Spectral Galerkin Methods for Riesz Space-Fractional Convection-Diffusion Equations

This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection-diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection-diffusion equations. The equations were solved by discretizing in space using the Galerkin-Legendre spectral approaches and in time using the Crank-Nicolson Leap-Frog (CNLF) scheme. In addition, the stability and convergence of semi-discrete and fully discrete schemes were analyzed. Secondly, we established a fully discrete form for the two-dimensional case with an additional complementary term on the left and then obtained the stability and convergence results for it. Finally, numerical simulations were performed, and the results demonstrate the effectiveness of our numerical methods.