Uniquely Solvable and Energy Stable Decoupled Numerical Schemes for the Cahn–Hilliard–Navier–Stokes–Darcy–Boussinesq System

In this article we propose the Cahn–Hilliard–Navier–Stokes–Darcy–Boussinesq system that models thermal convection of two-phase flows in a fluid layer overlying a porous medium. Based on operator splitting and pressure stabilization we propose a family of fully decoupled numerical schemes such that the Navier–Stokes equations, the Darcy equations, the heat equation and the Cahn–Hilliard equation are solved independently at each time step, thus significantly reducing the computational cost. We show that the schemes preserve the underlying energy law and hence are unconditionally long-time stable. Numerical results are presented to demonstrate the accuracy and stability of the algorithms.