A compact ADI Crank–Nicolson difference scheme for the two-dimensional time fractional subdiffusion equation

References

• D.A. Benson, S.W. Wheatcraft, and M.M. Meerchaert, Application of a fractional advection–dispersion equation, Water Resour. Res. 36 (2000), pp. 1403–1412.
   Web of Science ®Google Scholar
• A.H. Bhrawy, E.H. Doha, D. Baleanud, and S.S. Ezz-Eldien, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys. 293 (2015), pp. 142–156.
   Web of Science ®Google Scholar
• A. Chen and C.P. Li, A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions, Int. J. Comput. Math. 92 (2015), pp. 1–26.
   Google Scholar
• H.B. Chen, D. Xu, and Y.L Peng, An alternating direction implicit fractional trapezoidal rule type difference scheme for the two-dimensional fractional evolution equation, Int. J. Comput. Math. 92 (2015), pp. 2178–2197.
   Web of Science ®Google Scholar
• M. Cui, Compact alternating direction implicit method for two-dimensional time fractional diffusion equation, J. Comput. Phys. 231 (2012), pp. 2621–2633.
   Web of Science ®Google Scholar
• G. Fairweather, X. Yang, D. Xu, and H. Zhang, An ADI Crank–Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation, J. Sci. Comput. 65 (2015), pp. 1217–1239.
   Web of Science ®Google Scholar
• G.-H. Gao and Z.-Z. Sun, Compact difference schemes for heat equation with Neumann boundary conditions (II), Numer. Methods Partial Differential Equations 29 (2012), pp. 1459–1486.
   Web of Science ®Google Scholar
• R. Gorenflo, F. Mainardi, E. Scalas, and M. Raberto, Fractional calculus and continuous-time finance III: The diffusion limit, in Mathematical Finance, M. Kolhmann and S. Tang, eds., Trends in Math, Birkhäuser, Basel, 2001, pp. 171–180.
   Google Scholar
• C.C. Ji and Z.Z. Sun, The high-order compact numerical algorithms for the two-dimensional fractional sub-diffusion equation, Appl. Math. Comput. 269 (2015), pp. 775–791.
   Web of Science ®Google Scholar
• B. Jin, R. Lazarov, J. Pasciak, and Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA J. Numer. Anal. 35 (2015), pp. 561–582.
   Web of Science ®Google Scholar
• R.C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51 (1984), pp. 299–307.
   Web of Science ®Google Scholar
• X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal. 47 (2009), pp. 2108–2131.
   Web of Science ®Google Scholar
• L. Li, D. Xu, and M. Luo, Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation, J. Comput. Phys. 255 (2013), pp. 471–485.
   Web of Science ®Google Scholar
• H.-L. Liao and Z.-Z. Sun, Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations, Numer. Methods Partial Differential Equations 26 (2010), pp. 37–60.
   Web of Science ®Google Scholar
• M.M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection–dispersion flow equations, J. Comput. Appl. Math. 172 (2004), pp. 65–77.
   Web of Science ®Google Scholar
• M.M. Meerschaert, H.-P. Scheffler, and C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys. 211 (2006), pp. 249–261.
   Web of Science ®Google Scholar
• R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000), pp. 1–77.
   Web of Science ®Google Scholar
• A. Mohebbi and M. Abbaszadeh, Compact finite difference scheme for the solution of time fractional advection–dispersion equation, Numer. Algorithms 63 (2013), pp. 431–452.
   Web of Science ®Google Scholar
• K. Mustapha, Time-stepping discontinuous Galerkin methods for fractional diffusion problems, Numer. Math. 130 (2015), pp. 497–516.
   Web of Science ®Google Scholar
• D.W. Peaceman and H.H.J. Rachford, The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math. 3 (1955), pp. 28–41.
   Google Scholar
• E. Sousaa and C. Li, A weighted finite difference method for the fractional diffusion equation based on the Riemann–Liouville derivative, Appl. Numer. Math. 90 (2015), pp. 22–37.
   Web of Science ®Google Scholar
• W.Y. Tian, H. Zhou, and W.H. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput. 84 (2015), pp. 1703–1727.
   Web of Science ®Google Scholar
• S. Vong and Z.B. Wang, High order difference schemes for a time fractional differential equation with Neumann boundary conditions, East Asian J. Appl. Math. 4 (2014), pp. 222–241.
   Web of Science ®Google Scholar
• Z. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277 (2014), pp. 1–15.
   Web of Science ®Google Scholar
• Z.B. Wang and S. Vong, A high-order ADI scheme for the two-dimensional time fractional diffusion-wave equation, Int. J. Comput. Math. 92 (2015), pp. 970–979.
   Web of Science ®Google Scholar
• W. Wyss, The fractional diffusion equation, J. Math. Phys. 27 (1986), pp. 2782–2785.
   Web of Science ®Google Scholar
• S.B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal. 42 (2005), pp. 1862–1874.
   Web of Science ®Google Scholar
• Y.-N. Zhang and Z.-Z. Sun, Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, J. Sci. Comput. 59 (2014), pp. 104–128.
   Web of Science ®Google Scholar
• Y.-N. Zhang, Z.-Z. Sun, and H.-W. Wu, Error estimates of Crank–Nicolson-type difference schemes for the subdiffusion equation, SIAM J. Numer. Anal. 49 (2011), pp. 2302–2322.
   Web of Science ®Google Scholar
• S.Y. Zhai, X.L. Feng, and Z.F. Weng, New high-order compact ADI algorithms for 3D nonlinear time-fractional convection–diffusion equation, Math. Probl. Eng. 2013 (2013), article no. 246015.
   Google Scholar
• S.Y. Zhai, X.L. Feng, and Y.N. He, An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation, J. Comput. Phys. 269 (2014), pp. 138–155.
   Web of Science ®Google Scholar
• S.Y. Zhai, L.L. Wei, L.Y. Huang, and X.L. Feng, An efficient algorithm with high accuracy for time-space fractional heat equations, Numer. Heat Transf. B, Fund 67 (2015), pp. 550–562.
   Web of Science ®Google Scholar
• S.Y. Zhai, Z.F. Weng, D.W. Gui, and X.L. Feng, High-order compact operator splitting method for three-dimensional fractional equation with subdiffusion, Int. J. Heat Mass Transf. 84 (2015), pp. 440–447.
   Web of Science ®Google Scholar
• S.Y. Zhai, D.W. Gui, P.Z. Huang, and X.L. Feng, A novel high-order ADI method for 3D fractional convection–diffusion equations, Int. Commun. Heat Mass. Transf. 66 (2015), pp. 212–217.
   Web of Science ®Google Scholar