  The Reynolds-Averaged Navier-Stokes equations and the Large-Eddy Simulation
equations can be coupled using a transition function to switch from a set of
equations applied in some areas of a domain to the other set in the other part
of the domain. Following this idea, different time integration schemes can be
coupled. In this context, we developed a hybrid time integration scheme that
spatially couples the explicit scheme of Heun and the implicit scheme of Crank
and Nicolson using a dedicated transition function. This scheme is linearly
stable and second-order accurate. In this paper, an extension of this hybrid
scheme is introduced to deal with a temporal adaptive procedure. The idea is to
treat the time integration procedure with unstructured grids as it is performed
with Cartesian grids with local mesh refinement. Depending on its
characteristic size, each mesh cell is assigned a rank. And for two cells from
two consecutive ranks, the ratio of the associated time steps for time marching
the solutions is $2$. As a consequence, the cells with the lowest rank iterate
more than the other ones to reach the same physical time. In a finite-volume
context, a key ingredient is to keep the conservation property for the
interfaces that separate two cells of different ranks. After introducing the
different schemes, the paper recalls briefly the coupling procedure, and
details the extension to the temporal adaptive procedure. The new time
integration scheme is validated with the propagation of 1D wave packet, the
Sod's tube, and the transport of a bi-dimensional vortex in an uniform flow.
