  We consider a one-dimensional aggregation-diffusion equation, which is the
gradient flow in the Wasserstein space of a functional with competing
attractive-repulsive interactions.
  We prove that the fully deterministic particle approximations with piecewise
constant densities introduced in~\cite{Di Francesco-Rosini} starting from
general bounded initial densities converge strongly in $L^1$ to bounded weak
solutions of the PDE.
  In particular, the result is achieved in unbounded domains and for arbitrary
nonnegative bounded initial densities, thus extending the results in
\cite{Gosse-Toscani, Matthes-Osberger, Mathes-Soellner} (in which a no-vacuum
condition is required) and giving an alternative approach to
\cite{Carrillo-Craig-Patacchini} in the one-dimensional case, including also
subquadratic and superquadratic diffusions.
