# Stabilized energy factorization approach for Allen-Cahn equation with logarithmic Flory-Huggins potential

## TL;DR

This paper introduces a stabilized energy factorization method for the Allen-Cahn equation with logarithmic Flory-Huggins potential, ensuring linearity, stability, and ease of implementation while preserving key physical properties.

## Contribution

The paper develops a novel stabilized energy factorization scheme that handles the nonlinear Flory-Huggins potential efficiently and guarantees maximum principle and energy stability.

## Key findings

- The scheme is unconditionally energy stable.
- All nonlinear terms are treated semi-implicitly.
- Numerical results confirm stability and effectiveness.

## Abstract

The Allen--Cahn equation is one of fundamental equations of phase-field models, while the logarithmic Flory--Huggins potential is one of the most useful energy potentials in various phase-field models. In this paper, we consider numerical schemes for solving the Allen--Cahn equation with logarithmic Flory--Huggins potential. The main challenge is how to design efficient numerical schemes that preserve the maximum principle and energy dissipation law due to the strong nonlinearity of the energy potential function. We propose a novel energy factorization approach with the stability technique, which is called stabilized energy factorization approach, to deal with the Flory--Huggins potential. One advantage of the proposed approach is that all nonlinear terms can be treated semi-implicitly and the resultant numerical scheme is purely linear and easy to implement. Moreover, the discrete maximum principle and unconditional energy stability of the proposed scheme are rigorously proved using the discrete variational principle. Numerical results are presented to demonstrate the stability and effectiveness of the proposed scheme.