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

TL;DR

This paper introduces two novel decoupled numerical schemes for the Cahn-Hilliard-Stokes-Darcy system, ensuring unique solvability, energy stability, and mass conservation for two-phase flows in complex geometries, with demonstrated accuracy and efficiency.

Contribution

The paper develops two new fully decoupled, energy stable numerical schemes for the CHSD system, incorporating operator splitting and pressure stabilization techniques.

Findings

Both schemes are proven to be uniquely solvable.

Numerical results confirm the schemes' accuracy and efficiency.

The methods effectively handle two-phase flows in karstic geometries.

Abstract

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