Decoupled and energy stable schemes for phase-field surfactant model based on mobility operator splitting technique

In this paper, we investigate numerical methods for the phase-field surfactant (PFS) model, which is a gradient flow system consisting of two nonlinearly coupled Cahn-Hilliard type equations. The main challenge in developing high-order efficient energy stable methods for this system results from the nonlinearity and the strong coupling in the two variables in the free energy functional. We propose two fully decoupled, linear and energy stable schemes based on a linear stabilization approach and an operator splitting technique. We rigorously prove that both schemes can preserve the original energy dissipation law. The techniques employed in these schemes are then summarized into an innovative approach, which we call the mobility operator splitting (MOS), to design high-order decoupled energy stable schemes for a wide class of gradient flow systems. As a particular case, MOS allows different time steps for updating respective variables, leading to a multiple time-stepping strategy for fast-slow dynamics and thus serious improvement of computational efficiency. Various numerical experiments are presented to validate the accuracy, efficiency and other desired properties of the proposed schemes. In particular, detailed phenomena in thin-film pinch-off dynamics can be clearly captured by using the proposed schemes.