Skip to main content

A New Family of Difference Schemes for Space Fractional Advection Diffusion Equation

  • Can Li (a1) and Weihua Deng (a2)

The second order weighted and shifted Grünwald difference (WSGD) operators are developed in [Tian, Zhou and Deng, Math. Comput., 84 (2015), pp. 1703–1727] to solve space fractional partial differential equations. Along this direction, we further design a new family of second order WSGD operators; by properly choosing the weighted parameters, they can be effectively used to discretize space (Riemann-Liouville) fractional derivatives. Based on the new second order WSGD operators, we derive a family of difference schemes for the space fractional advection diffusion equation. By von Neumann stability analysis, it is proved that the obtained schemes are unconditionally stable. Finally, extensive numerical experiments are performed to demonstrate the performance of the schemes and confirm the convergence orders.

Corresponding author
*Corresponding author. Email: (C. Li), (W. H. Deng)
Hide All
[1] Baeumer B., Kovcs M. and Sankaranarayanan H., Higher order Grünwald approximations of fractional derivatives and fractional powers of operators, Trans. Amer. Math. Soc., 367 (2015), pp. 813834.
[2] Benson D. A., Wheatcraft S. W. and Meerschaert M. M., The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2000), pp. 14131423.
[3] ÇElik C. and Duman M., Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231 (2012), pp. 17431750.
[4] Chan T. F., Stability analysis of finite difference schemes for the advection-diffusion equation, SIAM J. Numer. Anal., 21 (1984), pp. 272284.
[5] Chen M. H., Deng W. H. and Wu Y. J., Superlinearly convergent algorithms for the two dimensional space-time Caputo-Riesz fractional diffusion equation, Appl. Numer. Math., 70 (2013), pp. 2241.
[6] Deng Z. Q., Singh V. P., Asce F. and Bengtsson L., Numerical solution of fractional advection-dispersion different equations, J. Hydral. Eng., 130 (2004), pp. 422431.
[7] Dyakonov E. G., Difference schemes of second-order accuracy with a splitting operator for parabolic equations without mixed derivatives, Zh. Vychisl. Mat. I Mat. Fiz., 4 (1964), pp. 935941.
[8] Fornberg B., Calculation of weights in finite difference formulas, SIAM Rev., 40 (1998), pp. 685691.
[9] Gustafsson B., Kreiss H.-O. and Oliger J., Time Dependent Problems and Difference Methods, Wiley Interscience, New York, 1995.
[10] Liu F., Ahn V. and Turner I., Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2004), pp. 209219.
[11] Lubich Ch., Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), pp. 704719.
[12] Lynch V. E., Carreras B. A., Del-Castillo-Negrete D., Ferreira-Mejias K. M. and Hicks H. R., Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys., 192 (2003), pp. 406421.
[13] Meerschaert M. M. and Tadjeran C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), pp. 6577.
[14] Meerschaert M. M., Scheffler H.-P. and Tadjeran C., Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006), pp. 249261.
[15] Meerschaert M. M. and Tadjeran C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), pp. 8090.
[16] Metzler R. and Klafter J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A Math. General, 37 (2004), pp. 161208.
[17] Metzler R. and Klafter J., The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), pp. 177.
[18] Oldham K. B. and Spanier J., The fractional calculus, Academic Press, New York, 1974.
[19] Ortigueira M. D., Riesz potential operators and inverses via fractional centred derivatives, Int. J. Math. Math. Sci., 2006 (2006), pp. 112.
[20] Pang H.-K. and Sun H. W., Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), pp. 693703.
[21] Peaceman D. W. and Rachford H. H. Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3 (1959), pp. 2841.
[22] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999.
[23] Song J., Yu Q., Liu F. and Turner I., A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation, Numer. Algorithms, 66 (2014), pp. 911932.
[24] Sousa E., Finite difference approximations for a fractional advection diffusion problem, J. Comput. Phys., 228 (2009), pp. 40384054.
[25] Sousa E. and Li C., A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative, Appl. Numer. Math., 90 (2015), pp. 2237.
[26] Strikwerda J. C., Finite Difference Schemes and Partial Differential Equations, SIAM, 2004.
[27] Sun Z. Z., Numerical Methods of Partial Differential Equations (in Chinese), Science Press, Beijing, 2005.
[28] Thomas J. W., Numerical Partial Differential Equations: Finite Difference Methods, Springer New York, 1995.
[29] Tuan V. K. and Gorenflo R., Extrapolation to the limit for numerical fractional differentiation, Z. Angew. Math. Mech., 75 (1995), pp. 646648.
[30] Tian W. Y., Zhou H. and Deng W. H., A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput., 84 (2015), pp. 17031727.
[31] Wang D. L., Xiao A. G. and Yang W., Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative, J. Comput. Phys., 242 (2013), pp. 670681.
[32] Wang H., Wang K. and Sircar T., A direct Nlog2N finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), pp. 80958104.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 78 *
Loading metrics...

Abstract views

Total abstract views: 316 *
Loading metrics...

* Views captured on Cambridge Core between 9th January 2017 - 22nd January 2018. This data will be updated every 24 hours.