  Improved uniform error bounds on time-splitting methods are rigorously proven
for the long-time dynamics of the weakly nonlinear Dirac equation (NLDE), where
the nonlinearity strength is characterized by a dimensionless parameter
$\varepsilon \in (0, 1]$ . We adopt a second order Strang splitting method to
discretize the NLDE in time and combine the Fourier pseudospectral method in
space for the full-discretization. By employing the {\sl regularity
compensation oscillation} (RCO) technique where the high frequency modes are
controlled by the regularity of the exact solution and the low frequency modes
are analyzed by phase cancellation and energy method, we establish improved
uniform error bounds at $O(\varepsilon^2\tau^2)$ and $O(h^{m-1}+
\varepsilon^2\tau^2)$ for the second-order Strang splitting semi-discretizaion
and full-discretization up to the long-time $T_{\varepsilon} = T/\varepsilon^2$
with $T>0$ fixed, respectively. Furthermore, the numerical scheme and error
estimates are extended to an oscillatory NLDE which propagates waves with
$O(\varepsilon^2)$ wavelength in time. Finally, numerical examples verifying
our analytical results are given.
