Implicit Runge-Kutta and spectral Galerkin methods for the two-dimensional nonlinear Riesz space fractional diffusion equation

A numerical method with high accuracy both in time and in space is proposed for the two-dimensional nonlinear Riesz space fractional diffusion equation. The main idea is based on a spectral Galerkin method in spatial direction and an s-stage implicit Runge-Kutta method in temporal direction. A rigorous stability and error analysis is performed for the proposed method. It is shown that the proposed method is stable and convergent. The optimal spatial error estimate is also derived. Numerical experiments are provided to illustrate the theoretical results.