Splitting spectral element method for fractional reaction-diffusion equations

In this paper, we propose a second-order operator splitting spectral element method for solving fractional reaction-diffusion equations. In order to achieve a fast second-order scheme in time, we decompose the original equation into linear and nonlinear sub-equations, and combine a quarter-time nonlinear solver and a half-time linear solver followed by final quarter-time nonlinear solver. The spatial discretization is eigen-decomposition based on spectral element method. Since this method gives a full diagonal representation of the fractional operator and gets an exponential convergence in space. We have an accurate and efficient approach for solving spacial fractional reaction-diffusion equations. Some numerical experiments are carried out to demonstrate the accuracy and efficiency of this method. Finally, we apply the proposed method to investigate the effect of the fractional order in the fractional reaction-diffusion equations.