  We propose a new fourth-order compact time-splitting ($S_\text{4c}$) Fourier
pseudospectral method for the Dirac equation by splitting the Dirac equation
into two parts together with using the double commutator between them to
integrate the Dirac equation at each time interval. The method is explicit,
fourth-order in time and spectral order in space. It is unconditional stable
and conserves the total density in the discretized level. It is called a
compact time-splitting method since, at each time step, the number of sub-steps
in $S_\text{4c}$ is much less than those of the standard fourth-order splitting
method and the fourth-order partitioned Runge-Kutta splitting method.
Comparison among $S_\text{4c}$ and many other existing time-splitting methods
for the Dirac equation are carried out in terms of accuracy and efficiency as
well as long time behavior. Numerical results demonstrate the advantage in
terms of efficiency and accuracy of the proposed $S_\text{4c}$. Finally we
report the spatial/temporal resolutions of $S_\text{4c}$ for the Dirac equation
in different parameter regimes including the nonrelativistic limit regime, the
semiclassical limit regime, and the simultaneously nonrelativisic and massless
limit regime.
