Convergence analysis of positivity-preserving finite difference scheme for the Flory-Huggins-Cahn-Hilliard equation with dynamical boundary condition
Yunzhuo Guo, Cheng Wang, Zhengru Zhang
TL;DR
This paper addresses the convergence analysis of fully discrete, positivity-preserving convex-splitting schemes for the Flory–Huggins–Cahn–Hilliard equation with dynamical boundary conditions on a square domain. A two-part mass-conservation strategy is developed: Fourier projection in the periodic x-direction and a carefully crafted auxiliary trigonometric function in y to enforce bulk mass conservation without impacting surface mass, enabling well-defined $H^{-1}$ error measures. The authors establish $ ext{ell}^ extfty(0,T;H_h^{-1}) igcap ext{ell}^2(0,T;H_h^1)$ convergence for the first-order scheme with rate $O( ext{Δt}+h^2)$, and extend the framework to a second-order, stabilized BDF2 scheme with a modified energy dissipation law. The auxiliary-correction approach preserves discrete mass conservation and energy dissipation, offering a robust path for high-fidelity simulations of CH-type systems with dynamical boundaries and potential generalization to other boundary geometries and higher-order schemes.
Abstract
The Cahn-Hilliard equation has a wide range of applications in many areas of physics and chemistry. To describe the short-range interaction between the solution and the boundary, scientists have constructed dynamical boundary conditions by introducing boundary energy. In this work, the dynamical boundary condition is located on two opposite edges of a square domain and is connected with bulk by a normal derivative. A convex-splitting numerical approach is proposed to enforce the positivity-preservation and energy dissipation, combined with the finite difference spatial approximation. The $\ell^\infty(0,T;H_h^{-1}) \cap \ell^2(0,T;H_h^1)$ convergence analysis and error estimate is theoretically established, with the first order accuracy in time and second order accuracy in space. The bulk and surface discrete mass conservation of the exact solution is required to reach the mean-zero property of the error function, so that the associated discrete $H_h^{-1}$ norm is well-defined. The mass conservation on the physical boundary is maintained by the classic Fourier projection. In terms of the mass conservation in bulk, we introduce a trigonometric auxiliary function based on the truncation error expansion, so that the bulk mass conservation is achieved, and it has no effect on the boundary. The smoothness of trigonometric function makes the Taylor expansion valid and maintains the convergence order of truncation error as well. As a result, the convergence analysis could be derived with a careful nonlinear error estimate.