A novel second-order linear scheme for the Cahn-Hilliard-Navier-Stokes equations

Abstract

In this paper, we consider the Cahn-Hilliard equation coupled with the incompressible Navier-Stokes equation, usually known as the Cahn-Hilliard-Navier-Stokes (CHNS) system. The CHNS system has been widely embraced to investigate the dynamics of a binary fluid mixture. By utilizing the modified leap-frog time-marching method, we propose a novel numerical algorithm for solving the CHNS system in an efficient and accurate manner. This newly proposed scheme has several advantages. First of all, the proposed scheme is linear in time and space, such that only a linear algebraic system needs to be solved at each time-marching step, making it extremely efficient. Also, the existence and uniqueness of numerical solutions are guaranteed for any time step size. In addition, the scheme is unconditionally energy stable with second-order accuracy in time and spectral accuracy in space, such that relatively large temporal and spatial mesh sizes can be used to obtain reliable numerical solutions. The rigorous proofs for the unconditional energy stable property and solution existence and uniqueness are given. Furthermore, we present several numerical examples to test the proposed numerical algorithm and illustrate its accuracy and efficiency. The differences of coarsening dynamics between the Cahn-Hilliard equation and the Cahn-Hilliard-Navier-Stokes equations have been investigated as well.