  A non-hydrostatic depth-averaged model for dry granular flows is proposed,
taking into account vertical acceleration. A variable friction coefficient
based on the $\mu(I)$ rheology is considered. The model is obtained from an
asymptotic analysis in a local reference system, where the non-hydrostatic
contribution is supposed to be small compared to the hydrostatic one. The
non-hydrostatic counterpart of the pressure may be written as the sum of two
terms: one corresponding to the stress tensor and the other to the vertical
acceleration. The model introduced here is weakly non-hydrostatic, in the sense
that the non-hydrostatic contribution related to the stress tensor is not taken
into account due to its complex implementation. A simple and efficient
numerical scheme is proposed. It consists of a three-step splitting procedure,
and it is based on a hydrostatic reconstruction. Two key points are: (i) the
friction force has to be taken into account before solving the non-hydrostatic
pressure. Otherwise, the incompressibility condition is not ensured; (ii) both
the hydrostatic and the non-hydrostatic pressure are taken into account when
dealing with the friction force. The model and numerical scheme are then
validated based on several numerical tests, including laboratory experiments of
granular collapse. The influence of non-hydrostatic terms and of the choice of
the coordinate system (Cartesian or local) is analyzed. We show that
non-hydrostatic models are less sensitive to the choice of the coordinate
system. In general, the non-hydrostatic model introduced here much better
reproduces granular collapse experiments compared to hydrostatic models. An
important result is that the simulated mass profiles up to the deposit and the
front velocity are greatly improved. As expected, the influence of the
non-hydrostatic pressure is shown to be larger for small values of the slope.
