Compact exponential scheme for the time fractional convection-diffusion reaction equation with variable coefficients

High-order compact exponential finite difference scheme for solving the time fractional convection-diffusion reaction equation with variable coefficients is considered in this paper. The convection, diffusion and reaction coefficients can depend on both the spatial and temporal variables. We begin with the one dimensional problem, and after transforming the original equation to one with diffusion coefficient unity, the new equation is discretized by a compact exponential finite difference scheme, with a high-order approximation for the Caputo time derivative. We prove the solvability of this fully discrete implicit scheme, and analyze its local truncation error. For the fractional equation with constant coefficients, we use Fourier method to prove the stability and utilize matrix analysis as a tool for the error estimate. Then we discuss the two dimensional problem, give the compact ADI scheme with the restriction that besides the time variable, the convection coefficients can only depend on the corresponding spatial variables, respectively. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm. The time fractional convection-diffusion-reaction equation is numerically solved.Compact exponential scheme for the variable coefficients equation is given.Fourier method and matrix analysis are used for the analysis.The compact exponential ADI scheme is given for the two dimensional problem.High accuracy and good efficiency of the proposed algorithm is demonstrated.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[2]  Y. Ge,et al.  A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems , 2007 .

[3]  Fawang Liu,et al.  New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Subdiffusion Equation , 2008, SIAM J. Numer. Anal..

[4]  Fawang Liu,et al.  Finite difference approximations for the fractional Fokker–Planck equation , 2009 .

[5]  Zhi‐zhong Sun,et al.  A fully discrete difference scheme for a diffusion-wave system , 2006 .

[6]  R. Hirsh,et al.  Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique , 1975 .

[7]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[8]  Mingrong Cui,et al.  A high-order compact exponential scheme for the fractional convection-diffusion equation , 2014, J. Comput. Appl. Math..

[9]  Zhibo Wang,et al.  A high-order exponential ADI scheme for two dimensional time fractional convection-diffusion equations , 2014, Comput. Math. Appl..

[10]  Fawang Liu,et al.  A Fourier method for the fractional diffusion equation describing sub-diffusion , 2007, J. Comput. Phys..

[11]  M. Dehghan,et al.  The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients , 2012 .

[12]  Z. F. Tian,et al.  High-order compact exponential finite difference methods for convection-diffusion type problems , 2007, J. Comput. Phys..

[13]  B. Henry,et al.  Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces. , 2010, Physical review letters.

[14]  D. Benson,et al.  Fractional calculus in hydrologic modeling: A numerical perspective. , 2013, Advances in water resources.

[15]  J. W. Thomas.  Numerical Partial Differential Equations: Finite Difference Methods , 1995 .

[16]  Changpin Li,et al.  Finite difference Methods for fractional differential equations , 2012, Int. J. Bifurc. Chaos.

[17]  I. Turner,et al.  Time fractional advection-dispersion equation , 2003 .

[18]  Mingrong Cui.  Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation , 2012, Numerical Algorithms.

[19]  Jun Zhang,et al.  An exponential high‐order compact ADI method for 3D unsteady convection–diffusion problems , 2013 .

[20]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[21]  Fawang Liu,et al.  Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term , 2009, J. Comput. Appl. Math..

[22]  G. Carey,et al.  High‐order compact scheme for the steady stream‐function vorticity equations , 1995 .

[23]  A. Pillai.  Fourth‐order exponential finite difference methods for boundary value problems of convective diffusion type , 2001 .

[24]  Yinnian He,et al.  An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation , 2014, J. Comput. Phys..

[25]  Graham F. Carey,et al.  Extension of high‐order compact schemes to time‐dependent problems , 2001 .

[26]  Qinwu Xu,et al.  Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient , 2014 .

[27]  Zhen F. Tian,et al.  A high-order exponential scheme for solving 1D unsteady convection-diffusion equations , 2011, J. Comput. Appl. Math..

[28]  Xuan Zhao,et al.  Compact Alternating Direction Implicit Scheme for the Two-Dimensional Fractional Diffusion-Wave Equation , 2012, SIAM J. Numer. Anal..

[29]  Mingrong Cui,et al.  Compact finite difference method for the fractional diffusion equation , 2009, J. Comput. Phys..

[30]  Fawang Liu,et al.  A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model , 2013, Comput. Math. Appl..

[31]  Zhi-Zhong Sun,et al.  A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications , 2014, J. Comput. Phys..

[32]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[33]  Fawang Liu,et al.  Numerical Schemes with High Spatial Accuracy for a Variable-Order Anomalous Subdiffusion Equation , 2010, SIAM J. Sci. Comput..

[34]  Zhi-Zhong Sun,et al.  Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation , 2011, J. Comput. Phys..

[35]  I. Podlubny.  Fractional differential equations , 1998 .

[36]  Zhi-Zhong Sun,et al.  A compact finite difference scheme for the fractional sub-diffusion equations , 2011, J. Comput. Phys..

[37]  R. Bhatia.  Positive Definite Matrices , 2007 .

[38]  Mingrong Cui,et al.  Compact alternating direction implicit method for two-dimensional time fractional diffusion equation , 2012, J. Comput. Phys..