Accurate modeling of flow instabilities requires computational tools
able to deal with several interacting scales, from the scale at which
fingers are triggered up to the scale at which their effects need
to be described. The Multiscale Finite Volume (MsFV) method offers
a framework to couple fine-and coarse-scale features by solving a
set of localized problems which are used both to define a coarse-scale
problem and to reconstruct the fine-scale details of the flow. The
MsFV method can be seen as an upscaling-downscaling technique, which
is computationally more efficient than standard discretization schemes
and more accurate than traditional upscaling techniques. We show
that, although the method has proven accurate in modeling density-driven
flow under stable conditions, the accuracy of the MsFV method deteriorates
in case of unstable flow and an iterative scheme is required to control
the localization error. To avoid large computational overhead due
to the iterative scheme, we suggest several adaptive strategies both
for flow and transport. In particular, the concentration gradient
is used to identify a front region where instabilities are triggered
and an accurate (iteratively improved) solution is required. Outside
the front region the problem is upscaled and both flow and transport
are solved only at the coarse scale. This adaptive strategy leads
to very accurate solutions at roughly the same computational cost
as the non-iterative MsFV method. In many circumstances, however,
an accurate description of flow instabilities requires a refinement
of the computational grid rather than a coarsening. For these problems,
we propose a modified iterative MsFV, which can be used as downscaling
method (DMsFV). Compared to other grid refinement techniques the
DMsFV clearly separates the computational domain into refined and
non-refined regions, which can be treated separately and matched
later. This gives great flexibility to employ different physical
descriptions in different regions, where different equations could
be solved, offering an excellent framework to construct hybrid methods.