  We establish the error bounds of fourth-order compact finite difference
(4cFD) methods for the Dirac equation in the massless and nonrelativistic
regime, which involves a small dimensionless parameter $0 < \varepsilon \le 1$
inversely proportional to the speed of light. In this regime, the solution
propagates waves with wavelength $O(\varepsilon)$ in time and $O(1)$ in space,
as well as with the wave speed $O(1/\varepsilon)$ rapid outgoing waves. We
adapt the conservative and semi-implicit 4cFD methods to discretize the Dirac
equation and rigorously carry out their error bounds depending explicitly on
the mesh size $h$, time step $\tau$ and the small parameter $\varepsilon$.
Based on the error bounds, the $\varepsilon$-scalability of the 4cFD methods is
$h = O(\varepsilon^{1/4})$ and $\tau = O(\varepsilon^{3/2})$, which not only
improves the spatial resolution capacity but also has superior accuracy than
classical second-order finite difference methods. Furthermore, physical
observables including the total density and current density have the same
conclusions. Numerical results are provided to validate the error bounds and
the dynamics of the Dirac equation with different potentials in 2D is
presented.
