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

Part of: Real functions Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  09 January 2017

Can Li  and
Weihua Deng
Can Li*
Affiliation:
Department of Applied Mathematics, School of Sciences, Xi'an University of Technology, Xi'an, Shaanxi 710054, China
Weihua Deng*
Affiliation:
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China
*
*Corresponding author. Email:mathlican@xaut.edu.cn (C. Li), dengwh@lzu.edu.cn (W. H. Deng)
*Corresponding author. Email:mathlican@xaut.edu.cn (C. Li), dengwh@lzu.edu.cn (W. H. Deng)
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Abstract

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.

Keywords

Riemann-Liouville fractional derivativeWSGD operatorfractional advection diffusion equationfinite difference approximationstability

MSC classification

Secondary: 26A33: Fractional derivatives and integrals 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 2 , April 2017 , pp. 282 - 306
Copyright
Copyright © Global-Science Press 2017 

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References

[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. 813–834.Google Scholar
[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. 1413–1423.Google Scholar
[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. 1743–1750.Google Scholar
[4] Chan, T. F., Stability analysis of finite difference schemes for the advection-diffusion equation, SIAM J. Numer. Anal., 21 (1984), pp. 272–284.Google Scholar
[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. 22–41.Google Scholar
[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. 422–431.CrossRefGoogle Scholar
[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. 935–941.Google Scholar
[8] Fornberg, B., Calculation of weights in finite difference formulas, SIAM Rev., 40 (1998), pp. 685–691.Google Scholar
[9] Gustafsson, B., Kreiss, H.-O. and Oliger, J., Time Dependent Problems and Difference Methods, Wiley Interscience, New York, 1995.Google Scholar
[10] Liu, F., Ahn, V. and Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2004), pp. 209–219.Google Scholar
[11] Lubich, Ch., Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), pp. 704–719.Google Scholar
[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. 406–421.CrossRefGoogle Scholar
[13] Meerschaert, M. M. and Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), pp. 65–77.Google Scholar
[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. 249–261.Google Scholar
[15] Meerschaert, M. M. and Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), pp. 80–90.Google Scholar
[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. 161–208.Google Scholar
[17] Metzler, R. and Klafter, J., The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), pp. 1–77.Google Scholar
[18] Oldham, K. B. and Spanier, J., The fractional calculus, Academic Press, New York, 1974.Google Scholar
[19] Ortigueira, M. D., Riesz potential operators and inverses via fractional centred derivatives, Int. J. Math. Math. Sci., 2006 (2006), pp. 1–12.Google Scholar
[20] Pang, H.-K. and Sun, H. W., Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), pp. 693–703.Google Scholar
[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. 28–41.Google Scholar
[22] Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.Google Scholar
[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. 911–932.Google Scholar
[24] Sousa, E., Finite difference approximations for a fractional advection diffusion problem, J. Comput. Phys., 228 (2009), pp. 4038–4054.Google Scholar
[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. 22–37.CrossRefGoogle Scholar
[26] Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations, SIAM, 2004.Google Scholar
[27] Sun, Z. Z., Numerical Methods of Partial Differential Equations (in Chinese), Science Press, Beijing, 2005.Google Scholar
[28] Thomas, J. W., Numerical Partial Differential Equations: Finite Difference Methods, Springer New York, 1995.Google Scholar
[29] Tuan, V. K. and Gorenflo, R., Extrapolation to the limit for numerical fractional differentiation, Z. Angew. Math. Mech., 75 (1995), pp. 646–648.Google Scholar
[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. 1703–1727.Google Scholar
[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. 670–681.Google Scholar
[32] Wang, H., Wang, K. and Sircar, T., A direct Nlog2N finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), pp. 8095–8104.Google Scholar