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In this paper, we focus on develop high-order and structure-preserving numerical algorithm for the two-dimensional nonlinear space fractional Schrödinger equations. By constructing a new generating function, we obtain a fourth-order numerical differential formula and use it to approximate the spatial Riesz derivative, while the Crank–Nicolson method is applied for the time derivative. Based on the energy method, the conservation, solvability and convergence of the numerical algorithm are proved. Finally, some numerical examples are used to verify the correctness of the theoretical analysis and the validity of the numerical algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a novel optimal fourth-order fractional-compact-type numerical differential formula for the Riesz derivatives is derived by constructing the appropriate generating function. Meanwhile, some interesting and important properties of the proposed formula are also presented and proved. Then, we take the nonlinear space fractional Ginzburg–Landau equations as an example and establish a high-order finite difference scheme by combining the compact integral factor method with Padé approximation. Furthermore, based on the discrete energy analysis method, the proposed difference scheme is proved to be uniquely solvable and convergent with order O (τ 2 + h 4) under different norms, where τ and h denote the temporal and spatial mesh step sizes, respectively. Finally, some numerical results are provided to illustrate the accuracy of numerical differential formula and the effectiveness of numerical method, and further demonstrate the correctness of theoretical analysis. • An optional fourth-order numerical differential formula is constructed. • A high-order two-level difference scheme is proposed. • The unique solvability and convergence of the proposed algorithm are analyzed. • The methodology can be extended to other nonlinear spatial fractional equations. [ABSTRACT FROM AUTHOR]