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

TL;DR

This work develops two MSAV-based, finite element schemes for the Abels-Garcke-Grün formulation of two-phase incompressible flows with variable density, delivering decoupled, linear, second-order-in-time methods with unconditional energy stability under a modified energy functional. One scheme uses a standard Galerkin saddle-point approach, while the other employs artificial compressibility to yield a time-independent pressure update, enabling fully decoupled solves. The authors prove unique solvability and energy stability, and validate the methods through comprehensive 2D and 3D tests—capillary waves, rising bubbles, and Rayleigh–Taylor instabilities—across large density contrasts and a range of We and Re numbers. The results demonstrate accurate interface dynamics, mass conservation, and robust performance, while highlighting practical aspects such as non-physical acoustic waves in the AC scheme and the influence of mobility models on accuracy. Overall, the proposed structure-preserving schemes offer scalable tools for complex phase-field simulations of two-phase flows with density variation, with clear paths for further analysis and extensions.

Abstract

In this paper, we present two multiple scalar auxiliary variable (MSAV)-based, finite element numerical schemes for the Abels-Garcke-Gr{ü}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.

Figures (18)

• Figure 1: Numerical errors, with $\tau=h$, at $T=0.5.$
• Figure 2: The profiles of $\phi$ computed by the AC+CM scheme, at a sequence of time instants, $t=0,0.4,0.5,0.6$ and $1$.
• Figure 3: Time evolution of discrete energy.
• Figure 4: Time evolution of the auxiliary variables and mass drift computed by AC+CM.
• Figure 5: Comparison of capillary wave amplitude versus time.

Theorems & Definitions (16)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 1
• Theorem 2
• Remark 5
• Theorem 3
• Remark 6
• Remark 7