A linearized spectral collocation method for Riesz space fractional nonlinear reaction–diffusion equations

In this work, we investigate an effective linearized spectral collocation method for two-dimensional (2D) Riesz space fractional nonlinear reaction–diffusion equations with homogeneous boundary conditions. The proposed method is based on the Jacobi–Gauss–Lobatto spectral collocation method for spatial discretization and the finite difference method for temporal discretization. The full implementation of the method is demonstrated in detail. The stability of the numerical scheme is rigorously discussed and the errors with benchmark solutions that show second-order convergence in time and spectral convergence in space are numerically analyzed. Finally, numerical simulations for 2D Riesz space fractional Allen–Cahn and FitzHugh–Nagumo models are carried out to illustrate the effectiveness of the developed method and its ability for long-time simulations.