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.
This content will become publicly available on July 1, 2025
We develop two totally decoupled, linear and second‐order accurate numerical methods that are unconditionally energy stable for solving the Cahn–Hilliard–Darcy equations for two phase flows in porous media or in a Hele‐Shaw cell. The implicit‐explicit Crank–Nicolson leapfrog method is employed for the discretization of the Cahn–Hiliard equation to obtain linear schemes. Furthermore the artificial compression technique and pressure correction methods are utilized, respectively, so that the Cahn–Hiliard equation and the update of the Darcy pressure can be solved independently. We establish unconditionally long time stability of the schemes. Ample numerical experiments are performed to demonstrate the accuracy and robustness of the numerical methods, including simulations of the Rayleigh–Taylor instability, the Saffman–Taylor instability (fingering phenomenon).
more » « less- Award ID(s):
- 2310340
- PAR ID:
- 10521008
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Numerical Methods for Partial Differential Equations
- Volume:
- 40
- Issue:
- 4
- ISSN:
- 0749-159X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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