Arbitrary High-Order Fully-Decoupled Numerical Schemes for Phase-Field Models of Two-Phase Incompressible Flows

Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows, it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations. One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them. The resulting schemes are only first-order accurate in time, and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version. In this paper, we employ the spectral deferred correction method to improve the temporal accuracy, based on the first-order decoupled and energy-stable scheme constructed by the stabilization idea. The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency. Within the framework of the spatially discretized local discontinuous Galerkin method, the resulting numerical schemes are fully decoupled, efficient, and high-order accurate in both time and space. Numerical experiments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.