A high-order accurate unconditionally stable bound-preserving numerical scheme for the Cahn-Hilliard-Navier-Stokes equations

TL;DR

This paper introduces a high-order, unconditionally stable, and bound-preserving numerical scheme for the Cahn-Hilliard-Navier-Stokes equations, ensuring accurate and stable simulations of complex fluid interfaces.

Contribution

The paper develops a novel high-order finite element scheme with mass lumping and convex splitting that guarantees stability, solvability, and bound-preservation for the coupled equations.

Findings

The scheme is unconditionally stable and uniquely solvable.

It preserves bounds for the solution, ensuring physical relevance.

Numerical experiments confirm the scheme's accuracy and stability.

Abstract

A high-order numerical method is developed for solving the Cahn-Hilliard-Navier-Stokes equations with the Flory-Huggins potential. The scheme is based on the $Q_k$ finite element with mass lumping on rectangular grids, the second-order convex splitting method, and the pressure correction method. The unique solvability, unconditional stability, and bound-preserving properties are rigorously established. The key to bound-preservation is the discrete $L^1$ estimate of the singular potential. Ample numerical experiments are performed to validate the desired properties of the proposed numerical scheme.