Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg–Landau equation

Space and time approximations for two-dimensional space fractional complex Ginzburg–Landau equation are examined. The schemes under consideration are discreted by the second-order backward differential formula (BDF2) in time and two classes of the fractional centered finite difference methods in space. A linearized technique is employed by the extrapolation. We prove the unique solvability and stability for both numerical methods. The convergence of both numerical methods is analyzed at length utilizing the energy argument, and the convergence orders under the optimal step size ratio are O ( τ 2 + h 2 ) and O ( τ 2 + h 4 ) in the sense of the discrete L 2 -norm, where τ is the time step size, h = max { h x , h y } , and h x , h y are spatial grid sizes in the x -direction and y -direction, respectively. In addition, we construct a multistep alternating direction implicit (ADI) scheme and a multistep compact ADI scheme based on BDF2 for the efficiently numerical implementation. Finally, numerical examples are carried out to verify our theoretical results.