  Dust grains are classically thought to form in the winds of asymptotic giant
branch (AGB) stars. However, there is increasing evidence today for dust
formation in supernovae (SNe). To establish the relative importance of these
two classes of stellar sources of dust, it is important to know the fraction of
freshly formed dust in SN ejecta that is able to survive the passage of the
reverse shock and be injected in the interstellar medium. We have developed a
new code (GRASH\_Rev) which follows the newly-formed dust evolution throughout
the supernova explosion until the merging of the forward shock with the
circumstellar ISM. We have considered four well studied SNe in the Milky Way
and Large Magellanic Cloud: SN1987A, CasA, the Crab Nebula, and N49. For all
the simulated models, we find good agreement with observations and estimate
that between 1 and 8$\%$ of the observed mass will survive, leading to a SN
dust production rate of $(3.9 \pm 3.7) \times 10^{-4}$ M$_{\odot}$yr$^{-1}$ in
the Milky Way. This value is one order of magnitude larger than the dust
production rate by AGB stars but insufficient to counterbalance the dust
destruction by SNe, therefore requiring dust accretion in the gas phase.
