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In this paper, we first construct an appropriate new generating function, and then based on this function, we establish a fourth-order numerical differential formula approximating the Riesz derivative with order γ ∈ (1 , 2 ]. Subsequently, we apply the formula to numerically study the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation and obtain a difference scheme with convergence order O τ 2 + h x 4 + h y 4 , where τ denotes the time step size, h x and h y denote the space step sizes, respectively. Furthermore, with the help of some newly derived discrete fractional Sobolev embedding inequalities, the unique solvability, the unconditional stability, and the convergence of the constructed numerical algorithm under different norms are proved by using the discrete energy method. Finally, some numerical results are presented to confirm the correctness of the theoretical results and verify the effectiveness of the proposed scheme. • A fourth-order numerical differential formula for the Riesz derivative is constructed. • An efficient high-order difference scheme is proposed. • The basic characteristics of the proposed scheme are analyzed under different norms. • The methodology can be extended to other spatial fractional differential equations. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to construct high-order numerical differential formulas approximating the Riesz derivative and apply them to the numerical solution of the nonlinear space fractional Ginzburg–Landau equations. Firstly, we introduce a novel second-order fractional central difference operator for the approximation of the Riesz derivative with order α ∈ (1 , 2 ]. Moreover, based on the difference operator and the compact technique, a novel fourth-order fractional compact difference operator is also derived. Secondly, using the fourth-order difference operator in space and the Crank–Nicolson method in time, a high-order difference scheme is proposed for the nonlinear space fractional Ginzburg–Landau equations. Thirdly, besides the standard energy method, some new techniques and important lemmas are developed to prove the unique solvability, stability and convergence in the sense of different norms. It is proved that the difference scheme is unconditionally stable and convergent with order O τ 2 + h 4 for α ∈ (1 , 1. 5) , where τ and h are the temporal step size and spatial step size, respectively. Finally, some numerical examples are given to show the efficiency and accuracy of the numerical differential formulas and finite difference scheme. • A fourth-order fractional compact numerical differential formula is constructed. • Some important inequalities are established. • The important properties of the coefficients are studied. • An implicit difference scheme is established. • The new techniques for analyzing stability and convergenceare proposed. [ABSTRACT FROM AUTHOR]