An improved scalar auxiliary variable (SAV) approach for the phase-field surfactant model

In this work, we develop a new linear, decoupled numerical scheme for the typical phase-field surfactant model. An improved scalar auxiliary variable (SAV) approach is used to discretize the governing equations in time. Different from the classical SAV approach, this improved form can calculate the phase field function ϕ, surfactant function ψ, and auxiliary variables in a step-by-step manner, i.e., the auxiliary variables are treated totally explicitly, thus we can directly calculate ϕ and ψ instead of computing the inner products. At each time step, the surfactant ψ can be directly obtained by an explicit way, then ϕ is updated by solving a linear system with constant coefficient. Therefore, the implementation of this improved SAV approach is easier than the classical SAV approach. The energy stability in first-order case can be analytically proved by using the our method. The numerical experiments show that our proposed method not only achieves desired first- and second-order accuracy but also satisfies the desired discrete energy dissipation law even if some larger time steps are used. Furthermore, the coarsening dynamics with different average concentrations can be well simulated by using our method. The co-continuous and drop patterns are generated in the even compositions case and uneven compositions case, respectively.