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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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This content will become publicly available on January 1, 2026
Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations
In this paper, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify the convergence and energy dissipation as well as demonstrate the accuracy and robustness of the proposed DRLM schemes.
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- PAR ID:
- 10617913
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Computational Physics
- Volume:
- 521
- Issue:
- P1
- ISSN:
- 0021-9991
- Page Range / eLocation ID:
- 113550
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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