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1. Efficient numerical scheme for a dendritic solidification phase field model with melt convection

   Chen, C; Yang, X. (March 2019, Journal of computational physics)

   In this paper, we consider numerical approximations for a dendritic solidification phase field model with melt convection in the liquid phase, which is a highly nonlinear system that couples the anisotropic Allen-Cahn type equation, the heat equation, and the weighted Navier-Stokes equations together. We first reformulate the model into a form which is suitable for numerical approximations and establish the energy dissipative law. Then, we develop a linear, decoupled, and unconditionally energy stable numerical scheme by combining the modified projection scheme for the Navier-Stokes equations, the Invariant Energy Quadratization approach for the nonlinear anisotropic potential, and some subtle explicit-implicit treatments for nonlinear coupling terms. Stability analysis and various numerical simulations are presented. 

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2. Dynamically Regularized Lagrange Multiplier Method for the Cahn‐Hilliard‐Navier‐Stokes System

   https://doi.org/10.1002/nme.70074  

   Doan, Cao‐Kha; Hoang, Thi‐Thao‐Phuong; Ju, Lili; Lan, Rihui (July 2025, International Journal for Numerical Methods in Engineering)

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

   https://doi.org/10.1016/j.jcp.2024.113550  

   Doan, Cao-Kha; Hoang, Thi-Thao-Phuong; Ju, Lili; Lan, Rihui (January 2025, Journal of Computational Physics)

   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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4. Conservative unconditionally stable decoupled numerical schemes for the Cahn–Hilliard–Navier–Stokes–Darcy–Boussinesq system

   https://doi.org/10.1002/num.22841  

   Chen, Wenbin; Han, Daozhi; Wang, Xiaoming; Zhang, Yichao (September 2021, Numerical Methods for Partial Differential Equations)

   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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5. Feedback control of transitional flows: A framework for controller verification using quadratic constraints

   https://doi.org/10.2514/6.2021-2825  

   Mushtaq, Talha; Seiler, Peter; Hemati, Maziar (July 2021, AIAA AVIATION 2021 FORUM)

   The dynamics of incompressible fluid flows are governed by a non-normal linear dynamical system in feedback with a static energy-conserving nonlinearity. These dynamics can be altered using feedback control but verifying performance of a given control law can be challenging. The conventional approach is to perform a campaign of high-fidelity direct numerical simulations to assess performance over a wide range of parameters and disturbance scenarios. In this paper, we propose an alternative simulation-free approach for controller verification. The incompressible Navier-Stokes equations are modeled as a linear system in feedback with a static and quadratic nonlinearity. The energy conserving property of this nonlinearity can be expressed as a set of quadratic constraints on the system, which allows us to perform a nonlinear stability analysis of the fluid dynamics with minimal complexity. In addition, the Reynolds number variations only influence the linear dynamics in the Navier-Stokes equations. Therefore, the fluid flow can be modeled as a parameter-varying linear system (with Reynolds number as the parameter) in feedback with a quadratic nonlinearity. The quadratic constraint framework is used to determine the range of Reynolds numbers over which a given flow will be stable, without resorting to numerical simulations. We demonstrate the framework on a reduced-order model of plane Couette flow. We show that our proposed method allows us to determine the critical Reynolds number, largest initial disturbance, and a range of parameter variations over which a given controller will stabilize the nonlinear dynamics. 

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