  We study the destruction of interstellar dust via sputtering in supernova
(SN) shocks using three-dimensional hydrodynamical simulations. With a novel
numerical framework, we follow both sputtering and dust dynamics governed by
direct collisions, plasma drag and betatron acceleration. Grain-grain
collisions are not included and the grain-size distribution is assumed to be
fixed. The amount of dust destroyed per SN is quantified for a broad range of
ambient densities and fitting formulae are provided. Integrated over the
grain-size distribution, nonthermal (inertial) sputtering dominates over
thermal sputtering for typical ambient densities. We present the first
simulations that explicitly follow dust sputtering within a turbulent
multiphase interstellar medium. We find that the dust destruction timescales
$\tau$ are 0.35 Gyr for silicate dust and 0.44 Gyr for carbon dust in solar
neighborhood conditions. The SN environment has an important impact on $\tau$.
SNe that occur in preexisting bubbles destroy less dust as the destruction is
limited by the amount of dust in the shocked gas. This makes $\tau$ about 2.5
times longer than the estimate based on results from a single SN explosion. We
investigate the evolution of the dust-to-gas mass ratio (DGR), and find that a
spatial inhomogeneity of $\sim$ 14\% develops for scales below 10 pc. It
locally correlates positively with gas density but negatively with gas
temperature even in the exterior of the bubbles due to incomplete gas mixing.
This leads to a $\sim$ 30\% lower DGR in the volume filling warm gas compared
to that in the dense clouds.
