  We establish improved uniform error bounds on time-splitting methods for the
long-time dynamics of the Dirac equation with small electromagnetic potentials
characterized by a dimensionless parameter $\varepsilon\in (0, 1]$ representing
the amplitude of the potentials. We begin with a semi-discritization of the
Dirac equation in time by a time-splitting method, and then followed by a
full-discretization in space by the Fourier pseudospectral method. Employing
the unitary flow property of the second-order time-splitting method for the
Dirac equation, we prove uniform error bounds at $C(t)\tau^2$ and
$C(t)(h^m+\tau^2)$ for the semi-discretization and full-discretization,
respectively, for any time $t\in[0,T_\varepsilon]$ with $T_\varepsilon =
T/\varepsilon$ for $T > 0$, which are uniformly for $\varepsilon \in (0, 1]$,
where $\tau$ is the time step, $h$ is the mesh size, $m\geq 2$ depends on the
regularity of the solution, and $C(t) = C_0 + C_1\varepsilon t\le C_0+C_1T$
grows at most linearly with respect to $t$ with $C_0\ge0$ and $C_1>0$ two
constants independent of $t$, $h$, $\tau$ and $\varepsilon$. Then by adopting
the regularity compensation oscillation (RCO) technique which controls the high
frequency modes by the regularity of the solution and low frequency modes by
phase cancellation and energy method, we establish improved uniform error
bounds at $O(\varepsilon\tau^2)$ and $O(h^m +\varepsilon\tau^2)$ for the
semi-discretization and full-discretization, respectively, up to the long-time
$T_\varepsilon$. Numerical results are reported to confirm our error bounds and
to demonstrate that they are sharp. Comparisons on the accuracy of different
time discretizations for the Dirac equation are also provided.
