  Lagrangian algorithms to simulate the evolution of cold dark matter (CDM) are
invaluable tools to generate large suites of mock halo catalogues. In this
paper, we first show that the main limitation of current semi-analytical
schemes to simulate the displacement of CDM is their inability to model the
evolution of overdensities in the initial density field, a limit that can be
circumvented by detecting halo particles in the initial conditions. We thus
propose `MUltiscale Spherical Collapse Lagrangian Evolution Using
Press-Schechter' (muscle-ups), a new scheme that reproduces the results from
Lagrangian perturbation theory on large scales, while improving the modelling
of overdensities on small scales. In muscle-ups, we adapt the extended Press
and Schechter (EPS) formalism to Lagrangian algorithms of the displacement
field. For regions exceeding a collapse threshold in the density smoothed at a
radius $R$, we consider all particles within a radius $R$ collapsed. Exploiting
a multi-scale smoothing of the initial density, we build a halo catalogue on
the fly by optimizing the selection of halo candidates. This allows us to
generate a density field with a halo mass function that matches one measured in
$N$-body simulations. We further explicitly gather particles in each halo
together in a profile, providing a numerical, Lagrangian-based implementation
of the halo model. Compared to previous semi-analytical Lagrangian methods, we
find that muscle-ups improves the recovery of the statistics of the density
field at the level of the probability density function (PDF), the power
spectrum, and the cross correlation with the $N$-body result.
