Second-order unconditionally stable time-filtered scheme for Cahn-Hilliard-Navier-Stokes system

TL;DR

This paper introduces a second-order, unconditionally energy-stable time-filtered scheme for the Cahn-Hilliard-Navier-Stokes system, improving accuracy with minimal modifications and supporting adaptive time-stepping.

Contribution

The paper develops a novel second-order time-filtered scheme for CHNS that maintains unconditional stability and is easy to implement, extending to adaptive time-stepping strategies.

Findings

The scheme achieves second-order accuracy in time.

Unconditional energy stability is rigorously proven.

Numerical tests confirm effectiveness and stability.

Abstract

In this work, we introduce the time filtering technique to develop several innovative semi-discrete schemes in time for the Cahn-Hilliard-Navier-Stokes (CHNS) system. These schemes achieve second-order temporal accuracy while maintaining unconditional energy stability. Our approach begins with the discretization of the CHNS system using the first-order semi-implicit method. Subsequently, by applying time filtering techniques, we improve the temporal accuracy from first-order to second-order. This improvement requires only minor modifications to the original first-order semi-implicit scheme, thereby enabling higher accuracy to be achieved at minimal cost. Moreover, we rigorously establish the unconditional energy stability of the proposed schemes through theoretical analysis. Additionally, we extend our work to develop semi-discrete schemes that incorporate variable and adaptive time-stepping strategies, enhancing the flexibility and efficiency of simulations. Numerical examples are presented to validate the theoretical results and demonstrate the effectiveness of the proposed methods.