Uniform error bounds of a time-splitting spectral method for the long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity

We establish uniform error bounds of time-splitting Fourier pseudospectral (TSFP) methods for the nonlinear Klein--Gordon equation (NKGE) with weak power-type nonlinearity and $O(1)$ initial data, while the nonlinearity strength is characterized by $\varepsilon^{p}$ with a constant $p \in \mathbb{N}^+$ and a dimensionless parameter $\varepsilon \in (0, 1]$, for the long-time dynamics up to the time at $O(\varepsilon^{-\beta})$ with $0 \leq \beta \leq p$. In fact, when $0 < \varepsilon \ll 1$, the problem is equivalent to the long-time dynamics of NKGE with small initial data and $O(1)$ nonlinearity strength, while the amplitude of the initial data (and the solution) is at $O(\varepsilon)$. By reformulating the NKGE into a relativistic nonlinear Schr\"{o}dinger equation, we adapt the TSFP method to discretize it numerically. By using the method of mathematical induction to bound the numerical solution, we prove uniform error bounds at $O(h^{m}+\varepsilon^{p-\beta}\tau^2)$ of the TSFP method with $h$ mesh size, $\tau$ time step and $m\ge2$ depending on the regularity of the solution. The error bounds are uniformly accurate for the long-time simulation up to the time at $O(\varepsilon^{-\beta})$ and uniformly valid for $\varepsilon\in(0,1]$. Especially, the error bounds are uniformly at the second order rate for the large time step $\tau = O(\varepsilon^{-(p-\beta)/2})$ in the parameter regime $0\le\beta <p$. Numerical results are reported to confirm our error bounds in the long-time regime. Finally, the TSFP method and its error bounds are extended to a highly oscillatory complex NKGE which propagates waves with wavelength at $O(1)$ in space and $O(\varepsilon^{\beta})$ in time and wave velocity at $O(\varepsilon^{-\beta})$.