Improved uniform error bounds for long-time dynamics of the high-dimensional nonlinear space fractional sine-Gordon equation with weak nonlinearity

Abstract

In this paper, we derive the improved uniform error bounds for the long-time dynamics of the d-dimensional $(d=2,3)$ nonlinear space fractional sine-Gordon equation (NSFSGE). The nonlinearity strength of the NSFSGE is characterized by ${\epsilon }^2$ where $0<\epsilon \le 1$ is a dimensionless parameter. The second-order time-splitting method is applied to the temporal discretization and the Fourier pseudo-spectral method is used for the spatial discretization. To obtain the explicit relation between the numerical errors and the parameter ε, we introduce the regularity compensation oscillation technique to the convergence analysis of fractional models. Then we establish the improved uniform error bounds $O\left({\epsilon }^2{\tau }^2\right)$ for the semi-discretization scheme and $O(h^m+{\epsilon }^2{\tau }^2)$ for the full-discretization scheme up to the long time at $O(1/{\epsilon }^2)$. Further, we extend the time-splitting Fourier pseudo-spectral method to the complex NSFSGE as well as the oscillatory complex NSFSGE, and the improved uniform error bounds for them are also given. Finally, extensive numerical examples in two-dimension or three-dimension are provided to support the theoretical analysis. The differences in dynamic behaviors between the fractional sine-Gordon equation and classical sine-Gordon equation are also discussed.