[Paper Review] A positivity-preserving and convergent numerical scheme for the binary fluid-surfactant system

Yuzhe Qin, Cheng Wang+1 more|arXiv (Cornell University)|Feb 16, 2021

Solidification and crystal growth phenomenaMaterials Science61 references26 citations

TL;DR

The paper develops a first-order-in-time convex-splitting scheme for a binary fluid–surfactant phase-field model, proving uniqueness, positivity preservation, unconditional energy stability, and optimal-rate convergence, with Newton iterations for the discrete system and numerical validation.

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.

Motivation & Objective

• Motivate and model interfacial dynamics with surfactants using a binary fluid-surfactant phase-field framework.
• Develop a numerical scheme that preserves positivity of the density and ensures energy stability at the discrete level.
• Prove unique solvability and obtain optimal-rate convergence for the scheme.
• Implement a Newton-iteration-based solver for the fully discrete system and validate stability and accuracy through numerical experiments.

Proposed method

• Formulate a convex-concave energy splitting G = G_c - G_e to enable a convex-splitting time discretization.
• Use a first-order in time semi-discrete scheme with implicit treatment of the convex part and explicit treatment of the concave part.
• Employ centered-difference spatial discretization and define discrete energy functionals E_c and E_e corresponding to the convex and concave parts.
• Derive discrete equations for φ and ρ and their chemical potentials μ_φ and μ_ρ, ensuring unique solvability and energy stability.
• Prove positivity-preserving property by leveraging the singular nature of logarithmic terms in the Flory-Huggins potential.
• Solve the resulting nonlinear system at each time step via Newton iteration and perform numerical experiments to confirm accuracy and stability.

Experimental results

Research questions

• RQ1 Can a convex-splitting based discretization yield unique solvability for the binary fluid-surfactant system?
• RQ2 Does the proposed scheme preserve the positivity of the density ρ at the discrete level?
• RQ3 Is the scheme unconditionally energy stable with respect to the discrete energy?
• RQ4 What is the optimal rate of convergence in time for the scheme, and under what conditions is it achieved?
• RQ5 Do numerical experiments confirm the theoretical positivity, stability, and convergence properties?

Key findings

• The proposed scheme is uniquely solvable.
• The scheme preserves positivity of the density variable ρ.
• The scheme is unconditionally energy stable (no time-step restriction for energy stability).
• An optimal rate of convergence in time is established for the scheme.
• Newton iteration is employed to solve the discrete nonlinear system, and numerical experiments validate accuracy and energy stability.