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

To discretize the distributed-order term of two-dimensional nonlinear Riesz space fractional diffusion equation, we consider the high accuracy Gauss-Legendre quadrature formula. By combining an s-stage implicit Runge-Kutta method in temporal direction with a spectral Galerkin method in spatial direction, we construct a numerical method with high global accuracy. If the nonlinear function satisfies the local Lipschitz condition, the s-stage implicit Runge-Kutta method with order p (p≥s+1) is coercive and algebraically stable, then we can prove that the proposed method is stable and convergent of order s+1 in time. In addition, we also derive the optimal error estimate for the discretization of distributed-order term and spatial term. Finally, numerical experiments are presented to verify the theoretical results.