A fourth-order compact difference scheme with efficient algorithms for space fractional nonlinear Schrödinger equations

In this paper, we develop a two-level Crank–Nicolson scheme combined with a fourth-order compact difference discretization for solving the space fractional nonlinear Schrödinger equation. The discrete mass and energy conservation properties are rigorously established using the discrete energy method. We prove that the numerical solution is bounded and converges with order O(τ2+h4) in a suitable discrete norm, where τ and h are the temporal and spatial step sizes, respectively. To efficiently solve the resulting complex linear systems, a modified generalized SOR method is proposed, together with two preconditioners designed to accelerate convergence. The coefficient matrix can be transformed into a tridiagonal form, which enables efficient implementation of the iteration. In particular, the proposed iterative solver remains stable even for indefinite coefficient matrices, while the preconditioners significantly accelerate convergence and ensure robust performance. Numerical results affirm both the theoretical findings and the computational performance of our algorithms.