  We study the fully nonlinear dynamics of black hole spontaneous
scalarizations in Einstein-Maxwell scalar theory with coupling function
$f(\phi)=e^{-b\phi^{2}}$, which can transform usual Reissner-Nordstr\"om
Anti-de Sitter (RN-AdS) black holes into hairy black holes. Fixing the
Arnowitt-Deser-Misner mass of the system, the initial scalar perturbation will
destroy the original RN-AdS black hole and turn it into a hairy black hole
provided that the constant $-b$ in the coupling function and the charge of the
original black hole are sufficiently large, while the cosmological constant is
small enough. In the scalarization process, we observe that the black hole
irreducible mass initially increases exponentially, then it approaches to and
finally saturates at a finite value. Choosing stronger coupling and larger
black hole charge, we find that the black hole mass exponentially grows earlier
and it takes a longer time for a hairy black hole to be developed and
stabilized. We further examine phase structure properties in the scalarization
process and confirm the observations in the non-linear dynamical study.
