# Linear and Unconditionally Energy Stable Schemes for the binary Fluid-Surfactant Phase Field Model

## TL;DR

This paper introduces linear, unconditionally energy stable numerical schemes for a complex binary fluid-surfactant phase field model, enabling efficient and stable simulations of coupled nonlinear equations with validated accuracy through numerical experiments.

## Contribution

The paper develops first and second order linear schemes using the Invariant Energy Quadratization approach that are unconditionally energy stable for the binary fluid-surfactant model.

## Key findings

- Schemes are linear and lead to symmetric positive definite systems.
- Proved unconditional energy stability of the schemes.
- Numerical experiments confirm accuracy and stability.

## Abstract

In this paper, we consider the numerical solution of a binary fluid-surfactant phase field model, in which the free energy contains a nonlinear coupling entropy, a Ginzburg-Landau double well potential, and a logarithmic Flory-Huggins potential. The resulting system consists of two coupled, nonlinear Cahn-Hilliard type equations. We develop a set of first and second order time marching schemes for this system using the "Invariant Energy Quadratization" approach, in particular, the system is transformed into an equivalent one by introducing appropriate auxiliary variables and all nonlinear terms are then treated semi-explicitly. Both schemes are linear and lead to symmetric positive definite systems at each time step, thus they can be efficiently solved. We further prove that these schemes are unconditionally energy stable in the discrete sense. Various 2D and 3D numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes.