Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation

In this paper, a new conservative difference scheme is proposed for solving the nonlinear space-fractional Schrödinger equation. The Riesz space-fractional derivative is approximated by the second-order accurate weighted and shifted Grünwald difference operator. Building on a careful analysis of this operator, the conservation properties including the mass and energy are analyzed and some norm equivalences are established. Then the numerical solution is shown to be bounded in various norms, uniformly with respect to the discretization parameters, and optimal order bounds on the global error of the scheme in the l2l2 norm, semi-Hα/2Hα/2 norm and l∞l∞ norm are derived. The uniform bounds on the numerical solutions as well as the error bounds hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed. Numerical tests are performed to support our theoretical results.