Discontinuous finite volume element method for a coupled Navier-Stokes-Cahn-Hilliard phase field model

Abstract

In this paper, we propose a discontinuous finite volume element method to solve a phase field model for two immiscible incompressible fluids. In this finite volume element scheme, discontinuous linear finite element basis functions are used to approximate the velocity, phase function, and chemical potential while piecewise constants are used to approximate the pressure. This numerical method is efficient, optimally convergent, conserving the mass, convenient to implement, flexible for mesh refinement, and easy to handle complex geometries with different types of boundary conditions. We rigorously prove the mass conservation property and the discrete energy dissipation for the proposed fully discrete discontinuous finite volume element scheme. Using numerical tests, we verify the accuracy, confirm the mass conservation and the energy law, test the influence of surface tension and small density variations, and simulate the driven cavity, the Rayleigh-Taylor instability.

An upwind DG scheme preserving the maximum principle for the convective Cahn-Hilliard model

“…Remark 3.5 We refer to (27) as a generalized bilinear form since it generalizes the definition of ( 11) considering the case where β may be discontinuous. If β is continuous both definitions are equivalent.…”

“…The first of them [3] is worth mentioning for the application of highresolution spectral Fourier schemes. We also underline the papers [7,27], based on finite volume approximations, and also [11,22], where finite difference techniques are applied for Navier-Stokes CH equations. Some authors have recently worked in DG methods for spatial discretization of the CCH problem.…”

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.…”

An upwind DG scheme preserving the maximum principle for the convective Cahn-Hilliard model

“…See e.g. [3] where finite-differences for space discretization is used and [23] where (discontinuous) finite volume method and DG methods are used for coupled Navier-Stokes-Cahn-Hilliard phase field models.…”