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Hydrodynamics coupled phase field models have intricate difficulties to solvenumerically as they feature high nonlinearity and great complexity in coupling.In this paper, we propose three second order, linear, unconditionally stabledecoupling methods based on the Crank--Nicolson leap-frog time discretizationfor solving the Allen--Cahn--Navier--Stokes (ACNS) phase field model oftwo-phase incompressible flows. The ACNS system is decoupled via the artificialcompression method and a splitting approach by introducing an exponentialscalar auxiliary variable.We prove all three algorithms are unconditionallylong time stable. Numerical examples are provided to verify the convergencerate, unconditional stability, and computational efficiency.

Three linear, unconditionally stable, second order decoupling methods for the Allen--Cahn--Navier--Stokes phase field model