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 ${\epsilon }^p$ with a constant $p \in $\mathbb{N}$^+$andadimensionlessparameter$\varepsilon \in (0, 1]$, for the long-time dynamics up to the time at $O({\epsilon }^{-\beta })$ with $0 \leq \beta \leq p$\mathrm{.}Infact,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 $O(\epsilon )$. 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 $O(h^m+{\epsilon }^{p-\beta }{\tau }^2)$ of the TSFP method with $h$ mesh size, $\tau$ time step and $m\ge 2$ depending on the regularity of the solution. The error bounds are uniformly accurate for the long-time simulation up to the time at $O({\epsilon }^{-\beta })$ and uniformly valid for $\epsilon \in (0,1]$. Especially, the error bounds are uniformly at the second order rate for the large time step $\tau =O({\epsilon }^{-(p-\beta )\mathrm{/}2})$ 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 $O({\epsilon }^{\beta })$ in time and wave velocity at $O({\epsilon }^{-\beta })$.