2014, 4(4): 317-325. doi: 10.3934/naco.2014.4.317

A note on the stability of a second order finite difference scheme for space fractional diffusion equations

1. 

Department of Mathematics, University of Macau, Macau, China

2. 

Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macau

Received  August 2013 Revised  October 2014 Published  December 2014

The unconditional stability of a second order finite difference scheme for space fractional diffusion equations is proved theoretically for a class of variable diffusion coefficients. In particular, the scheme is unconditionally stable for all one-sided problems and problems with Riesz fractional derivative. For problems with general smooth diffusion coefficients, numerical experiments show that the scheme is still stable if the space step is small enough.
Citation: Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317
References:
[1]

C. Celik and M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231 (2011), 1743-1750. doi: 10.1016/j.jcp.2011.11.008.

[2]

M. Chen and W. Deng, Fourth order accurate scheme for the space fractional diffusion equations, SIAM J Numer. Anal., 52 (2013), 1418-1438. doi: 10.1137/130933447.

[3]

M. Chen, W. Deng and Y. Wu, Superlinearly convergent algorithms for the two-dimensional space-time Caputo-Riesz fractional diffusion equation, Appl. Numer. math., 70 (2013), 22-41. doi: 10.1016/j.apnum.2013.03.006.

[4]

W. Chen and S. Wang, A finite difference method for pricing European and American options under a geometric Levy process, J. Ind. Manag. Optim., 11 (2015), 241-264. doi: 10.3934/jimo.2015.11.241.

[5]

W. Deng and M. Chen, Efficient numerical algorithms for three-dimensional fractional partial differential equations, Journal of Computational Mathematics, 32 (2014), 371-391. doi: 10.4208/jcm.1401-m3893.

[6]

R. Horn and C. Johnson, Topics on Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.

[7]

S. Lei and H. Sun, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013), 715-725. doi: 10.1016/j.jcp.2013.02.025.

[8]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77. doi: 10.1016/j.cam.2004.01.033.

[9]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), 80-90. doi: 10.1016/j.apnum.2005.02.008.

[10]

H. Pang and H. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693-703. doi: 10.1016/j.jcp.2011.10.005.

[11]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[12]

E. Sousa and C. Li, A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative,, Applied Numerical Mathematics, ().  doi: 10.1016/j.apnum.2014.11.007.

[13]

E. Sousa, A second order explicit finite difference method for the fractional advection diffusion equation, Comput. Math. Appl., 64 (2012), 3141-3152. doi: 10.1016/j.camwa.2012.03.002.

[14]

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), 205-213. doi: 10.1016/j.jcp.2005.08.008.

[15]

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

[16]

W. Tian, H. Zhou and W. Deng, A class of second order difference approximations for solving space fractional diffusion equations,, Math. Comput., (). 

[17]

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), 50-63. doi: 10.1016/j.jcp.2013.06.040.

[18]

H. Wang, K. Wang and T. Sircar, A direct O(N log2N) finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), 8095-8104. doi: 10.1016/j.jcp.2010.07.011.

[19]

H. Wang and K. Wang, An O(N log2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations, J. Comput. Phys., 230 (2011), 7830-7839. doi: 10.1016/j.jcp.2011.07.003.

[20]

H. Wang and T. S. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444-A2458. doi: 10.1137/12086491X.

show all references

References:
[1]

C. Celik and M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231 (2011), 1743-1750. doi: 10.1016/j.jcp.2011.11.008.

[2]

M. Chen and W. Deng, Fourth order accurate scheme for the space fractional diffusion equations, SIAM J Numer. Anal., 52 (2013), 1418-1438. doi: 10.1137/130933447.

[3]

M. Chen, W. Deng and Y. Wu, Superlinearly convergent algorithms for the two-dimensional space-time Caputo-Riesz fractional diffusion equation, Appl. Numer. math., 70 (2013), 22-41. doi: 10.1016/j.apnum.2013.03.006.

[4]

W. Chen and S. Wang, A finite difference method for pricing European and American options under a geometric Levy process, J. Ind. Manag. Optim., 11 (2015), 241-264. doi: 10.3934/jimo.2015.11.241.

[5]

W. Deng and M. Chen, Efficient numerical algorithms for three-dimensional fractional partial differential equations, Journal of Computational Mathematics, 32 (2014), 371-391. doi: 10.4208/jcm.1401-m3893.

[6]

R. Horn and C. Johnson, Topics on Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.

[7]

S. Lei and H. Sun, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013), 715-725. doi: 10.1016/j.jcp.2013.02.025.

[8]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77. doi: 10.1016/j.cam.2004.01.033.

[9]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), 80-90. doi: 10.1016/j.apnum.2005.02.008.

[10]

H. Pang and H. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693-703. doi: 10.1016/j.jcp.2011.10.005.

[11]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[12]

E. Sousa and C. Li, A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative,, Applied Numerical Mathematics, ().  doi: 10.1016/j.apnum.2014.11.007.

[13]

E. Sousa, A second order explicit finite difference method for the fractional advection diffusion equation, Comput. Math. Appl., 64 (2012), 3141-3152. doi: 10.1016/j.camwa.2012.03.002.

[14]

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), 205-213. doi: 10.1016/j.jcp.2005.08.008.

[15]

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

[16]

W. Tian, H. Zhou and W. Deng, A class of second order difference approximations for solving space fractional diffusion equations,, Math. Comput., (). 

[17]

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), 50-63. doi: 10.1016/j.jcp.2013.06.040.

[18]

H. Wang, K. Wang and T. Sircar, A direct O(N log2N) finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), 8095-8104. doi: 10.1016/j.jcp.2010.07.011.

[19]

H. Wang and K. Wang, An O(N log2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations, J. Comput. Phys., 230 (2011), 7830-7839. doi: 10.1016/j.jcp.2011.07.003.

[20]

H. Wang and T. S. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444-A2458. doi: 10.1137/12086491X.

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