Second order linear decoupled energy dissipation rate preserving schemes for the Cahn-Hilliard-extended-Darcy model

Abstract

The thermodynamically consistent Cahn-Hilliard-Extended-Darcy (CHED) model has been used to describe transient motion of a binary incompressible fluid flow in porous media. In this paper, we develop a series of linear, second-order, energy-dissipation-rate preserving numerical algorithms for the CHED model based on the energy quadratization strategy. We first extend the incompressible CHED model into a weakly compressible, thermodynamically consistent one using the generalized Onsager principle. Guided by the weakly compressible model, we then devise a couple of linear, second-order, decoupled, semi-discrete, temporal algorithms in the form of projection and the energy quadratization (EQ) method. The fully discrete algorithms are obtained by the use of the second-order finite difference method on staggered grids in space. We show theoretically that the obtained numerical algorithms respect the energy-dissipation-rate and the volume conservation property at the discrete level for any time steps, making them unconditionally energy stable. Mesh refinement tests, coarsening dynamics of binary fluids, and the buoyancy-driven binary fluid motion in porous media are investigated numerically. In buoyancy-driven flow simulations, a new set of inflow and outflow boundary conditions are devised using the model. The numerical results compare well with the results in the literature.