Communications in Mathematical Sciences

Volume 18 (2020)

Number 2

Numerical study of compressible Navier–Stokes–Cahn–Hilliard system

Pages: 571 – 591



Qiaolin He (School of Mathematics, Sichuan University, Chengdu, China)

Xiaoding Shi (Department of Mathematics, School of Science, Beijing University of Chemical Technology, Beijing, China)


In this paper, we present a relaxation scheme coupled with a stabilized method to numerically study the compressible Navier–Stokes–Cahn–Hilliard system, which has energy-decaying property under certain conditions. Numerical simulations are carried out to verify the stability of the scheme. Numerical results show that the average concentration difference for the two components of the initial state determines the long-time behavior of the diffusive interface for the two-phase flow, which are consistent with theoretical asymptotic stability results established by the authors.


relaxation scheme, Navier–Stokes, Cahn–Hilliard, phase transition, large-time behavior

2010 Mathematics Subject Classification

35L65, 65D30, 65M06, 76T10

The authors’ research was supported in part by National Key R & D Program of China (2018YFC0830300) and NSFC (No. 11201322, No. 11671027, No. 11971020).

Received 3 January 2019

Accepted 8 November 2019

Published 20 June 2022