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

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 two 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 both algorithms are unconditionally long time stable. Numerical examples are provided to verify the convergence rate and unconditional stability.