  We propose finite-volume schemes for the Cahn-Hilliard equation which
unconditionally and discretely preserve the boundedness of the phase field and
the dissipation of the free energy. Our numerical framework is applicable to a
variety of free-energy potentials, including Ginzburg-Landau and Flory-Huggins,
to general wetting boundary conditions, and to degenerate mobilities. Its
central thrust is the upwind methodology, which we combine with a semi-implicit
formulation for the free-energy terms based on the classical convex-splitting
approach. The extension of the schemes to an arbitrary number of dimensions is
straightforward thanks to their dimensionally split nature, which allows to
efficiently solve higher-dimensional problems with a simple parallelisation.
The numerical schemes are validated and tested through a variety of examples,
in different dimensions, and with various contact angles between droplets and
substrates.
