October  2021, 26(10): 5495-5508. doi: 10.3934/dcdsb.2020355

High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation

1. 

Department of Mathematics and Applications, Faculty of Sciences, University of Mohaghegh Ardabili, 56199-11367 Ardabil, Iran

2. 

Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6-95125 Catania, Italy

3. 

RUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia

* Corresponding author: Maria Alessandra Ragusa

Received  June 2020 Revised  September 2020 Published  October 2021 Early access  December 2020

In this paper, by combining of fractional centered difference approach with alternating direction implicit method, we introduce a mixed difference method for solving two-dimensional Riesz space fractional advection-dispersion equation. The proposed method is a fourth order centered difference operator in spatial directions and second order Crank-Nicolson method in temporal direction. By reviewing the consistency and stability of the method, the convergence of the proposed method is achieved. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed technique.

Citation: Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5495-5508. doi: 10.3934/dcdsb.2020355
References:
[1]

S. Abdi-MazraehM. Lakestani and M. Dehghan, The construction of operational matrices of integral and fractional integral using the flatlet oblique multiwavelets, J. Vib. Control, 21 (2015), 818-832.  doi: 10.1177/1077546313490430.

[2]

E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer: 2. Spatial moment analysis, Water Resour. Res., 28 (1992), 3293-3307.  doi: 10.1029/92WR01757.

[3]

B. BaeumerD. A. BensonM. M. Meerschaert and S. W. Wheatcraft, Subordinated advection-dispersion equation for contaminant transport, Water Resour. Res., 37 (2001), 1543-1550.  doi: 10.1029/2000WR900409.

[4]

D. A. BensonS. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), 1403-1412.  doi: 10.1029/2000WR900031.

[5]

C. Çelik and M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231 (2012), 1743-1750.  doi: 10.1016/j.jcp.2011.11.008.

[6]

S. Chen and F. Liu, ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation, J. Appl. Math. Comput., 26 (2008), 295-311.  doi: 10.1007/s12190-007-0013-4.

[7]

H. Ding and C. Li, High-order numerical algorithms for Riesz derivatives via constructing new generating functions, J. Sci. Comput., 71 (2017), 759-784.  doi: 10.1007/s10915-016-0317-3.

[8]

H. DingC. Li and Y. Chen, High-order algorithms for Riesz derivative and their applications (II), J. Comput. Phys., 293 (2015), 218-237.  doi: 10.1016/j.jcp.2014.06.007.

[9]

H.-F. Ding and Y.-X. Zhang, New numerical methods for the Riesz space fractional partial differential equations, Comput. Math. Appl., 63 (2012), 1135-1146.  doi: 10.1016/j.camwa.2011.12.028.

[10]

S. GalaQ. Liu and M. A. Ragusa, A new regularity criterion for the nematic liquid crystal flows, Appl. Anal., 91 (2012), 1741-1747.  doi: 10.1080/00036811.2011.581233.

[11]

S. Gala and M. A. Ragusa, Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices, Appl. Anal., 95 (2016), 1271-1279.  doi: 10.1080/00036811.2015.1061122.

[12]

R. Gorenflo and F. Mainardi, Approximation of Lévy-Feller diffusion by random walk models, Z. Anal. Anwendungen, 18 (1999), 231-246.  doi: 10.4171/ZAA/879.

[13]

S. Irandoust-PakchinM. DehghanS. Abdi-Mazraeh and M. Lakestani, Numerical solution for a class of fractional convection-diffusion equations using the flatlet oblique multiwavelets, J. Vib. Control, 20 (2014), 913-924.  doi: 10.1177/1077546312470473.

[14]

M. LakestaniM. Dehghan and S. Irandoust-Pakchin, The construction of operational matrix of fractional derivatives using B-spline functions, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1149-1162.  doi: 10.1016/j.cnsns.2011.07.018.

[15]

J. Manafian and M. Lakestani, A new analytical approach to solve some of the fractional-order partial differential equations, Indian J. Phys., 91 (2017), 243-258.  doi: 10.1007/s12648-016-0912-z.

[16]

C. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000.

[17]

S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM J. Control Optim., 44 (2006), 1950-1972.  doi: 10.1137/S036301290444263X.

[18]

D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3 (1955), 28-41.  doi: 10.1137/0103003.

[19] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999. 
[20]

S. Polidoro and M. A. Ragusa, Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term, Rev. Mat. Iberoam., 24 (2008), 1011-1046.  doi: 10.4171/RMI/565.

