Energy-stable decoupled numerical approximation with practical correction technique for the binary phase field Darcy fluid system

In this article, we propose a practical numerical scheme for two-phase creeping flows governed by the Cahn-Hilliard-Darcy system. The proposed method achieves second-order temporal accuracy, while remaining linear and fully decoupled throughout the entire computational procedure. Moreover, it is particularly well suited for modeling incompressible two-phase fluid behavior in confined geometries. A simple-solving Scalar Auxiliary Variables (SAV) approach serves as the backbone of our scheme, where all the nonlinear terms and auxiliary variables are explicit. The original complex governing equations would be transformed to equivalent simpler form involving two time-dependent auxiliary variables. We employ a novel variant of the SAV method combined with a projection-based method to enable the quantites to be completely decoupled during the step-by-step computation. To further ensure stability and analytical consistency, we introduce correction to the two auxiliary variables correspondingly. We prove the unique solvability and the feature of energy dissipation of our method. Benchmark tests are conducted in two dimensions, including: i ) phase-separation ii ) viscous fingering iii ) interfacial pinchoff. In the three-dimensional setting, we consider: i ) phase-separation ii ) droplet dynamics in a free-falling process.