Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn-Hilliard equation

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