A uniquely solvable and positivity-preserving finite difference scheme for the Flory-Huggins-Cahn-Hilliard equation with dynamical boundary condition

TL;DR

The paper addresses the Flory–Huggins–Cahn–Hilliard system with dynamical boundary conditions by developing a positivity-preserving, uniquely solvable finite-difference scheme based on convex splitting. The interior and boundary logarithmic energies are treated implicitly, while the concave part is explicit, leading to a discrete energy $F_h^n$ whose minimization yields the update, and a discrete energy-dissipation law $E_h(\phi^{n+1})\le E_h(\phi^n)$. The authors establish rigorous positivity bounds $-1<\phi^{n+1}<1$ and mass conservation, and demonstrate unconditional energy stability via discrete summation-by-parts. Numerical experiments, including convergence tests, spinodal decomposition, and boundary-influenced droplet dynamics, confirm both the theoretical properties and the practical effectiveness of the scheme, highlighting how dynamical boundaries affect wetting behavior and evolution rates. These results provide a robust framework for simulating complex interfacial phenomena with singular Flory–Huggins potentials on domains with evolving boundaries.

Abstract

In this paper we propose and analyze a finite difference numerical scheme for the Flory-Huggins-Cahn-Hilliard equation with dynamical boundary condition. The singular logarithmic potential is included in the Flory-Huggins energy expansion. Meanwhile, a dynamical evolution equation for the boundary profile corresponds to a lower-dimensional singular energy potential. In turn, a theoretical analysis for the coupled system becomes very challenging, since it contains nonlinear and singular energy potentials for both the interior region and on the boundary. In the numerical design, a convex splitting approach is applied to the chemical potential associated with the energy both at the interior region and on the boundary: implicit treatments for the singular and logarithmic terms, as well as the surface diffusion terms, combined with an explicit treatment for the concave expansive term. In addition, the discrete boundary condition for the phase variable is coupled with the evolutionary equation of the boundary profile. The resulting numerical system turns out to be highly nonlinear, singular and coupled. A careful finite difference approximation and convexity analysis reveals that such a numerical system could be represented as a minimization of a discrete numerical energy functional, which contains both the interior and boundary integrals. More importantly, all the singular terms correspond to a discrete convex functional. As a result, a unique solvability and positivity-preserving analysis could be theoretically justified, based on the subtle fact that the singular nature of the logarithmic terms around the singular limit values prevent the numerical solutions reaching these values. The total energy stability analysis could be established by a careful estimate over the finite difference inner product. Some numerical results are presented in this article.

Figures (7)

• Figure 1: The phase evolution of $\phi$ at several time instants with initial data \ref{['rand_initial']}.
• Figure 2: (a) Left: the time evolution of the energy. (b) Right: the time evolution of the mass difference.
• Figure 3: Snapshots of solution for droplet evolution at selected times.
• Figure 4: Initial profile of the fusion process of two circular droplets.
• Figure 5: (a): The evolution of droplet fusion process, with the homogeneous Neumann boundary condition, at time instants $t=1\times 10^{-4}$, $1\times 10^{-3}$, $3\times 10^{-3}$, $1\times 10^{-2}$. (b) The evolution of droplet fusion process, with the dynamical boundary condition at the bottom boundary, at the same time instants.

Theorems & Definitions (16)

• Lemma 2.1
• Proposition 2.1
• Definition 3.1
• Proposition 3.1
• Lemma 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Lemma 3.2: chen19b