Fourth‐Order Discretization of the Fractional Laplacian and Integrating Factor Time‐Stepping for Coupled Ginzburg–Landau Equations

ABSTRACT In this paper, we present a detailed theoretical analysis and numerical study on the coupled space‐fractional Ginzburg–Landau equations. These types of equations serve as a fundamental model for describing dynamic processes in media with fractal geometric structures, where the fractional dispersion effect often plays a critical role. Due to the nonlocality of the fractional Laplacian operator and the presence of strong nonlinear coupling terms in the equation, both rigorous mathematical analysis and efficient numerical simulation of this model face significant challenges. To address these difficulties effectively, our work first focuses on investigating the well‐posedness of the model. By establishing a series of a priori estimates, we provide a strict mathematical proof for the existence and uniqueness of weak solutions. These theoretical results lay a solid foundation for the subsequent design of numerical methods. After establishing the theoretical basis, the focus of our research naturally shifts to the construction of efficient numerical schemes. By introducing a novel generating function, we systematically derive a fourth‐order finite difference approximation for the fractional Laplacian operator. For the discretization in the temporal direction, we adopt an approach that combines the implicit integration factor method with the Padé approximation. This strategy not only achieves second‐order accuracy in time, but also constructs a two‐level scheme that retains the self‐starting property, thus avoiding the common initialization difficulty encountered in traditional multistep methods. Based on the above spatial and temporal discretization techniques, the resulting fully discrete scheme achieves fourth‐order accuracy in space and second‐order accuracy in time. Furthermore, we conduct a comprehensive theoretical analysis on the proposed numerical scheme, confirming that it possesses unique solvability, unconditional stability, and convergence. Meanwhile, to solve the system of nonlinear algebraic equations generated at each time step, we design a dedicated iterative algorithm and conduct a rigorous analysis of its convergence. Finally, we present a series of numerical experiments. These numerical examples not only verify the theoretical convergence order of the proposed algorithm, but also further demonstrate its performance in practical computations.