A decoupled unconditionally stable numerical scheme for the Cahn-Hilliard-Hele-Shaw system

TL;DR

This paper introduces a new decoupled, unconditionally stable numerical scheme for simulating two-phase flow in Hele-Shaw cells governed by the Cahn-Hilliard-Hele-Shaw system, improving computational efficiency and stability.

Contribution

The paper presents a novel decoupled, unconditionally stable numerical scheme that separates the nonlinear Cahn-Hilliard equation from pressure updates using operator-splitting and stabilization techniques.

Findings

The scheme is unconditionally stable.

Numerical results confirm high accuracy.

Method reduces computational complexity.

Abstract

We propose a novel decoupled unconditionally stable numerical scheme for the simulation of two-phase flow in a Hele-Shaw cell which is governed by the Cahn-Hilliard-Hele-Shaw system (CHHS) with variable viscosity. The temporal discretization of the Cahn-Hilliard equation is based on a convex-splitting of the associated energy functional. Moreover, the capillary forcing term in the Darcy equation is separated from the pressure gradient at the time discrete level by using an operator-splitting strategy. Thus the computation of the nonlinear Cahn-Hilliard equation is completely decoupled from the update of pressure. Finally, a pressure-stabilization technique is used in the update of pressure so that at each time step one only needs to solve a Poisson equation with constant coefficient. We show that the scheme is unconditionally stable. Numerical results are presented to demonstrate the accuracy and efficiency of our scheme.