New efficient time-stepping schemes for the Navier–Stokes–Cahn–Hilliard equations

Abstract In this paper, we construct and analyze a class of efficient time-stepping schemes for the Navier–Stokes-Cahn-Hilliard equation system stemming from phase field modelling of two-phase incompressible flows. The proposed schemes are based on an auxiliary variable approach for the Cahn–Hilliard equation and a delicate treatment of the terms coupling the Navier–Stokes equation and the Cahn–Hilliard equation. The former has been found to be an efficient tool for numerical integration of the Cahn–Hilliard equation in recent years. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. In the theoretical aspect we rigorously prove that the designed schemes are unconditionally stable in the sense that some kind of energy remains bounded during the time stepping. In the implementation, we show how the schemes can be reformulated into six linear second-order elliptic equations with constant coefficients. The efficiency of the proposed algorithm is verified through several numerical examples.