Galerkin‐Legendre spectral method for the nonlinear Ginzburg‐Landau equation with the Riesz fractional derivative

In this paper, we first construct a linearized Galerkin‐Legendre spectral method for the one‐dimensional nonlinear fractional Ginzburg‐Landau equation, where a three‐level linearized Crank‐Nicolson scheme is used for time discretization. The unique solvability and boundedness properties of the fully discrete scheme are analyzed. It is shown that the method is unconditionally convergent in the maximum norm with second‐order accuracy in time and spectral accuracy in space. Then, two‐dimensional problems are considered and a split‐step alternating direction implicit Galerkin‐Legendre spectral method is introduced without theoretical analysis. Finally, some numerical examples are presented to illustrate the effectiveness of the two proposed schemes.