We present the fourth-order compact finite difference (4cFD) discretizations for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE), while the nonlinearity strength is characterized by $\varepsilon^p$ with a constant $p \in \mathbb{N}^+$ and a dimensionless parameter $\varepsilon \in (0, 1]$. Based on analytical results of the life-span of the solution, rigorous error bounds of the 4cFD methods are carried out up to the time at $O(\varepsilon^{-p})$. We pay particular attention to how error bounds depend explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $\varepsilon \in (0, 1]$, which indicate that, in order to obtain `correct' numerical solutions up to the time at $O(\varepsilon^{-p})$, the $\varepsilon$-scalability (or meshing strategy requirement) of the 4cFD methods should be taken as: $h = O(\varepsilon^{p/4})$ and $\tau = O(\varepsilon^{p/2})$. It has better spatial resolution capacity than the classical second order central difference methods. By a rescaling in time, it is equivalent to an oscillatory NKGE whose solution propagates waves with wavelength at $O(1)$ in space and $O(\varepsilon^p)$ in time. It is straightforward to get the error bounds of the oscillatory NKGE in the fixed time. Finally, numerical results are provided to confirm our theoretical analysis.

中文翻译：

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弱非线性非线性Klein-Gordon方程四阶紧致有限差分法的长时间误差分析

$h = O(\varepsilon^{p/4})$ 和 $\tau = O(\varepsilon^{p/2})$。它比经典的二阶中心差分方法具有更好的空间分辨率能力。通过时间上的重新缩放，它相当于一个振荡的 NKGE，其解在空间中传播波长为 $O(1)$ 且时间为 $O(\varepsilon^p)$ 的波。在固定时间内获得振荡 NKGE 的误差范围很简单。最后，提供了数值结果来证实我们的理论分析。在固定时间内获得振荡 NKGE 的误差范围很简单。最后，提供了数值结果来证实我们的理论分析。在固定时间内获得振荡 NKGE 的误差范围很简单。最后，提供了数值结果来证实我们的理论分析。