Improved uniform error bounds for long-time dynamics of the high-dimensional nonlinear space fractional sine-Gordon equation with weak nonlinearity
<|reference_start|>Improved uniform error bounds for long-time dynamics of the high-dimensional nonlinear space fractional sine-Gordon equation with weak nonlinearity: 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 $\varepsilon^2$ where $0<\varepsilon \le 1$ 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 $\varepsilon$, we introduce the regularity compensation oscillation technique to the convergence analysis of fractional models. Then we establish the improved uniform error bounds $O\left(\varepsilon^2 \tau^2\right)$ for the semi-discretization scheme and $O\left(h^m+\varepsilon^2 \tau^2\right)$ for the full-discretization scheme up to the long time at $O(1/\varepsilon^2)$. 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.<|reference_end|>
@article{jia2024improved,
    title={Improved uniform error bounds for long-time dynamics of the
  high-dimensional nonlinear space fractional sine-Gordon equation with weak
  nonlinearity},
    author={Junqing Jia and Xiaoqing Chi and Xiaoyun Jiang},
    journal={arXiv preprint arXiv:2402.18071},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.18071},
    primaryClass={math.NA cs.NA}
}