We establish uniform error bounds of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity whose strength is characterized by $\varepsilon^2$ with $\varepsilon \in (0, 1]$ a dimensionless parameter. When $0 < \varepsilon \ll 1$, the problem is equivalent to the long-time dynamics of the NKGE with small initial data (and $O(1)$ cubic nonlinearity), while the amplitude of the initial data (and the solution) is at $O(\varepsilon)$. For the long-time dynamics of the NKGE up to the time at $O(1/\varepsilon^{2})$, the resolution and error bounds of the classical numerical methods depend significantly on the small parameter $\varepsilon$, which causes severe numerical burdens as $\varepsilon \to 0^+$. The EWI-FP method is fully explicit, symmetric in time and has many superior properties in solving wave equations. By adapting the energy method combined with the method of mathematical induction, we rigorously carry out the uniform error bounds of the EWI-FP discretization at $O(h^{m_0} + \varepsilon^{2-\beta}\tau^2)$ up to the time at $O(1/\varepsilon^{\beta})$ with $0 \leq \beta \leq 2$, mesh size $h$, time step $\tau$ and $m_0$ an integer depending on the regularity of the solution. By a rescaling in time, our results are straightforwardly extended to the error bounds and $\varepsilon$-scalability (or meshing strategy requirement) of the EWI-FP method for an oscillatory NKGE, whose solution propagates waves with wavelength at $O(1)$ and $O(\varepsilon^{\beta})$ in space and time, respectively, and wave speed at $O(\varepsilon^{-\beta})$. Finally, extensive numerical results are reported to confirm our error estimates.

中文翻译：

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非线性 Klein-Gordon 方程长期动力学的指数波积分器的均匀误差界

时间对称，在求解波动方程方面具有许多优越的性质。通过采用能量法与数学归纳法相结合的方法，我们在$O(h^{m_0} + \varepsilon^{2-\beta}\tau^2处严格执行EWI-FP离散化的统一误差界)$ 到 $O(1/\varepsilon^{\beta})$ 的时间，$0 \leq \beta \leq 2$，网格尺寸 $h$，时间步长 $\tau$ 和 $m_0$ 一个整数取决于溶液的规律性。通过及时重新缩放，我们的结果直接扩展到 EWI-FP 方法的误差界限和 $\varepsilon$-可扩展性（或网格策略要求），用于振荡 NKGE，其解决方案传播波长为 $O(1 )$ 和 $O(\varepsilon^{\beta})$ 分别在空间和时间上，以及在 $O(\varepsilon^{-\beta})$ 处的波速。最后，