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The aim of this paper is to establish improved uniform error bounds under H α / 2 -norm ( 1 < α ≤ 2 ) for the long-time dynamics of the high-dimensional nonlinear space fractional sine-Gordon equation (NSFSGE) by a Lawson-type exponential integrator Fourier pseudo-spectral (LEI-FP) method. Firstly, a Lawson-type exponential integrator method is used to discretize the time direction. Then, the Fourier pseudo-spectral method is applied to discretize the space direction. We rigorously prove that the equation is energy conservation in a continuous state. Regularity compensation oscillation (RCO) technique is employed to strictly prove the improved uniform error bounds at O ε 2 τ in temporal semi-discretization and O h m + ε 2 τ in full-discretization up to the long-time T ε = T / ε 2 ( T > 0 fixed), respectively. To obtain the convergence order h m in space, we only need to directly prove it instead of proving that the numerical solution is H m + α / 2 -norm bounded as before. Complex NSFSGE and oscillatory NSFSGE are also discussed. This is the novel work to construct the improved uniform error bounds for the long-time dynamics of the high-dimensional nonlinear space fractional Klein-Gordon equation with non-polynomial nonlinearity. Finally, numerical examples in two-dimension and three-dimension are provided to confirm the improved error bounds, and we find drastically different evolving patterns between NSFSGE and the classical sine-Gordon equation.

This paper presents a high-order structure-preserving difference scheme for the nonlinear space fractional sine-Gordon equation with damping, employing the triangular scalar auxiliary variable approach. The original equation is reformulated into an equivalent system that satisfies a modified energy conservation or dissipation law, significantly reducing the computational complexity of nonlinear terms. Temporal discretization is achieved using a second-order difference method, while spatial discretization utilizes a simple and easily implementable discrete approximation for the fractional Laplacian operator. The boundedness and convergence of the proposed numerical scheme under the maximum norm are rigorously analyzed, demonstrating its adherence to discrete energy conservation or dissipation laws. Numerical experiments validate the scheme’s effectiveness, structure-preserving properties, and capability for long-time simulations for both one- and two-dimensional problems. Additionally, the impact of the parameter ε on error dynamics is investigated.