Projection Method for Steady states of Cahn-Hilliard Equation with the Dynamic Boundary Condition

TL;DR

This paper introduces a computationally efficient projection method for finding steady states of the Cahn-Hilliard equation with dynamic boundary conditions, preserving key physical properties and validated through numerical experiments.

Contribution

It presents a novel projection method that reduces computational cost by using the $L^2$ metric for the variational derivative, while maintaining mass conservation and energy dissipation.

Findings

Method effectively locates steady states with reduced computational effort.

Numerical experiments confirm the method's accuracy and physical property preservation.

Applicable to two-dimensional Ginzburg-Landau free energy models.

Abstract

The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of two-phase flows or binary mixtures. In recent years, the dynamic boundary conditions for the Cahn-Hilliard equation have been proposed and analyzed. Our first goal in this article is to present a projection method to locate the steady state of the CH equation with dynamic boundary conditions. The main feature of this method is that it only uses the variational derivative in the metric $L^2$ and not that in the metric $H^{-1}$, thus significantly reducing the computational cost. In addition, the projected dynamics fulfill the important physical properties: mass conservation and energy dissipation. In the temporal construction of the numerical schemes, the convex splitting method is used to ensure a large time step size. Numerical experiments for the two-dimensional Ginzburg-Landau free energy, where the surface potential is the double well potential or the moving contact line potential, are conducted to demonstrate the effectiveness of this projection method.