Efficient finite element schemes for a phase field model of two-phase incompressible flows with different densities

TL;DR

This paper introduces two efficient, decoupled finite element schemes based on MSAV for simulating two-phase incompressible flows with different densities, ensuring stability and ease of implementation.

Contribution

The paper develops two novel MSAV-based finite element schemes for the AGG model, featuring decoupling, linearity, second-order accuracy, and proven energy stability.

Findings

Schemes are unconditionally energy stable.

Numerical results confirm robustness and efficiency.

Methods are straightforward to implement.

Abstract

In this paper, we present two multiple scalar auxiliary variable (MSAV)-based, finite element numerical schemes for the Abels-Garcke-Gr{\"u}n (AGG) model, which is a thermodynamically consistent phase field model of two-phase incompressible flows with different densities. Both schemes are decoupled, linear, second-order in time, and the numerical implementation turns out to be straightforward. The first scheme solves the Navier-Stokes equations in a saddle point formulation, while the second one employs the artificial compressibility method, leading to a fully decoupled structure with a time-independent pressure update equation. In terms of computational cost, only a sequence of independent elliptic or saddle point systems needs to be solved at each time step. At a theoretical level, the unique solvability and unconditional energy stability (with respect to a modified energy functional) of the proposed schemes are established. In addition, comprehensive numerical simulations are performed to verify the effectiveness and robustness of the proposed schemes.