A high-order compact exponential scheme for the fractional convection-diffusion equation

A high-order compact exponential finite difference scheme for solving the fractional convection-diffusion equation is considered in this paper. The convection and diffusion terms are approximated by a compact exponential finite difference scheme, with a high-order approximation for the Caputo time derivative. For this fully discrete implicit scheme, the local truncation error is analyzed and the Fourier method is used to discuss the stability. The error estimate is given by the discrete energy method. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.