A new decoupled unconditionally stable scheme and its optimal error analysis for the Cahn-Hilliard-Navier-Stokes equations

TL;DR

This paper introduces a novel, fully discrete, unconditionally energy stable scheme for the coupled Cahn-Hilliard-Navier-Stokes equations, enabling efficient and stable simulations with proven optimal error estimates.

Contribution

The paper develops a decoupled, first-order scheme with unconditional energy stability and optimal error analysis for the Cahn-Hilliard-Navier-Stokes system.

Findings

The scheme is unconditionally stable both semi-discretely and fully discretely.

Numerical experiments confirm the theoretical stability and error estimates.

The method effectively decouples the Cahn-Hilliard and Navier-Stokes computations.

Abstract

We construct a decoupled, first-order, fully discrete, and unconditionally energy stable scheme for the Cahn-Hilliard-Navier-Stokes equations. The scheme is divided into two main parts. The first part involves the calculation of the Cahn-Hilliard equations, and the other part is calculating the Navier-Stokes equations subsequently by utilizing the phase field and chemical potential values obtained from the above step. Specifically, the velocity in the Cahn-Hilliard equation is discretized explicitly at the discrete time level, which enables the computation of the Cahn-Hilliard equations is fully decoupled from that of Navier-Stokes equations. Furthermore, the pressure-correction projection method, in conjunction with the scalar auxiliary variable approach not only enables the discrete scheme to satisfy unconditional energy stability, but also allows the convective term in the Navier-Stokes equations to be treated explicitly. We subsequently prove that the time semi-discrete scheme is unconditionally stable and analyze the optimal error estimates for the fully discrete scheme. Finally, several numerical experiments validate the theoretical results.