The construction of an optimal fourth-order fractional-compact-type numerical differential formula of the Riesz derivative and its application.

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]