The Explicit‐Invariant Energy Quadratization Method for the Cahn‐Hilliard‐Navier‐Stokes Phase‐Field Model

ABSTRACT In this paper, we propose an efficient numerical method based on explicit invariant energy quadratization (EIEQ) for the Cahn‐Hilliard‐Navier‐Stokes model. By introducing local and nonlocal auxiliary variables, we linearize the nonlinear terms. Combining the pressure‐correction method to handle the velocity‐pressure coupling, and using the intermediate velocity (IV) method and the zero‐energy‐contribution (ZEC) method respectively to solve the coupling between the velocity and the phase‐field variable, we construct two fully decoupled, linear, and unconditionally energy stable numerical schemes. Through rigorous theoretical analysis, we prove that both schemes possess unconditional energy stability. Finally, we verify their temporal convergence order and the dissipative property of the discrete energy through numerical experiments. The research results indicate that the proposed methods can efficiently and stably simulate multiphase flow phenomena, providing a reliable numerical tool for complex interfacial dynamics problems.