July  2015, 20(5): 1427-1441. doi: 10.3934/dcdsb.2015.20.1427

Fast finite volume methods for space-fractional diffusion equations

1. 

Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

2. 

School of Mathematics, Shandong University, Jinan, 250100, China, China

Received  September 2013 Revised  January 2015 Published  May 2015

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that is characterized by a heavy tail or an inverse power law decay, which cannot be modeled accurately by second-order diffusion equations that is well known to model Brownian motions that are characterized by an exponential decay. However, fractional differential equations introduce new mathematical and numerical difficulties that have not been encountered in the context of traditional second-order differential equations. For instance, because of the nonlocal property of fractional differential operators, the corresponding numerical methods have full coefficient matrices which require storage of $O(N^2)$ and computational cost of $O(N^3)$ where $N$ is the number of grid points.
    We develop a fast locally conservative finite volume method for a time-dependent variable-coefficient conservative space-fractional diffusion equation. This method requires only a computational cost of $O(N \log N)$ at each iteration and a storage of $O(N)$. Numerical experiments are presented to investigate the performance of the method and to show the strong potential of these methods.
Citation: Hong Wang, Aijie Cheng, Kaixin Wang. Fast finite volume methods for space-fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1427-1441. doi: 10.3934/dcdsb.2015.20.1427
References:
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V. J. Ervin, N. Heuer and J. P. Roop, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation,, SIAM J. Numer. Anal., 45 (2007), 572.  doi: 10.1137/050642757.  Google Scholar

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M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations,, J. Comput. Appl. Math., 172 (2004), 65.  doi: 10.1016/j.cam.2004.01.033.  Google Scholar

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M. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations,, Appl. Numer. Math., 56 (2006), 80.  doi: 10.1016/j.apnum.2005.02.008.  Google Scholar

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C. Tadjeran, M. M. Meerschaert and H. P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation,, J. Comput. Phys., 213 (2006), 205.  doi: 10.1016/j.jcp.2005.08.008.  Google Scholar

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H. Wang and T. S. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations,, SIAM J. Sci. Comput., 34 (2012).  doi: 10.1137/12086491X.  Google Scholar

[35]

H. Wang and N. Du, A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations,, J. Comput. Phys., 240 (2013), 49.  doi: 10.1016/j.jcp.2012.07.045.  Google Scholar

[36]

H. Wang and N. Du, A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation,, J. Comput. Phys., 253 (2013), 50.  doi: 10.1016/j.jcp.2013.06.040.  Google Scholar

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H. Wang and N. Du, Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations,, J. Comput. Phys., 258 (2014), 305.  doi: 10.1016/j.jcp.2013.10.040.  Google Scholar

[38]

H. Wang and K. Wang, An $O(N \log^2 N)$ alternating-direction finite difference method for two-dimensional fractional diffusion equations,, J. Comput. Phys., 230 (2011), 7830.  doi: 10.1016/j.jcp.2011.07.003.  Google Scholar

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H. Wang, K. Wang and T. Sircar, A direct $O(N\log^2 N)$ finite difference method for fractional diffusion equations,, J. Comput. Phys., 229 (2010), 8095.  doi: 10.1016/j.jcp.2010.07.011.  Google Scholar

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H. Wang and D. Yang, Wellposedness of variable-coefficient conservative fractional elliptic differential equations,, SIAM J. Numer. Anal., 51 (2013), 1088.  doi: 10.1137/120892295.  Google Scholar

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H. Wang, D. Yang and S. Zhu, Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations,, SIAM J. Numer. Anal., 52 (2014), 1292.  doi: 10.1137/130932776.  Google Scholar

show all references

References:
[1]

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. M. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods,, SIAM, (1994).  doi: 10.1137/1.9781611971538.  Google Scholar

[2]

T. S. Basu and H. Wang, A fast second-order finite difference method for space-fractional diffusion equations,, Int'l J. Numer. Anal. Modeling, 9 (2012), 658.   Google Scholar

[3]

B. Beumer, M. Kovàcs and M. M. Meerschaert, Numerical solutions for fractional reaction-diffusion equations,, Computers & Mathematics with Applications, 55 (2008), 2212.  doi: 10.1016/j.camwa.2007.11.012.  Google Scholar

[4]

D. Benson, S. W. Wheatcraft and M. M. Meerschaert, The fractional-order governing equation of Lévy motion,, Water Resour. Res., 36 (2000), 1413.   Google Scholar

[5]

A. Böttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices,, Springer, (1999).  doi: 10.1007/978-1-4612-1426-7.  Google Scholar

[6]

M. Cui, Compact finite difference method for the fractional diffusion equation,, J. Comput. Phys., 228 (2009), 7792.  doi: 10.1016/j.jcp.2009.07.021.  Google Scholar

[7]

P. J. Davis, Circulant Matrices,, Wiley-Intersciences, (1979).   Google Scholar

[8]

W. Deng, Finite element method for the space and time fractional Fokker-Planck equation,, SIAM J. Numer. Anal., 47 (2008), 204.  doi: 10.1137/080714130.  Google Scholar

[9]

V. J. Ervin, N. Heuer and J. P. Roop, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation,, SIAM J. Numer. Anal., 45 (2007), 572.  doi: 10.1137/050642757.  Google Scholar

[10]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation,, Numer. Methods Partial Differential Eq, 22 (2005), 558.  doi: 10.1002/num.20112.  Google Scholar

[11]

V. J. Ervin and J. P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in $\mathbbR^d$,, Numer. Methods Partial Differential Eq., 23 (2007), 256.  doi: 10.1002/num.20169.  Google Scholar

[12]

