Efficient Numerical Schemes for the Flow-Coupled Binary Phase-Field Surfactant System Based on the Supplementary Variable Method

In this work, we study a flow-coupled phase-field surfactant system governed by a coupled system of two Cahn-Hilliard equations and the Navier-Stokes equation. To design a second-order time-accurate numerical scheme that preserves the original energy dissipation property at the discrete level, we employ the structure-preserving supplementary variable method. By introducing an auxiliary variable and combining stabilization techniques with a pressure-projection scheme, we decompose the system into biharmonic-type subsystems with constant coefficient matrices. This decomposition significantly reduces computational costs. Benchmark numerical examples demonstrate the accuracy, energy stability, and efficiency of the proposed scheme.