Second‐order, fully decoupled, linearized, and unconditionally stable scalar auxiliary variable schemes for Cahn–Hilliard–Darcy system

Abstract

In this paper, we establish the fully decoupled numerical methods by utilizing scalar auxiliary variable approach for solving Cahn-Hilliard-Darcy system. We exploit the operator splitting technique to decouple the coupled system and Galerkin finite element method in space to construct the fully discrete formulation. The developed numerical methods have the features of second order accuracy, totally decoupling, linearization, and unconditional energy stability. The unconditionally stability of the two proposed decoupled numerical schemes are rigorously proved. Abundant numerical results are reported to verify the accuracy and effectiveness of proposed numerical methods.

Fully decoupled unconditionally stable Crank–Nicolson leapfrog numerical methods for the Cahn–Hilliard–Darcy system

“…The CHD system is a strongly coupled nonlinear system that models interfacial phenomena with sharp transitions in narrow layers (stiffness). There have been abundant numerical works addressing these challenges [14,16,22,26,27,35,46,47]. Feng and Wise [14] analyzed a fully discrete implicit finite element method for studying the CHD system, establishing unconditional unique solvability and convergence of the numerical scheme.…”

“…In recent years a class of Lagrange multiplier approaches are developed for the design of high-order, unconditionally stable, linear and decoupled time-stepping methods for gradient flow models. Popular methods in this class include the scalar auxiliary variable (SAV) method [16], the invariant energy quadratization (IEQ) method [5], and many other variants. For the CHD equations Yang [47] constructed a fully-decoupled second-order linear numerical scheme in which zero-energy-contribution idea is introduced to break the coupling of velocity and phase-field variable.…”

A fully decoupled numerical method for Cahn–Hilliard–Navier–Stokes–Darcy equations based on auxiliary variable approaches

“…There are several successful techniques that can well handle the stiffness of phase field model, such as the convex-splitting strategy [8,41,66,83,89,101], the stabilization method [35,56,92,100,109], the Invariant Energy Quadratization (IEQ) approach [62,99,104,107,111], the scalar auxiliary variable (SAV) approach [1,34,38,61,67,82,90,91], generalized positive auxiliary variable (gPAV) approach [68,69,108], and the zero-energy-contribution approach [102,103,105,106,110,111]. Among these options, the SAV and gPAV approaches only need the bounded below restriction of free energy.…”

A linear convex splitting scheme for the Cahn–Hilliard equation with a high‐order polynomial free energy

“…For a sake of simplicity, we impose the periodic boundary condition, but other boundary conditions can be adopted. The CH equation was developed to model the phase separation phenomenon in heterogeneous binary mixtures 1,2 and has been applied to investigate the dynamics of such materials as spinodal decomposition, 3,4 multiphase flows, [5][6][7][8][9][10] tumor growth, 11,12 medical image processing, 13 topology optimization, 14,15 microstructures with elastic inhomogeneity, 16 block copolymers, 17,18 damage field of sensitivity-uncertainty quantification framework, 19 just to name a few. The CH equation is a H −1 -gradient flow (𝜙 t = −(−Δ)𝜇 = Δ(𝛿∕𝛿𝜙)) of the following Ginzburg-Landau free energy functional,…”

“…For a sake of simplicity, we impose the periodic boundary condition, but other boundary conditions can be adopted. The CH equation was developed to model the phase separation phenomenon in heterogeneous binary mixtures 1,2 and has been applied to investigate the dynamics of such materials as spinodal decomposition, 3,4 multiphase flows, 5‐10 tumor growth, 11,12 medical image processing, 13 topology optimization, 14,15 microstructures with elastic inhomogeneity, 16 block copolymers, 17,18 damage field of sensitivity‐uncertainty quantification framework, 19 just to name a few. The CH equation is a $$$ {H}^{-1} $$$‐gradient flow ($$$ {\phi}_t=-\left(-\Delta \right)\mu =\Delta \left(\delta \mathit{\mathcal{E}}/\delta \phi \right) $$$) of the following Ginzburg–Landau free energy functional, $$$$ \mathcal{E}\left(\phi \right)=\underset{\Omega}{\int}\left(F\left(\phi \right)+\frac{\epsilon^2}{2}{\left|\nabla \phi \right|}^2\right)\mathrm{d}\mathbf{x}.\kern0.5em $$$$ There are two important properties of the gradient flow in $$$ {H}^{-1}\left(\Omega \right) $$$, energy dissipation and mass conservation over time.…”