Second-Order Decoupled Energy-Stable Schemes for Cahn-Hilliard-Navier-Stokes Equations


The Cahn-Hilliard-Navier-Stokes (CHNS) equations represent the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Due to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation, the CHNS system is non-trivial to be solved numerically. Traditionally, a numerical extrapolation for the coupling terms is used. However, such brute-force extrapolation usually destroys the intrinsic thermodynamic structures of this CHNS system. This paper proposes a new strategy to reformulate the CHNS system into a constraint gradient flow formulation, where the reversible and irreversible structures are clearly revealed. This guides us to propose operator splitting schemes that have several advantageous properties. First of all, the proposed schemes lead to several decoupled systems in smaller sizes to be solved at each time marching step. This significantly reduces computational costs. Secondly, the proposed schemes still guarantee the thermodynamic laws of the CHNS system at the discrete level. In addition, unlike the recently populated IEQ or SAV approaches using auxiliary variables, our resulting energy laws are formulated in the original variables. This is a significant improvement, as the modified energy laws with auxiliary variables sometimes deviate from the original energy law. Our proposed framework lays a foundation for designing decoupled and energy stable numerical algorithms for hydrodynamic phase-field models. Furthermore, various numerical algorithms can be obtained given different splitting steps, making this framework rather general. The proposed numerical algorithms are implemented. Their second-order temporal and spatial accuracy are verified numerically. Some numerical examples and benchmark problems are calculated to verify the effectiveness of the proposed schemes.


Mathematics and Statistics


National Science Foundation, Grant DMS-1816783

Keywords and Phrases

Cahn-Hilliard-Navier-Srtokes; Decoupled Scheme; Energy Stable; Hydrodynamics; Phase Field

International Standard Serial Number (ISSN)

1090-2716; 0021-9991

Document Type

Article - Journal

Document Version


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© 2021 Elsevier, All rights reserved.

Publication Date

15 Oct 2021