Maximum-norm error analysis of a difference scheme for the space fractional CNLS

The difference method for the space fractional coupled nonlinear Schrödinger equations (CNLS) is studied. The fractional centered difference is used to approximate the space fractional Laplacian. This scheme conserves the discrete mass and energy. Due to the nonlocal nature of fractional Laplacian, in the classic Sobolev space, it is hard to obtain the error estimation in l∞. To overcome this difficulty, the fractional Sobolev space Hα/2 and a fractional norm equivalence in Hα/2 are introduced. Then the convergence of order O(h2+τ2) in l∞ is proved by fractional Sobolev inequality, where h is the mesh size and τ is the time step. Numerical examples are given to illustrate the theoretical results at last.