A fourth‐order accurate numerical method for the distributed‐order Riesz space fractional diffusion equation: A fourth-order accurate numerical method for the distributed-order Riesz space fractional diffusion equation

In this article, a novel fourth‐order accurate difference method is derived for the distributed‐order Riesz space fractional diffusion equation in one‐dimensional (1D) and two‐dimensional (2D) cases, respectively. First, the distributed integral terms are discretized by using the Simpson quadrature rule into the multi‐term Riesz space fractional diffusion equations. Then, a fourth‐order accurate difference scheme is presented to approximate the multi‐term Riesz fractional diffusion equations. Moreover, the proposed difference schemes are proved to be unconditionally stable and convergent in norm for both 1D and 2D cases. Finally, numerical experiments are given to verify the efficiency of the schemes.