Structure-preserving weighted BDF2 methods for Anisotropic Cahn-Hilliard model: uniform/variable-time-steps

TL;DR

This paper introduces novel uniform and variable-time-step weighted BDF2 methods for the anisotropic Cahn-Hilliard model, ensuring energy stability and mass conservation, validated through extensive numerical simulations.

Contribution

The paper develops and proves energy stability of new weighted BDF2 schemes for anisotropic Cahn-Hilliard models with both uniform and variable time steps, incorporating SAV and stabilization techniques.

Findings

The methods are energy-stable for both uniform and variable time steps.

The schemes are mass-conservative.

Numerical simulations confirm stability and accuracy.

Abstract

In this paper, we innovatively develop uniform/variable-time-step weighted and shifted BDF2 (WSBDF2) methods for the anisotropic Cahn-Hilliard (CH) model, combining the scalar auxiliary variable (SAV) approach with two types of stabilized techniques. Using the concept of $G$-stability, the uniform-time-step WSBDF2 method is theoretically proved to be energy-stable. Due to the inapplicability of the relevant G-stability properties, another technique is adopted in this work to demonstrate the energy stability of the variable-time-step WSBDF2 method. In addition, the two numerical schemes are all mass-conservative.Finally, numerous numerical simulations are presented to demonstrate the stability and accuracy of these schemes.