We propose two mass and heat energy conservative, unconditionally stable, decoupled numerical algorithms for solving the Cahn-Hilliard-Navier-Stokes-Darcy-Boussinesq system that models thermal convection of two-phase flows in superposed free flow and porous media. The schemes totally decouple the computation of the Cahn-Hilliard equation, the Darcy equations, the heat equation, the Navier-Stokes equations at each time step, and thus significantly reducing the computational cost. We rigorously show that the schemes are conservative and energy-law preserving. Numerical results are presented to demonstrate the accuracy and stability of the algorithms.


Mathematics and Statistics

Publication Status

Early View: Online Version of Record before inclusion in an issue


Daozhi Han acknowledges support from NSF-DMS-1912715.

First Published 10 Sep 2021

Keywords and Phrases

Convection; Phase Field Model; Two-Phase Flow; Unconditional Stability

International Standard Serial Number (ISSN)

1098-2426; 0749-159X

Document Type

Article - Journal

Document Version

Final Version

File Type





© 2021 The Authors, All rights reserved.

Creative Commons Licensing

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

Publication Date

10 Sep 2021