Efficient and structure-preserving time-dependent auxiliary variable method for a conservative Allen–Cahn type surfactant system

In this study, we establish a phase-field two-phase surfactant system using two conservative Allen–Cahn type equations. Two nonlocal Lagrange multipliers are used to achieve the mass conservations. Comparing with the Cahn–Hilliard-type binary surfactant models which consist of two fourth-order nonlinear partial differential equations, the present model is easier because we solve two second-order nonlinear equations. In phase-field surfactant models, the existences of nonlinear terms lead to high challenges in energy estimation and numerical computation. To deal with these problems, we present first- and second-order time-accurate methods using a new time-dependent auxiliary variable approach. Due to the introduction of a new auxiliary variable, all nonlinear terms are explicitly solved. The energy dissipation law can be proved. To achieve linear and totally decoupled computation, we describe an efficient splitting algorithm. Various two- and three-dimensional computational tests are presented to show that our proposed schemes have desired accuracy, energy dissipation property, and work well for surfactant-laden coarsening.