  Hydrodynamics coupled phase field models have intricate difficulties to solve
numerically as they feature high nonlinearity and great complexity in coupling.
In this paper, we propose three second order, linear, unconditionally stable
decoupling methods based on the Crank--Nicolson leap-frog time discretization
for solving the Allen--Cahn--Navier--Stokes (ACNS) phase field model of
two-phase incompressible flows. The ACNS system is decoupled via the artificial
compression method and a splitting approach by introducing an exponential
scalar auxiliary variable.We prove all three algorithms are unconditionally
long time stable. Numerical examples are provided to verify the convergence
rate, unconditional stability, and computational efficiency.
