Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations

• Published in: Journal of computational physics 2025-01, Vol.521, p.113550, Article 113550
• Main Authors: Doan, Cao-Kha, Hoang, Thi-Thao-Phuong, Ju, Lili, Lan, Rihui
• Format: Article
• Language: English
• Subjects: Dynamic regularization; Energy stability; Incompressible Navier-Stokes equations; Lagrange multiplier; Marker-and-Cell method
• ISSN: 0021-9991
• DOI: 10.1016/j.jcp.2024.113550

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. •A new formulation involves the kinetic energy, a Lagrange multiplier, and a regularization parameter.•The regularization parameter allows using large time step sizes while ensuring the uniqueness of the Lagrange multiplier.•The proposed schemes require solving two linear Stokes systems and a scalar quadratic equation at each time step.•Numerical results in two and three dimensions confirm convergence, energy dissipation and robustness of the DRLM schemes.