richtmyer, R. D., Survey of the stability of linear finite difference equations, In Selected Papers Volume I, Springer, New York, NY. 125-151, (2005).
[11] Magin, R. L., Fractional calculus in bioengineering begell house publishers, Inc., Connecticut. (2006).
[12] Tadjeran, C., Meerschaert, M. M., & Scheffler, H. P., A second-order accurate numerical approximation for the fractional diffusion equation, Journal of computational physics. 213(1), 205-213, (2006).
[13] Meerschaert, M. M., & Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Applied numerical mathematics. 56(1), 80-90, (2006).
[14] Meerschaert, M. M., & Tadjeran, C., Finite difference approximations for fractional advection–dispersion flow equations, Journal of computational and applied mathematics. 172(1), 65-77, (2004).
[15] Podlubny, I., Fractional differential equations, mathematics in science and engineering, (1999).
[16] Pang, H. K., & Sun, H. W., Multigrid method for fractional diffusion equations, Journal of Computational Physics. 231(2), 693-703, (2012).
[17] Saad, Y., & Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on scientific and statistical computing. 7(3), 856-869, (1986).
[18] Saad, Y., Iterative methods for sparse linear systems, Society for Industrial and Applied Mathematics. (2003)
Downloads
Published
2025-09-22
Issue
Section
Research Articles
License
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
How to Cite
Raeisi, B. ., & Khoshsiar Ghaziani, R. (2025). Preconditioned iterative methods for solving a fractional advection-diffusion equation. Boletim Da Sociedade Paranaense De Matemática, 43, 1-13. https://doi.org/10.5269/bspm.64142
