# Decoupled, Energy Stable Scheme for Hydrodynamic Allen-Cahn Phase Field Moving Contact Line Model

## TL;DR

This paper introduces a novel, energy-stable, decoupled numerical scheme for simulating hydrodynamic phase field models with contact lines, combining Allen-Cahn equations and Navier-Stokes with finite element methods.

## Contribution

The paper develops a linear, decoupled, energy-stable scheme for a phase field model with contact line conditions, using Allen-Cahn equations and a projection method.

## Key findings

- The scheme is proven to be energy stable both semi- and fully discretized.
- Numerical experiments confirm the scheme's efficiency and accuracy.
- Finite element discretization effectively captures the contact line dynamics.

## Abstract

In this paper, we present an efficient energy stable scheme to solve a phase field model incorporating contact line condition. Instead of the usually used Cahn-Hilliard type phase equation, we adopt the Allen-Cahn type phase field model with the static contact line boundary condition that coupled with incompressible Navier-Stokes equations with Navier boundary condition. The projection method is used to deal with the Navier-Stokes equa- tions and an auxiliary function is introduced for the non-convex Ginzburg-Landau bulk potential. We show that the scheme is linear, decoupled and energy stable. Moreover, we prove that fully discrete scheme is also energy stable. An efficient finite element spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes. Numerical results show that the proposed scheme is very efficient and accurate