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A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation. (English) Zbl 1349.76213
Summary: We propose a novel second order in time numerical scheme for Cahn-Hilliard-Navier-Stokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn-Hilliard equation and pressure-projection for the Navier-Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, we prove that the scheme is unconditionally uniquely solvable at each time step by exploring the monotonicity associated with the scheme. Thanks to the simple coupling of the scheme, we design an efficient Picard iteration procedure to further decouple the computation of Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by the mixed finite element method. Ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids 82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
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