A high-order scheme for time-space fractional diffusion equations with Caputo-Riesz derivatives

In this paper, we present a high-order approach for solving one- and two-dimensional time-space fractional diffusion equations (FDEs) with Caputo-Riesz derivatives. To design the scheme, the Caputo temporal derivative is approximated using a high-order method, and the spatial Riesz derivative is discretized by the second-order weighted and shifted Grunwald difference (WSGD) method. It is proved that the scheme is unconditionally stable and convergent with the order of O (tau(alpha)h(2) + tau(4) ), where tau and h are time and space step sizes, respectively. We illustrate the accuracy and effectiveness of the method by providing several numerical examples.