[21]

M. Popolizio, A matrix approach for partial differential equations with Riesz space fractional derivatives, Eur. Phys. J. Special Topics, 222 (2013), 1975-1985.  doi: 10.1140/epjst/e2013-01978-8.

[22]

Y. Povstenko, T. Kyrylych and G. Rygał, Fractional diffusion in a solid with mass absorption, Entropy, 19 (2017), 203. doi: 10.3390/e19050203.

[23]

M. RahmanA. Mahmood and M. Younis, Improved and more feasible numerical methods for Riesz space fractional partial differential equations, Appl. Math. Comput., 237 (2014), 264-273.  doi: 10.1016/j.amc.2014.03.103.

[24]

S. ShenF. LiuV. Anh and I. Turner, The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation, IMA J. Appl. Math., 73 (2008), 850-872.  doi: 10.1093/imamat/hxn033.

[25]

C. Tadjeran and M. M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys., 220 (2007), 813-823.  doi: 10.1016/j.jcp.2006.05.030.

[26]

J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, 22. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4899-7278-1.

[27]

F. J. Valdes-ParadaJ. A. Ochoa-Tapia and J. Alvarez-Ramirez, Effective medium equations for fractional Fick's law in porous media, Physica A: Statistical Mechanics and its Applications, 373 (2007), 339-353.  doi: 10.1016/j.physa.2006.06.007.

[28]

S. Valizadeh and A. Borhanifar, Numerical solution for Riesz fractional diffusion equation via fractional centered difference scheme, Walailak J. Sci. Tech., 2020, Accepted.

[29]

F. ZengF. LiuC. LiK. BurrageI. Turner and V. Anh, A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space farctional nonlinear reaction-diffusion equation, SIAM J. Numer. Anal., 52 (2014), 2599-2622.  doi: 10.1137/130934192.

[30]

Y. Zhang and H. Ding, Improved matrix transform method for the Riesz space fractional reaction dispersion equation, J. Comput. Appl. Math., 260 (2014), 266-280.  doi: 10.1016/j.cam.2013.09.040.

[31]

Y. Zhang and H. Ding, High-order algorithm for the two-dimension Riesz space-fractional diffusion equation, Int. J. Comput. Math., 94 (2017), 2063-2073.  doi: 10.1080/00207160.2016.1274746.

show all references

References:
[1]

S. Abdi-MazraehM. Lakestani and M. Dehghan, The construction of operational matrices of integral and fractional integral using the flatlet oblique multiwavelets, J. Vib. Control, 21 (2015), 818-832.  doi: 10.1177/1077546313490430.

[2]

E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer: 2. Spatial moment analysis, Water Resour. Res., 28 (1992), 3293-3307.  doi: 10.1029/92WR01757.

[3]

B. BaeumerD. A. BensonM. M. Meerschaert and S. W. Wheatcraft, Subordinated advection-dispersion equation for contaminant transport, Water Resour. Res., 37 (2001), 1543-1550.  doi: 10.1029/2000WR900409.

[4]

D. A. BensonS. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), 1403-1412.  doi: 10.1029/2000WR900031.

[5]

C. Çelik and M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231 (2012), 1743-1750.  doi: 10.1016/j.jcp.2011.11.008.

[6]

S. Chen and F. Liu, ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation, J. Appl. Math. Comput., 26 (2008), 295-311.  doi: 10.1007/s12190-007-0013-4.

[7]

H. Ding and C. Li, High-order numerical algorithms for Riesz derivatives via constructing new generating functions, J. Sci. Comput., 71 (2017), 759-784.  doi: 10.1007/s10915-016-0317-3.

[8]

H. DingC. Li and Y. Chen, High-order algorithms for Riesz derivative and their applications (II), J. Comput. Phys., 293 (2015), 218-237.  doi: 10.1016/j.jcp.2014.06.007.

[9]

H.-F. Ding and Y.-X. Zhang, New numerical methods for the Riesz space fractional partial differential equations, Comput. Math. Appl., 63 (2012), 1135-1146.  doi: 10.1016/j.camwa.2011.12.028.

[10]

S. GalaQ. Liu and M. A. Ragusa, A new regularity criterion for the nematic liquid crystal flows, Appl. Anal., 91 (2012), 1741-1747.  doi: 10.1080/00036811.2011.581233.

