Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach

We present in this note a unified approach on how to design simple, efficient and energy stable time discretization schemes for the Allen-Cahn or Cahn-Hilliard Navier-Stokes system which models twophase incompressible flows with matching or non-matching density. Special emphasis is placed on designing schemes which only require solving linear systems at each time step while satisfy discrete energy laws that mimic the continuous energy laws. We construct the time discretization schemes in weak formulations so that they can be used with any consistent Galerkin type spacial discretization schemes such as finite element methods and spectral/spectral-element methods.