A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation

TL;DR

This paper introduces a second order, unconditionally stable and uniquely solvable numerical scheme for the coupled Cahn-Hilliard-Navier-Stokes equations, combining convex-splitting and pressure projection methods.

Contribution

It presents a novel second order scheme with proven stability, mass conservation, and unique solvability, along with an efficient decoupling iteration for phase field and fluid flow.

Findings

The scheme is unconditionally stable and mass-conservative.

Numerical experiments confirm high accuracy and efficiency.

The method effectively decouples the coupled equations for practical computation.

Abstract

We propose a novel second order in time numerical scheme for Cahn-Hilliard-Navier- Stokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn-Hilliard equation and pressure-projection for the Navier-Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, we prove that the scheme is uncondition- ally uniquely solvable at each time step by exploring the monotonicity associated with the scheme. Thanks to the weak coupling of the scheme, we design an efficient Picard iteration procedure to further decouple the computation of Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by the mixed finite element method. Ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme.