Efficiently linear and unconditionally energy-stable time-marching schemes with energy relaxation for the phase-field surfactant model

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In this article, we consider consistent, efficient, and accurate numerical approximations for a recently developed Cahn–Hilliard type phase-field surfactant model enjoying the bounded-from-below condition of total energy. The phase-field surfactant model naturally satisfies the energy dissipation property in time. Based on this feature, we construct linearly decoupled and unconditionally energy-stable time-marching schemes based on an efficient variant of scalar auxiliary variable approach. To enhance the consistency of discrete modified energy and discrete original energy, a practical energy relaxation technique is adopted. We analytically proved the unique solvability and unconditional energy stability of the proposed schemes. In each time step, only two linear elliptic type equations with constants coefficients need to be solved and several algebraic operations need to be performed. Therefore, the proposed schemes are highly efficient and easy to implement. The space is discretized by the finite difference method and the linear multigrid algorithm is used to accelerate the convergence. The numerical results indicate that the proposed method not only has desired accuracy in time but also satisfies the energy dissipation law even if larger time steps are used. Furthermore, the surfactant laden spinodal decompositions and surfactant absorptions can be well simulated.