Efficient Energy-Stable Discontinuous Galerkin Scheme for the Non-Isothermal Cahn–Hilliard–Navier–Stokes Two-Phase Fluid Flow System

Abstract

In this article, we propose a novel numerical framework for the non-isothermal Cahn–Hilliard–Navier–Stokes two-phase flow system, which couples the incompressible Navier–Stokes equations, the Cahn–Hilliard phase-field equation, and the heat transport equation to capture temperature-dependent two-phase flow dynamics. The proposed scheme achieves three major advances: (i) unconditional energy stability through a combined scalar auxiliary variable (SAV) and zero-energy-contribution (ZEC) approach, (ii) linearity and full decoupling of all variables while using a second-order temporal discretization, and (iii) efficient implementation via discontinuous Galerkin (DG) spatial discretization together with a second-order projection method for the Navier–Stokes equations. We rigorously prove the unconditional energy stability of the scheme and present key details of its decoupled implementation. Extensive 2D and 3D simulations, including droplet deformation, bubble coalescence, and interfacial instabilities in stratified binary fluids, are presented to demonstrate the accuracy, efficiency, and robustness of the proposed numerical method, thereby confirming its effectiveness for energy-stable simulation of non-isothermal two-phase incompressible flows.