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 & 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. doi: 10.1016/j.jcp.2011.11.008. Google Scholar

[2]

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

[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. doi: 10.1016/j.apnum.2013.03.006. Google Scholar

[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. doi: 10.3934/jimo.2015.11.241. Google Scholar

[5]

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

[6]

R. Horn and C. Johnson, Topics on Matrix Analysis,, Cambridge University Press, (1991). Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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. Google Scholar

[13]

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

[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. doi: 10.1016/j.jcp.2005.08.008. Google Scholar

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

[16]

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

[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. doi: 10.1016/j.jcp.2013.06.040. Google Scholar

[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. doi: 10.1016/j.jcp.2010.07.011. Google Scholar

[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. doi: 10.1016/j.jcp.2011.07.003. Google Scholar

[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). doi: 10.1137/12086491X. Google Scholar

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. doi: 10.1016/j.jcp.2011.11.008. Google Scholar

[2]

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

[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. doi: 10.1016/j.apnum.2013.03.006. Google Scholar

[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. doi: 10.3934/jimo.2015.11.241. Google Scholar

[5]

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

[6]

R. Horn and C. Johnson, Topics on Matrix Analysis,, Cambridge University Press, (1991). Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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. Google Scholar

[13]

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

[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. doi: 10.1016/j.jcp.2005.08.008. Google Scholar

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

[16]

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

[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. doi: 10.1016/j.jcp.2013.06.040. Google Scholar

[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. doi: 10.1016/j.jcp.2010.07.011. Google Scholar

[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. doi: 10.1016/j.jcp.2011.07.003. Google Scholar

[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). doi: 10.1137/12086491X. Google Scholar

[1]

Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025

[2]

Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control & Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016

[3]

Imen Manoubi. Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2837-2863. doi: 10.3934/dcdsb.2014.19.2837

[4]

Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558

[5]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007

[6]

Jaemin Shin, Yongho Choi, Junseok Kim. An unconditionally stable numerical method for the viscous Cahn--Hilliard equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1737-1747. doi: 10.3934/dcdsb.2014.19.1737

[7]

Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033

[8]

Wenbin Chen, Wenqiang Feng, Yuan Liu, Cheng Wang, Steven M. Wise. A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 149-182. doi: 10.3934/dcdsb.2018090

[9]

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

[10]

Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037

[11]

Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644

[12]

Georgios T. Kossioris, Georgios E. Zouraris. Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1845-1872. doi: 10.3934/dcdsb.2013.18.1845

[13]

Jiyu Zhong, Shengfu Deng. Two codimension-two bifurcations of a second-order difference equation from macroeconomics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1581-1600. doi: 10.3934/dcdsb.2018062

[14]

Bi Ping, Maoan Han. Oscillation of second order difference equations with advanced argument. Conference Publications, 2003, 2003 (Special) : 108-112. doi: 10.3934/proc.2003.2003.108

[15]

Junxiong Jia, Jigen Peng, Jinghuai Gao, Yujiao Li. Backward problem for a time-space fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (3) : 773-799. doi: 10.3934/ipi.2018033

[16]

Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035

[17]

Cheng Wang, Xiaoming Wang, Steven M. Wise. Unconditionally stable schemes for equations of thin film epitaxy. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 405-423. doi: 10.3934/dcds.2010.28.405

[18]

Meng Zhao, Aijie Cheng, Hong Wang. A preconditioned fast Hermite finite element method for space-fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3529-3545. doi: 10.3934/dcdsb.2017178

[19]

Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583

[20]

Gabriela Marinoschi. Well posedness of a time-difference scheme for a degenerate fast diffusion problem. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 435-454. doi: 10.3934/dcdsb.2010.13.435

 Impact Factor: 

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]