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.