Numerical Approximations and Error Analysis of the Cahn-Hilliard Equation with Dynamic Boundary Conditions

TL;DR

This paper introduces a new linear, energy-stable numerical scheme for the Cahn-Hilliard equation with dynamic boundary conditions, providing error analysis and validation through numerical experiments.

Contribution

It proposes a first-order, linear, energy-stable scheme based on stabilized implicit methods, with proven stability and error estimates.

Findings

The scheme is energy stable and first-order accurate in time.

Numerical experiments confirm the scheme's stability and accuracy.

Comparison with previous methods shows improved performance.

Abstract

We consider the numerical approximations of the Cahn-Hilliard equation with dynamic boundary conditions (C. Liu et. al., Arch. Rational Mech. Anal., 2019). We propose a first-order in time, linear and energy stable numerical scheme, which is based on the stabilized linearly implicit approach. The energy stability of the scheme is proved and the semi-discrete-in-time error estimates are carried out. Numerical experiments, including the comparison with the former work, the accuracy tests with respect to the time step size and the shape deformation of a droplet, are performed to validate the accuracy and the stability of the proposed scheme.