Second-Order Decoupled Linear Energy-Law Preserving gPAV Numerical Schemes for Two-Phase Flows in Superposed Free Flow and Porous Media

We propose second-order numerical methods based on the generalized positive auxiliary variable (gPAV) framework for solving the Cahn–Hilliard–Navier–Stokes–Darcy model in superposed free flow and porous media. In the gPAV-reformulated system, we introduce an auxiliary variable according to the modified energy law and take account into the interface conditions between the two subdomains. By implicit-explicit temporal discretization, we develop fully decoupled linear gPAV-CNLF and gPAV-BDF2 numerical methods effected with the Galerkin finite element method. The fully discrete schemes satisfy a modified energy law irrespective of time step size. Plentiful numerical experiments are performed to validate the methods and demonstrate the robustness. The application in filtration systems, the influence of viscous instability, general permeability, curve interface, and different densities are discussed in details to further illustrate the compatibility and applicability of our developed gPAV numerical methods.