Comparison of Structure-Preserving Methods for the Cahn-Hilliard-Navier-Stokes Equations

TL;DR

This paper introduces structure-preserving discontinuous Galerkin methods for the Cahn-Hilliard-Navier-Stokes equations, ensuring stability, mass conservation, and energy dissipation, with improved computational efficiency demonstrated through numerical validation.

Contribution

It develops new DG methods with enhanced stability and efficiency for coupled Cahn-Hilliard-Navier-Stokes equations, including proofs of coercivity and optimal convergence.

Findings

Methods preserve mass, energy, and maximum principle.

Achieve optimal convergence rates.

Significant computational savings on adaptive meshes.

Abstract

We develop structure-preserving discontinuous Galerkin methods for the Cahn-Hilliard-Navier-Stokes equations with degenerate mobility. The proposed SWIPD-L and SIPGD-L methods incorporate parametrized mobility fluxes with edge-wise mobility treatments for enhanced coercivity-stability control. We prove coercivity for the generalized trilinear form and demonstrate optimal convergence rates while preserving mass conservation, energy dissipation, and the discrete maximum principle. Comparisons with existing SIPG-L and SWIP-L methods confirm similar stability. Validation on $hp$-adaptive meshes for both standalone Cahn-Hilliard and coupled systems shows significant computational savings without accuracy loss.