Second order stabilized semi-implicit scheme for the Cahn-Hilliard model with dynamic boundary conditions

TL;DR

This paper introduces a second-order, energy-stable semi-implicit numerical scheme for the Cahn-Hilliard equation with dynamic boundary conditions, validated through theoretical analysis and numerical experiments.

Contribution

It extends previous stabilized methods to second order in time, providing a new, more accurate scheme with proven stability and convergence for this class of problems.

Findings

The scheme is energy stable and convergent.

Numerical experiments confirm accuracy and effectiveness.

The method outperforms first-order approaches.

Abstract

We study the numerical algorithm and error analysis for the Cahn-Hilliard equation with dynamic boundary conditions. A second-order in time, linear and energy stable scheme is proposed, which is an extension of the first-order stabilized approach. The corresponding energy stability and convergence analysis of the scheme are derived theoretically. Some numerical experiments are performed to verify the effectiveness and accuracy of the second-order numerical scheme, including numerical simulations under various initial conditions and energy potential functions, and comparisons with the literature works.