  In this paper we present a numerical approach to solve the Navier-Stokes
equations on moving domains with second-order accuracy. The space
discretization is based on the ghost-point method, which falls under the
category of unfitted boundary methods, since the mesh does not adapt to the
moving boundary. The equations are advanced in time by using Crank-Nicholson.
The momentum and continuity equations are solved simultaneously for the
velocity and the pressure by adopting a proper multigrid approach. To avoid the
checkerboard instability for the pressure, a staggered grid is adopted, where
velocities are defined at the sides of the cell and the pressure is defined at
the centre. The lack of uniqueness for the pressure is circumvented by the
inclusion of an additional scalar unknown, representing the average divergence
of the velocity, and an additional equation to set the average pressure to
zero. Several tests are performed to simulate the motion of an incompressible
fluid around a moving object, as well as the lid-driven cavity tests around
steady objects. The object is implicitly defined by a level-set approach, that
allows a natural computation of geometrical properties such as distance from
the boundary, normal directions and curvature. Different shapes are tested:
circle, ellipse and flower. Numerical results show the second order accuracy
for the velocity and the divergence (that decays to zero with second order) and
the efficiency of the multigrid, that is comparable with the tests available in
literature for rectangular domains without objects, showing that the presence
of a complex-shaped object does not degrade the performance.