[11]

S. Gala and M. A. Ragusa, Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices, Appl. Anal., 95 (2016), 1271-1279.  doi: 10.1080/00036811.2015.1061122.

[12]

R. Gorenflo and F. Mainardi, Approximation of Lévy-Feller diffusion by random walk models, Z. Anal. Anwendungen, 18 (1999), 231-246.  doi: 10.4171/ZAA/879.

[13]

S. Irandoust-PakchinM. DehghanS. Abdi-Mazraeh and M. Lakestani, Numerical solution for a class of fractional convection-diffusion equations using the flatlet oblique multiwavelets, J. Vib. Control, 20 (2014), 913-924.  doi: 10.1177/1077546312470473.

[14]

M. LakestaniM. Dehghan and S. Irandoust-Pakchin, The construction of operational matrix of fractional derivatives using B-spline functions, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1149-1162.  doi: 10.1016/j.cnsns.2011.07.018.

[15]

J. Manafian and M. Lakestani, A new analytical approach to solve some of the fractional-order partial differential equations, Indian J. Phys., 91 (2017), 243-258.  doi: 10.1007/s12648-016-0912-z.

[16]

C. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000.

[17]

S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM J. Control Optim., 44 (2006), 1950-1972.  doi: 10.1137/S036301290444263X.

[18]

D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3 (1955), 28-41.  doi: 10.1137/0103003.

[19] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999. 
[20]

S. Polidoro and M. A. Ragusa, Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term, Rev. Mat. Iberoam., 24 (2008), 1011-1046.  doi: 10.4171/RMI/565.

[21]

M. Popolizio, A matrix approach for partial differential equations with Riesz space fractional derivatives, Eur. Phys. J. Special Topics, 222 (2013), 1975-1985.  doi: 10.1140/epjst/e2013-01978-8.

[22]

Y. Povstenko, T. Kyrylych and G. Rygał, Fractional diffusion in a solid with mass absorption, Entropy, 19 (2017), 203. doi: 10.3390/e19050203.

[23]

M. RahmanA. Mahmood and M. Younis, Improved and more feasible numerical methods for Riesz space fractional partial differential equations, Appl. Math. Comput., 237 (2014), 264-273.  doi: 10.1016/j.amc.2014.03.103.

[24]

S. ShenF. LiuV. Anh and I. Turner, The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation, IMA J. Appl. Math., 73 (2008), 850-872.  doi: 10.1093/imamat/hxn033.

[25]

C. Tadjeran and M. M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys., 220 (2007), 813-823.  doi: 10.1016/j.jcp.2006.05.030.

[26]

J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, 22. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4899-7278-1.

[27]

F. J. Valdes-ParadaJ. A. Ochoa-Tapia and J. Alvarez-Ramirez, Effective medium equations for fractional Fick's law in porous media, Physica A: Statistical Mechanics and its Applications, 373 (2007), 339-353.  doi: 10.1016/j.physa.2006.06.007.

[28]

S. Valizadeh and A. Borhanifar, Numerical solution for Riesz fractional diffusion equation via fractional centered difference scheme, Walailak J. Sci. Tech., 2020, Accepted.

[29]

F. ZengF. LiuC. LiK. BurrageI. Turner and V. Anh, A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space farctional nonlinear reaction-diffusion equation, SIAM J. Numer. Anal., 52 (2014), 2599-2622.  doi: 10.1137/130934192.

[30]

Y. Zhang and H. Ding, Improved matrix transform method for the Riesz space fractional reaction dispersion equation, J. Comput. Appl. Math., 260 (2014), 266-280.  doi: 10.1016/j.cam.2013.09.040.

[31]

Y. Zhang and H. Ding, High-order algorithm for the two-dimension Riesz space-fractional diffusion equation, Int. J. Comput. Math., 94 (2017), 2063-2073.  doi: 10.1080/00207160.2016.1274746.

