The equation is:
$\partial f/\partial t + \nabla \cdot (v f) = 0$ $, \;\; f \in [0,1] $
and $v$ is a velocity known at every grid cell. A more precise constraint is that $f$ is either 0 or 1 but I can also live with it being in the domain [0,1]. The geometry is 2D cylindrical R-Z coordinates.
For the Initial condition, $f(R, Z)$ is known for all zones in the $R-Z$ domain and is not a simple function. Note that $v$ is evolving in time and space i.e. $v$ is really $v(R, Z, t)$. I do not want the solution to be diffusive.