In this work, we are concerned with a time-splitting Fourier pseudospectral
(TSFP) discretization for the Klein-Gordon (KG) equation, involving a dimensionless
parameter ε∈(0,1]. In the nonrelativistic limit regime, the small ε produces high oscillations
in exact solutions with wavelength of O(ε^{2}) in time. The key idea behind the
TSFP is to apply a time-splitting integrator to an equivalent first-order system in time,
with both the nonlinear and linear subproblems exactly integrable in time and, respectively,
Fourier frequency spaces. The method is fully explicit and time reversible.
Moreover, we establish rigorously the optimal error bounds of a second-order TSFP
for fixed ε = O(1), thanks to an observation that the scheme coincides with a type of
trigonometric integrator. As the second task, numerical studies are carried out, with
special efforts made to applying the TSFP in the nonrelativistic limit regime, which are
geared towards understanding its temporal resolution capacity and meshing strategy
for O(ε^{2})-oscillatory solutions when 0 < ε ≪ 1. It suggests that the method has uniform
spectral accuracy in space, and an asymptotic O(ε^{−2}∆t^{2}) temporal discretization
error bound (∆t refers to time step). On the other hand, the temporal error bounds for
most trigonometric integrators, such as the well-established Gautschi-type integrator
in [6], are O(ε^{−4}∆t^{2}). Thus, our method offers much better approximations than the
Gautschi-type integrator in the highly oscillatory regime. These results, either rigorous
or numerical, are valid for a splitting scheme applied to the classical relativistic NLS
reformulation as well.