Table 1.  The maximum errors and convergence rates for the modified Crank-Nicolson ADI method for solving two dimensional RSFADE with halved spatial step sizes and $ \mathit{k}_{t} = 0.001 $
Maximum Estimated
$ \mathit{h}_{x}=\mathit{h}_{y} $ Absolute Error Convergence Rate
$ 0.10000 $ $ 3.19826e-003 $ -
$ 0.05000 $ $ 2.61740e-004 $ $ 3.61108 $
$ 0.02500 $ $ 1.90572e-005 $ $ 3.77973 $
$ 0.01250 $ $ 1.33477e-006 $ $ 3.83567 $
$ 0.00625 $ $ 8.92357e-008 $ $ 3.90283 $
Maximum Estimated
$ \mathit{h}_{x}=\mathit{h}_{y} $ Absolute Error Convergence Rate
$ 0.10000 $ $ 3.19826e-003 $ -
$ 0.05000 $ $ 2.61740e-004 $ $ 3.61108 $
$ 0.02500 $ $ 1.90572e-005 $ $ 3.77973 $
$ 0.01250 $ $ 1.33477e-006 $ $ 3.83567 $
$ 0.00625 $ $ 8.92357e-008 $ $ 3.90283 $
Table 2.  The maximum errors and convergence rates for the modified Crank-Nicolson ADI method for solving two dimensional RSFADE with halved temporal step sizes and $ \mathit{h}_{x} = \mathit{h}_{y} = 0.001 $
Maximum Estimated
$ \mathit{k}_{t} $ Absolute Error Convergence Rate
$ 0.10000 $ $ 3.77425e-003 $ -
$ 0.05000 $ $ 1.20417e-003 $ $ 1.64815 $
$ 0.02500 $ $ 3.55418e-004 $ $ 1.76045 $
$ 0.01250 $ $ 1.02360e-004 $ $ 1.79586 $
$ 0.00625 $ $ 2.69907e-005 $ $ 1.92312 $
Maximum Estimated
$ \mathit{k}_{t} $ Absolute Error Convergence Rate
$ 0.10000 $ $ 3.77425e-003 $ -
$ 0.05000 $ $ 1.20417e-003 $ $ 1.64815 $
$ 0.02500 $ $ 3.55418e-004 $ $ 1.76045 $
$ 0.01250 $ $ 1.02360e-004 $ $ 1.79586 $
$ 0.00625 $ $ 2.69907e-005 $ $ 1.92312 $
Table 3.  The maximum errors and convergence rates for the modified Crank-Nicolson ADI method for solving two dimensional RSFADE with halved spatial step sizes and $ \mathit{k}_{t} = 0.001 $
Maximum Estimated
$ \mathit{h}_{x}=\mathit{h}_{y} $ Absolute Error Convergence Rate
$ 0.10000\pi $ $ 3.26587e-004 $ -
$ 0.05000\pi $ $ 2.60038e-005 $ $ 3.65067 $
$ 0.02500\pi $ $ 1.81670e-006 $ $ 3.83933 $
$ 0.01250\pi $ $ 1.26448e-007 $ $ 3.84471 $
$ 0.00625\pi $ $ 8.13028e-009 $ $ 3.95909 $
Maximum Estimated
$ \mathit{h}_{x}=\mathit{h}_{y} $ Absolute Error Convergence Rate
$ 0.10000\pi $ $ 3.26587e-004 $ -
$ 0.05000\pi $ $ 2.60038e-005 $ $ 3.65067 $
$ 0.02500\pi $ $ 1.81670e-006 $ $ 3.83933 $
$ 0.01250\pi $ $ 1.26448e-007 $ $ 3.84471 $
$ 0.00625\pi $ $ 8.13028e-009 $ $ 3.95909 $
Table 4.  The maximum errors and convergence rates for the modified Crank-Nicolson ADI method for solving two dimensional RSFADE with halved temporal step sizes and $ \mathit{h}_{x} = \mathit{h}_{y} = 0.001\pi $
Maximum Estimated
$ \mathit{k}_{t} $ Absolute Error Convergence Rate
$ 0.10000 $ $ 4.79240e-003 $ -
$ 0.05000 $ $ 1.46420e-003 $ $ 1.71063 $
$ 0.02500 $ $ 4.21902e-004 $ $ 1.79513 $
$ 0.01250 $ $ 1.14998e-004 $ $ 1.87530 $
$ 0.00625 $ $ 2.95945e-005 $ $ 1.95821 $
Maximum Estimated
$ \mathit{k}_{t} $ Absolute Error Convergence Rate
$ 0.10000 $ $ 4.79240e-003 $ -
$ 0.05000 $ $ 1.46420e-003 $ $ 1.71063 $
$ 0.02500 $ $ 4.21902e-004 $ $ 1.79513 $
$ 0.01250 $ $ 1.14998e-004 $ $ 1.87530 $
$ 0.00625 $ $ 2.95945e-005 $ $ 1.95821 $
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