A pressure-correction and bound-preserving discretization of the phase-field method for variable density two-phase flows

In this paper, we present an efficient numerical algorithm for solving the time-dependent Cahn--Hilliard--Navier--Stokes equations that model the flow of two phases with different densities. The pressure-correction step in the projection method consists of a Poisson problem with a modified right-hand side. Spatial discretization is based on discontinuous Galerkin methods with piecewise linear or piecewise quadratic polynomials. Flux and slope limiting techniques successfully eliminate the bulk shift, overshoot and undershoot in the order parameter, which is shown to be bound-preserving. Several numerical results demonstrate that the proposed numerical algorithm is effective and robust for modeling two-component immiscible flows in porous structures and digital rocks.