R. M. Gray, Toeplitz and circulant matrices: A review,, Foundations and Trends in Communications and Information Theory, 2 (2006), 155.  doi: 10.1561/0100000006.  Google Scholar

[13]

J. Jia, C. Wang and H. Wang, A fast locally refined method for a space-fractional diffusion equation,, submitted., ().   Google Scholar

[14]

T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation,, J. Comput. Phys., 205 (2005), 719.  doi: 10.1016/j.jcp.2004.11.025.  Google Scholar

[15]

C. Li and F. Zeng, Finite difference methods for fractional differential equations,, Int'l J. Bifurcation Chaos, 22 (2012).  doi: 10.1142/S0218127412300145.  Google Scholar

[16]

X. Li and C. Xu, The existence and uniqueness of the week solution of the space-time fractional diffusion equation and a spectral method approximation,, Commun. Comput. Phys., 8 (2010), 1016.  doi: 10.4208/cicp.020709.221209a.  Google Scholar

[17]

R. Lin, F. Liu, V. Anh and I. Turner, Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation,, Appl. Math. Comp., 212 (2009), 435.  doi: 10.1016/j.amc.2009.02.047.  Google Scholar

[18]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation,, J. Comput. Phys., 225 (2007), 1533.  doi: 10.1016/j.jcp.2007.02.001.  Google Scholar

[19]

F. Liu, V. Anh and I. Turner, Numerical solution of the space fractional Fokker-Planck equation,, J. Comput. Appl. Math., 166 (2004), 209.  doi: 10.1016/j.cam.2003.09.028.  Google Scholar

[20]

F. Liu, P. Zhuang, I. Turner, K. Burrage and V. Anh, A new fractional finite volume method for solving the fractional diffusion equation,, Appl. Math. Modeling, 38 (2014), 3871.  doi: 10.1016/j.apm.2013.10.007.  Google Scholar

[21]

C. Lubich, Discretized fractional calculus,, SIAM J. Math. Anal., 17 (1986), 704.  doi: 10.1137/0517050.  Google Scholar

[22]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations,, J. Comput. Appl. Math., 172 (2004), 65.  doi: 10.1016/j.cam.2004.01.033.  Google Scholar

[23]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations,, Appl. Numer. Math., 56 (2006), 80.  doi: 10.1016/j.apnum.2005.02.008.  Google Scholar

[24]

R. Metler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach,, Phys. Reports, 339 (2000), 1.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[25]

R. Metler and J. Klafter, The restaurant at the end of random walk: Recent developments in the description of anomalous transport by fractional dynamics,, J. Phys. A, 37 (2004).  doi: 10.1088/0305-4470/37/31/R01.  Google Scholar

[26]

K. B. Oldham and J. Spanier, The Fractional Calculus,, Academic Press, (1974).   Google Scholar

[27]

H.-K. Pang and H.-W. Sun, Multigrid method for fractional diffusion equations,, J. Comput. Phys., 231 (2012), 693.  doi: 10.1016/j.jcp.2011.10.005.  Google Scholar

[28]

I. Podlubny, Fractional Differential Equations,, Academic Press, (1999).   Google Scholar

[29]

E. Sousa, Finite difference approximates for a fractional advection diffusion problem,, J. Comput. Phys., 228 (2009), 4038.  doi: 10.1016/j.jcp.2009.02.011.  Google Scholar

[30]

L. Su, W. Wang and Z. Yang, Finite difference approximations for the fractional advection diffusion equation,, Physics Letters A, 373 (2009), 4405.  doi: 10.1016/j.physleta.2009.10.004.  Google Scholar

[31]

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.  doi: 10.1016/j.jcp.2006.05.030.  Google Scholar

[32]

C. Tadjeran, M. M. Meerschaert and H. P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation,, J. Comput. Phys., 213 (2006), 205.  doi: 10.1016/j.jcp.2005.08.008.  Google Scholar

[33]

R. S. Varga, Matrix Iterative Analysis,, Second Edition, (2000).  doi: 10.1007/978-3-642-05156-2.  Google Scholar

[34]

H. Wang and T. S. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations,, SIAM J. Sci. Comput., 34 (2012).  doi: 10.1137/12086491X.  Google Scholar

[35]

H. Wang and N. Du, A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations,, J. Comput. Phys., 240 (2013), 49.  doi: 10.1016/j.jcp.2012.07.045.  Google Scholar

[36]

H. Wang and N. Du, A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation,, J. Comput. Phys., 253 (2013), 50.  doi: 10.1016/j.jcp.2013.06.040.  Google Scholar

[37]

H. Wang and N. Du, Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations,, J. Comput. Phys., 258 (2014), 305.  doi: 10.1016/j.jcp.2013.10.040.  Google Scholar

[38]

H. Wang and K. Wang, An $O(N \log^2 N)$ alternating-direction finite difference method for two-dimensional fractional diffusion equations,, J. Comput. Phys., 230 (2011), 7830.  doi: 10.1016/j.jcp.2011.07.003.  Google Scholar

[39]

H. Wang, K. Wang and T. Sircar, A direct $O(N\log^2 N)$ finite difference method for fractional diffusion equations,, J. Comput. Phys., 229 (2010), 8095.  doi: 10.1016/j.jcp.2010.07.011.  Google Scholar

[40]

H. Wang and D. Yang, Wellposedness of variable-coefficient conservative fractional elliptic differential equations,, SIAM J. Numer. Anal., 51 (2013), 1088.  doi: 10.1137/120892295.  Google Scholar

[41]

H. Wang, D. Yang and S. Zhu, Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations,, SIAM J. Numer. Anal., 52 (2014), 1292.  doi: 10.1137/130932776.  Google Scholar

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