An energy conservative difference scheme for the nonlinear fractional Schrödinger equations

In this paper, an energy conservative Crank–Nicolson difference scheme for nonlinear Riesz space-fractional Schrödinger equations is studied. We give a rigorous analysis of the conservation properties, including mass conservation and energy conservation in the discrete sense. Based on Brouwer fixed point theorem, the existence of the difference solution is proved. By virtue of the energy method, the difference solution is shown to be unique and convergent at the order of O(τ2+h2)O(τ2+h2) in the l2l2-norm with time step τ and mesh size h. Finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results.