High-order algorithm for the two-dimension Riesz space-fractional diffusion equation

In this paper, applying a novel second-order numerical approximation formula for the Riesz derivative and Crank-Nicolson technique for the temporal derivative, a numerical algorithm is constructed for the two-dimensional spatial fractional diffusion equation with convergence order , where , and are the temporal and spatial step sizes, respectively. It is proved that the proposed algorithm is unconditionally stable and convergent by using the energy method. Meanwhile, by adding the high-order perturbation items for the above numerical scheme, an alternating direction implicit difference scheme is also constructed. Finally, some numerical results are presented to demonstrate the validity of theoretical analysis and show the accuracy and effectiveness of the method described herein.