We construct a fully-discrete finite element numerical scheme for the Cahn–Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity. The scheme is linear, decoupled, and unconditionally energy stable. Its key idea is to combine the penalty method of the Navier–Stokes equations with the Strang operator splitting method, and introduce several nonlocal variables and their ordinary differential equations to process coupled nonlinear terms. The scheme is highly efficient and it only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn–Hilliard equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability and solvability of the scheme and carry out numerous accuracy/stability examples and various benchmark numerical simulations in 2D and 3D, including the Rayleigh–Taylor instability and rising/coalescence dynamics of bubbles to demonstrate the effectiveness of the scheme, numerically.
Keywords: Variable density, fully-decoupled, phase-field, Cahn–Hilliard, Navier–Stokes, unconditional energy stability
@article{M2AN_2021__55_5_2323_0,
author = {Chen, Chuanjun and Yang, Xiaofeng},
title = {Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the {Cahn{\textendash}Hilliard} phase-field model of two-phase incompressible flow system with variable density and viscosity},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2323--2347},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {5},
doi = {10.1051/m2an/2021056},
mrnumber = {4328497},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021056/}
}
TY - JOUR AU - Chen, Chuanjun AU - Yang, Xiaofeng TI - Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the Cahn–Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosity JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 2323 EP - 2347 VL - 55 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021056/ DO - 10.1051/m2an/2021056 LA - en ID - M2AN_2021__55_5_2323_0 ER -
%0 Journal Article %A Chen, Chuanjun %A Yang, Xiaofeng %T Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the Cahn–Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosity %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 2323-2347 %V 55 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021056/ %R 10.1051/m2an/2021056 %G en %F M2AN_2021__55_5_2323_0
Chen, Chuanjun; Yang, Xiaofeng. Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the Cahn–Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosity. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 5, pp. 2323-2347. doi: 10.1051/m2an/2021056
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