# Analysis of a time-discrete scheme for the Navier-Stokes/Allen-Cahn model

## TL;DR

This paper develops and analyzes a stable, implicit time-discrete scheme for simulating two-fluid flows with different densities and viscosities using the Navier-Stokes/Allen-Cahn model, including convergence and numerical validation.

## Contribution

It introduces a novel stabilized fixed point algorithm for the coupled Navier-Stokes/Allen-Cahn system, ensuring existence, uniqueness, and maximum principle in a semi-discrete setting.

## Key findings

- Proved well-posedness and convergence of the scheme.
- Established maximum principle for the numerical solution.
- Numerical experiments demonstrate the effectiveness of the iterative process.

## Abstract

This paper address the approximation of the dynamic of two fluids with non matching densities and viscosities modeled by the Allen-Cahn equation coupled with the time dependent Navier-Stokes equations. Existence, uniqueness and a maximum principle are obtained for a totally implicit semi-discrete in time formulation. These results are based on an original stabilized fixed point algorithm for which well posedness and convergence is analyzed. Numerical experiments are performed to show the influence of the iterative process.