# Efficient Second Order Unconditionally Stable Schemes for a Phase-field Moving Contact Line Model Using Invariant Energy Quadratization Approach

## TL;DR

This paper introduces two efficient, second-order, unconditionally energy stable numerical schemes for a complex phase-field model involving Navier-Stokes and Cahn-Hilliard equations with moving contact lines, using invariant energy quadratization.

## Contribution

The paper develops novel linear, second-order accurate schemes that are unconditionally energy stable for a coupled Navier-Stokes and Cahn-Hilliard phase-field model with moving contact lines.

## Key findings

- Schemes are proven to be well-posed and energy stable.
- Numerical results verify accuracy and efficiency.
- Spectral-Galerkin discretization confirms effectiveness.

## Abstract

We consider the numerical approximations for a phase field model consisting of incompressible Navier--Stokes equations with a generalized Navier boundary condition, and the Cahn-Hilliard equation with a dynamic moving contact line boundary condition. A crucial and challenging issue for solving this model numerically is the time marching problem, due to the high order, nonlinear, and coupled properties of the system. We solve this issue by developing two linear, second order accurate, and energy stable schemes based on the projection method for the Navier--Stokes equations, the invariant energy quadratization for the nonlinear gradient terms in the bulk and boundary, and a subtle implicit-explicit treatment for the stress and convective terms. The well-posedness of the semidiscretized system and the unconditional energy stabilities are proved. Various numerical results based on a spectral-Galerkin spatial discretization are presented to verify the accuracy and efficiency of the proposed schemes.