Efficient second-order, linear, decoupled and unconditionally energy stable schemes of the Cahn-Hilliard-Darcy equations for the Hele-Shaw flow

In this paper, we consider the numerical approximations for a hydrodynamical model of Cahn-Hilliard-Darcy equations. We develop two linear, decoupled, energy stable, and second-order time-marching schemes based on the "Invariant Energy Quadratization" method for nonlinear terms in the Cahn-Hilliard equation, and the projection method for the Darcy equations. Moreover, we prove the well-posedness of the linear system and their unconditional energy stabilities rigorously. We also construct a linear, decoupled, energy stable, and second-order time marching scheme by using the "Scalar Auxiliary Variable" method. Various numerical tests are presented to illustrate the stability and the accuracy of the numerical schemes and simulate the process of coarsening in binary fluid and investigate the effect of the rotating and the gravity on the Hele-Shaw cell in 2D and 3D.