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We establish improved uniform error bounds on time-splitting methods for thelong-time dynamics of the nonlinear Klein--Gordon equation (NKGE) with weakcubic nonlinearity, whose strength is characterized by $\varepsilon^2$ with $0< \varepsilon \leq 1$ a dimensionless parameter. Actually, when $0 <\varepsilon \ll 1$, the NKGE with $O(\varepsilon^2)$ nonlinearity and $O(1)$initial data is equivalent to that with $O(1)$ nonlinearity and small initialdata of which the amplitude is at $O(\varepsilon)$. We begin with asemi-discretization of the NKGE by the second-order time-splitting method, andfollowed by a full-discretization via the Fourier spectral method in space.Employing the regularity compensation oscillation (RCO) technique whichcontrols the high frequency modes by the regularity of the exact solution andanalyzes the low frequency modes by phase cancellation and energy method, wecarry out the improved uniform error bounds at $O(\varepsilon^2\tau^2)$ and$O(h^m+\varepsilon^2\tau^2)$ for the second-order semi-discretization andfull-discretization up to the long time $T_\varepsilon = T/\varepsilon^2$ with$T$ fixed, respectively. Extensions to higher order time-splitting methods andthe case of an oscillatory complex NKGE are also discussed. Finally, numericalresults are provided to confirm the improved error bounds and to demonstratethat they are sharp.

Improved uniform error bounds on time-splitting methods for long-time dynamics of the nonlinear Klein--Gordon equation with weak nonlinearity