  To investigate universal behavior and effects of long-range temporal
correlations in kinetic roughening, we perform extensive simulations on the
Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise based on
pseudospectral (PS) and one of the improved finite-difference (FD) schemes. We
find that scaling properties are affected by long-range temporal correlations
within the effective temporally correlated regions. Our results are consistent
with each other using these two independent numerical schemes, three
characteristic roughness exponents (global roughness exponent $\alpha$, local
roughness exponent $\alpha_{loc}$, and spectral roughness exponent
$\alpha_{s}$) are approximately equal within the small temporally correlated
regime, and satisfy $\alpha_{loc} \approx \alpha<\alpha_{s}$ for the large
temporally correlated regime, and the difference between $\alpha_{s}$ and
$\alpha$ increases with increasing the temporal correlation exponent $\theta$.
Our results also show that PS and the improved FD schemes could effectively
suppress numerical instabilities in the temporally correlated KPZ growth
equation. Furthermore, our investigations suggest that when the effects of
long-range temporal correlation are present, the continuum and discrete growth
systems do not belong to the same universality class with the same temporal
correlation exponent.
