A Second Order in Time, Decoupled, Unconditionally Stable Numerical Scheme for the Cahn–Hilliard–Darcy System

We propose a novel second order in time, fully decoupled and unconditionally stable numerical scheme for solving the Cahn–Hilliard–Darcy system which models two-phase flow in porous medium or in a Hele–Shaw cell. The scheme is based on the ideas of second order convex-splitting for the Cahn–Hilliard equation and pressure-correction for the Darcy equation. The computation of order parameter, pressure and velocity is completely decoupled in our scheme. We show that the scheme is uniquely solvable, unconditionally energy stable and mass-conservative. Ample numerical results are presented to gauge the efficiency and robustness of our scheme.