  In this work we have used for the first time pseudo-spectral methods to
perform numerical simulations of spherically symmetric black hole formations on
a Friedman-Robertson-Walker universe. With these methods, the differential
equations describing the gravitational collapse are partially solved
algebraically. With our publicly available code we then independently check,
and confirm, previous numerical estimations of the thresholds to form
primordial black holes. By using an excision technique and analytical
estimations of accretion rates, we were also able to estimate the black holes
mass even in the case of large deviations from the threshold. There, we
confirm, with an explicit example, that the estimation of the black hole mass
via the self-similar scaling law is only accurate up to $O(15\%)$, for the
largest allowed mass.
