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Abstract 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.
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This content will become publicly available on July 3, 2026
Dynamically Regularized Lagrange Multiplier Method for the Cahn‐Hilliard‐Navier‐Stokes System
ABSTRACT This paper is concerned with efficient and accurate numerical schemes for the Cahn‐Hilliard‐Navier‐Stokes phase field model of binary immiscible fluids. By introducing two Lagrange multipliers for each of the Cahn‐Hilliard and Navier‐Stokes parts, we reformulate the original model problem into an equivalent system that incorporates the energy evolution process. Such a nonlinear, coupled system is then discretized in time using first‐ and second‐order backward differentiation formulas, in which all nonlinear terms are treated explicitly and no extra stabilization term is imposed. The proposed dynamically regularized Lagrange multiplier (DRLM) schemes are mass‐conserving and unconditionally energy‐stable with respect to the original variables. In addition, the schemes are fully decoupled: Each time step involves solving two biharmonic‐type equations and two generalized linear Stokes systems, together with two nonlinear algebraic equations for the Lagrange multipliers. A key feature of the DRLM schemes is the introduction of the regularization parameters which ensure the unique determination of the Lagrange multipliers and mitigate the time step size constraint without affecting the accuracy of the numerical solution, especially when the interfacial width is small. Various numerical experiments are presented to illustrate the accuracy and robustness of the proposed DRLM schemes in terms of convergence, mass conservation, and energy stability.
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- PAR ID:
- 10612789
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- International Journal for Numerical Methods in Engineering
- Volume:
- 126
- Issue:
- 13
- ISSN:
- 0029-5981
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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