Unconditional Bound-Preserving and Energy-Dissipating Finite-Volume Schemes for the Cahn-Hilliard Equation (Communications in Computational Physics, 2023)
We propose finite-volume schemes for the Cahn-Hilliard equation whichunconditionally and discretely preserve the boundedness of the phase field and thedissipation of the free energy. Our numerical framework is applicable to a varietyof free-energy potentials, including Ginzburg-Landau and Flory-Huggins, to generalwetting 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 theschemes 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 testedthrough a variety of examples, in different dimensions, and with various contact anglesbetween droplets and substrates.
Keywords: Cahn-Hilliard equation; diffuse interface theory; gradient flow; finite-volume method; bound preservation; energy dissipation.