  We model the coagulation and fragmentation of dust grains during the
protostellar collapse with our newly developed shark code. It solves the
gas-dust hydrodynamics in a spherical geometry and the
coagulation/fragmentation equation. It also computes the ionization state of
the cloud and the Ohmic, ambipolar and Hall resistivities. We find that the
dust size distribution evolves significantly during the collapse, large grain
formation being controlled by the turbulent differential velocity. When
turbulence is included, only ambipolar diffusion remains efficient at removing
the small grains from the distribution, brownian motion is only efficient as a
standalone process. The macroscopic gas-dust drift is negligible for grain
growth and only dynamically significant near the first Larson core. At high
density, we find that the coagulated distribution is unaffected by the initial
choice of dust distribution. Strong magnetic fields are found to enhance the
small grains depletion, causing an important increase of the ambipolar
diffusion. This hints that the magnetic field strength could be regulated by
the small grain population during the protostellar collapse. Fragmentation
could be effective for bare silicates, but its modeling relies on the choice of
ill-constrained parameters. It is also found to be negligible for icy grains.
When fragmentation occurs, it strongly affects the magnetic resistivities
profiles. Dust coagulation is a critical process that needs to be fully taken
into account during the protostellar collapse. The onset and feedback of
fragmentation remains uncertain and its modeling should be further
investigated.
