A Second Order in Time, Uniquely Solvable, Unconditionally Stable Numerical Scheme for Cahn-Hilliard-Navier-Stokes Equation


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.


Mathematics and Statistics


This work was completed while Han was supported as a Research Assistant on an NSF Grant ( DMS1312701 ). The authors also acknowledge the support of NSF DMS1008852 , a planning grant and a multidisciplinary support grant from the Florida State University .

Keywords and Phrases

Cahn-Hilliard-Navier-Stokes; Diffuse interface model; Energy law preserving; Mixed finite element; Pressure-projection; Unique solvability

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Article - Journal

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Publication Date

01 Jun 2015