  Different efficient and accurate numerical methods have recently been
proposed and analyzed for the nonlinear Klein-Gordon equation (NKGE) with a
dimensionless parameter $\varepsilon\in (0,1]$, which is inversely proportional
to the speed of light. In the nonrelativestic limit regime, i.e.
$0<\varepsilon\ll1$, the solution of the NKGE propagates waves with wavelength
at $O(1)$ and $O(\varepsilon^2)$ in space and time, respectively, which brings
significantly numerical burdens in designing numerical methods. We compare
systematically spatial/temporal efficiency and accuracy as well as
$\varepsilon$-resolution (or $\varepsilon$-scalability) of different numerical
methods including finite difference time domain methods, time-splitting method,
exponential wave integrator, limit integrator, multiscale time integrator,
two-scale formulation method and iterative exponential integrator. Finally, we
adopt the multiscale time integrator to study the convergence rates from the
NKGE to its limiting models when $\varepsilon\to0^+$.
