Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions

In this paper, we propose a conservative and effective difference scheme for solving the two dimensional nonlinear space-fractional Schrödinger equation with the Riesz fractional derivative. The scheme is constructed by combining the split-step method for handling the nonlinearity with the alternating direction implicit (ADI) method for resolving the multi-dimensions difficulty. The Riesz space-fractional derivative is approximated by the second order accurate fractional centered difference. Based on matrix analysis, we show that in the discrete sense the scheme conserves the mass and energy for linear problems and conserves the mass for nonlinear problems. The unconditional convergence is proved rigorously in the linear case. Numerical tests are performed to support our theoretical results and show the efficiency of the proposed scheme.