Combined compact difference scheme for the time fractional convection-diffusion equation with variable coefficients

Fourth-order combined compact finite difference scheme is given for solving the time fractional convection–diffusion–reaction equation with variable coefficients. We introduce the flux as a new variable and transform the original equation into a system of two equations. Compact difference is used as a high-order approximation for spatial derivatives of integer order in the coupled partial differential equations. The Caputo fractional derivative is dis-cretized by a high-order approximation. Both Dirichlet and Robin boundary conditions are discussed. Convergence analysis is given for the problem of integer order with constant coefficients under some assumption. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm. Fractional differential equations(FDEs) can be used to in mathematical models for physical, biological, geological and financial systems, and the review article [1] and monograph [2] have given detailed discussions on fractional differential equations. In the recent years, many numerical methods have been proposed and studied for solving FDEs up to now, e.g., [3–5]. Compact finite difference schemes have high-order of accuracy and the desirable tridiagonal nature of the finite-difference equations (see [6,7]), and one-dimensional fractional sub-diffusion equation was recently solved by the compact finite difference scheme with convergence order Oðs þ h 4 Þ in [8,9], a higher order one for the temporal variable in [10], and for two dimensional problems, ADI schemes with error analysis were given in papers [11,12], with the compact schemes for the time fractional convection–diffusion equations with constant coefficients considered in [13]. While high order methods for partial different equations with constant coefficients have been discussed by many authors, it is not so easy to give corresponding results for variable coefficients. One way to give high order difference schemes is to start from lower order scheme, then follow a substitution of the lower order truncation error by replacing the derivatives using the successive derivation of the original continuous problem. Thus high order schemes for variable coefficients must be examined carefully as many derivatives appear, and our aim in this paper is to give some easily implemented schemes for problems with variable coefficients. We must admit for problems with variable coefficients, following this way would be a tedious task. As pointed in [14], the idea of posing a boundary-value problem in the form of a first-order system as a governing conservation equation and an associated constitutive equation, and then using a mixed approximation scheme, has been