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.

翻译：本文为Cahn-Hilliard-Navier-Stokes方程构造了一种解耦的、一阶、全离散且无条件能量稳定的数值格式。该格式主要分为两部分。第一部分涉及Cahn-Hilliard方程的计算，第二部分则利用上一步获得的相场和化学势值，随后计算Navier-Stokes方程。具体而言，Cahn-Hilliard方程中的速度在离散时间层上被显式离散，这使得Cahn-Hilliard方程的计算完全与Navier-Stokes方程的计算解耦。此外，压力修正投影法与标量辅助变量方法相结合，不仅使离散格式满足无条件能量稳定性，还允许对Navier-Stokes方程中的对流项进行显式处理。随后，我们证明了时间半离散格式是无条件稳定的，并分析了全离散格式的最优误差估计。最后，通过若干数值实验验证了理论结果。