TY - JOUR
T1 - A Physics-Informed Structure-Preserving Numerical Scheme for the Phase-Field Hydrodynamic Model of Ternary Fluid Flows
AU - Hong , Qi
AU - Gong , Yuezheng
AU - Zhao , Jia
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 565
EP - 596
PY - 2023
DA - 2023/08
SN - 16
DO - http://doi.org/10.4208/nmtma.OA-2023-0007 
UR - https://global-sci.org/intro/article_detail/nmtma/21958.html
KW - Energy stable, phase field, Cahn-Hilliard-Navier-Stokes, supplementary variable method.
AB - <p style="text-align: justify;">Phase-field models are widely used in studying multiphase flow dynamics. Given the complexity and strong nonlinearity, designing accurate, efficient, and
stable numerical algorithms to solve these models has been an active research field
for decades. This paper proposes a novel numerical scheme to solve a highly cited
and used phase field hydrodynamic model for simulating ternary phase fluid flows.
The main novelty is the introduction of a supplementary variable to reformulate the
original problem into a constrained optimization problem. This reformulation leads
to several advantages for our proposed numerical algorithms compared with many
existing numerical techniques for solving this model. First, the developed schemes
allow more straightforward calculations for the hydrodynamic phase-field models
by solving a few decoupled Helmholtz or Poisson-type systems with a constant precomputable coefficient matrix, remarkably reducing the computational cost. Secondly, the numerical schemes can maintain mass conservation and energy dissipation at the discrete level. Additionally, the developed scheme based on the second-order backward difference formula respects the original energy dissipation law that
differs from many existing schemes, such as the IEQ, SAV, and Lagrange multiplier
approaches for which a modified energy dissipation law is respected. Furthermore,
rigorous proof of energy stability and practical implementation strategies are provided. We conduct adequate 2D and 3D numerical tests to demonstrate the proposed
schemes’ accuracy and effectiveness.</p>