We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is $O(\varepsilon)$ with $0 < \varepsilon \ll 1$ a dimensionless parameter up to the time at $O(1/\varepsilon^2)$. The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discreization at $O(\varepsilon^2\tau)$ instead of $O(\tau)$ according to classical error estimates and at $O(h^m+\varepsilon^2\tau)$ for the full-discretization up to the time $T_{\varepsilon} = T/\varepsilon^2$ with $T>0$ fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the NKGE with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with $O(\varepsilon^2)$ wavelength in time and $O(\varepsilon^{-2})$ wave speed, which indicates that the temporal error is independent of $\varepsilon$ when the time step size is chosen as $O(\varepsilon^2)$. Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp.

翻译：我们用劳森型指数化集成器 Fleier伪光谱(LEI-FP) 方法为Srein-Gordon方程式的长期动态建立更好的统一误差框, 初始数据的振幅值为$O( varepsilon)$, 美元为1美元, 一个无维度的参数, 直至 $O( 1/\ varepsilon) 美元。 数字方案将劳森型指数化集成器与 Freyer 偏差( FleI- FFP) 的 Freier 伪光谱法相结合, 由于快速的 Fourier 变换, 在实际的计算中该方法完全清晰和高效。 通过将线性部分从正弦函数中分离出来, 使用正统性补偿( RCOO) 技术, 通过阶段取消来处理多维度不直线性参数, 我们将半分解的误差值( $( varepsilon) 2\\\\toau) 美元( tau) modeal- mocial disal disal disal) mode, 美元为全时值( =xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx