  Two important enhanced sampling algorithms, simulated (ST) and parallel (PT)
tempering, are commonly used when ergodic simulations may be hard to achieve,
e.g, due to a phase space separated by large free-energy barriers. This is so
for systems around first-order phase transitions, a case still not fully
explored with such approaches in the literature. In this contribution we make a
comparative study between the PT and ST for the Ising (a lattice-gas in the
fluid language) and the BEG (a lattice-gas with vacancies) models at phase
transition regimes. We show that although the two methods are equivalent in the
limit of sufficiently long simulations, the PT is more advantageous than the ST
with respect to all the analysis performed: convergence towards the
stationarity; frequency of tunneling between phases at the coexistence; and
decay of time-displaced correlation functions of thermodynamic quantities.
Qualitative arguments for why one may expect better results from the PT than
the ST near phase transitions conditions are also presented.
