An efficient dimension splitting-based multi-threaded simulation approach for the phase-field model of two-phase incompressible flows

This paper presents a study on the fast numerical simulation of the phase-field model for two-phase incompressible flow, which comprises a coupled system of the Cahn-Hilliard and Navier-Stokes equations. To address the practical challenges posed by high storage demands and computational complexity, we aim to introduce a numerical approach that leverages dimension splitting for parallel and multi-threaded implementation. Specifically, we develop a novel splitting method: First, a projection method with a dimension splitting effect is incorporated to solve the phase-variable-coupled Navier-Stokes equation in parallel. Second, the convective Cahn-Hilliard equation is tackled using a space-time operator splitting scheme. It is confirmed that the proposed method can effectively reduce the huge amount of computation and storage in solving two- and three-dimensional problems. At the same time, it also has the advantages of linearity, space-time second-order accuracy, mass conservation, parallel implementation, and easy programming. The mass conservation property, time complexity, and storage requirement are analyzed. The parallel efficiency is shown by numerical verification. A large number of interesting numerical simulations, such as phase separation, two-phase cavity flow, bubble rising, viscous droplet falling, Kelvin-Helmholtz, and Rayleigh-Taylor instabilities, are performed to show the performance of the method and investigate complex two-phase interface problems.