High-order numerical algorithm and error analysis for the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation.

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]