  We present the first 2D hydrodynamical finite volume simulations in which
dust is fully coupled with the gas, including its back-reaction onto it, and at
the same time the dust size is evolving according to coagulation and
fragmentation based on a sub-grid model. The aim of this analysis is to present
the differences occurring when dust evolution is included relative to
simulations with fixed dust size, with and without an embedded Jupiter-mass
planet that triggers gap formation. We use the two-fluid polar Godunov-type
code RoSSBi developed by Surville et al. 2016 combined with a new local
sub-grid method for dust evolution based on the model by Birnstiel et al. 2012.
We find striking differences between simulations with variable and fixed dust
sizes. The timescales for dust depletion differ significantly and yield a
completely different evolution of the dust surface density. In general sharp
features such as pile-ups of dust in the inner disk and near gap edges, when a
massive planet is present, become much weaker. This has important implications
on the interpretation of observed substructure in disks, suggesting that the
presence of a massive planet does not necessarily cause sharp gaps and rings in
the dust component. Also, particles with different dust sizes show a different
distribution, pointing to the importance of multi-wavelength synthetic
observations in order to compare with observations by ALMA and other
instruments. We also find that simulations adopting fixed intermediate particle
sizes, in the range $10^{-2} - 10^{-1}$ cm, best approximate the surface
density evolution seen in simulations with dust evolution.
