Improved Error Estimates on Time‐Splitting Method for the Space Fractional Klein–Gordon–Schrödinger Equation With the Small Coupling Constant

ABSTRACT This study aims to present the improved long‐time error estimates of the space fractional Klein–Gordon–Schrödinger equation (SFKSGE) with a small coupling parameter . We first turn the SFKGSE into a nonlinear coupled space fractional Schrödinger system for numerical discretization. By the Strang splitting method, the time‐splitting scheme with second‐order accuracy is proposed to discretize the time domain, and it is a time‐symmetric scheme. Then, the Fourier pseudo‐spectral method is adopted to discretize the space domain. Through strict derivation and with the help of the regularity compensation oscillation technique, we establish the improved uniform error bound for the fully discrete scheme up to the long time at , indicating an explicit expression relating the numerical errors and the coupling parameter . Combined with numerical examples, we verify that the improved uniform error bounds are sharp. The numerical results also show that the time‐splitting Fourier pseudo‐spectral scheme can preserve the conservation of discrete mass and have long‐time stability of discrete energy. Moreover, we simulate the dynamic evolution of different soliton solutions in the SFKGSE, which further illustrates the intrinsic nonlocal interactions and heavy‐tailed behavior inherent in fractional diffusion processes.