A positivity-preserving and convergent numerical scheme for the binary fluid-surfactant system

TL;DR

This paper introduces a first-order, positivity-preserving, and unconditionally energy-stable numerical scheme for the binary fluid-surfactant system, with proven convergence and validated through numerical experiments.

Contribution

It presents the first optimal rate convergence analysis for a numerical scheme solving the binary fluid-surfactant system, ensuring stability and positivity.

Findings

The scheme is proven to be unique solvable.

It maintains positivity and energy stability unconditionally.

Numerical experiments confirm accuracy and stability.

Abstract

In this paper, we develop a first order (in time) numerical scheme for the binary fluid surfactant phase field model. The free energy contains a double-well potential, a nonlinear coupling entropy and a Flory-Huggins potential. The resulting coupled system consists of two Cahn-Hilliard type equations. This system is solved numerically by finite difference spatial approximation, in combination with convex splitting temporal discretization. We prove the proposed scheme is unique solvable, positivity-preserving and unconditionally energy stable. In addition, an optimal rate convergence analysis is provided for the proposed numerical scheme, which will be the first such result for the binary fluid-surfactant system. Newton iteration is used to solve the discrete system. Some numerical experiments are performed to validate the accuracy and energy stability of the proposed scheme.