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We propose and analyze two novel decoupled numerical schemes for solving theCahn-Hilliard-Stokes-Darcy (CHSD) model for two-phase flows in karsticgeometry. In the first numerical scheme, we explore a fractional step method(operator splitting) to decouple the phase-field (Cahn-Hilliard equation) fromthe velocity field (Stokes-Darcy fluid equations). To further decouple theStokes-Darcy system, we introduce a first order pressure stabilization term inthe Darcy solver in the second numerical scheme so that the Stokes system isdecoupled from the Darcy system and hence the CHSD system can be solved in afully decoupled manner. We show that both decoupled numerical schemes areuniquely solvable, energy stable, and mass conservative. Ample numericalresults are presented to demonstrate the accuracy and efficiency of ourschemes.

Uniquely solvable and energy stable decoupled schemes for Cahn-Hilliard-Stokes-Darcy system for two-phase flows in karstic geometry