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We present the fourth-order compact finite difference (4cFD) discretizationsfor the long time dynamics of the nonlinear Klein-Gordon equation (NKGE), whilethe 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 errorbounds of the 4cFD methods are carried out up to the time at$O(\varepsilon^{-p})$. We pay particular attention to how error bounds dependexplicitly on the mesh size $h$ and time step $\tau$ as well as the smallparameter $\varepsilon \in (0, 1]$, 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 methodsshould be taken as: $h = O(\varepsilon^{p/4})$ and $\tau =O(\varepsilon^{p/2})$. It has better spatial resolution capacity than theclassical second order central difference methods. By a rescaling in time, itis equivalent to an oscillatory NKGE whose solution propagates waves withwavelength at $O(1)$ in space and $O(\varepsilon^p)$ in time. It isstraightforward to get the error bounds of the oscillatory NKGE in the fixedtime. Finally, numerical results are provided to confirm our theoreticalanalysis.

Long time error analysis of the fourth-order compact finite difference methods for the nonlinear Klein-Gordon equation with weak nonlinearity