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1. SECOND ORDER, LINEAR, AND UNCONDITIONALLY ENERGY STABLE SCHEMES FOR A HYDRODYNAMIC MODEL OF SMECTIC-A LIQUID CRYSTALS

   Chen, R ; Yang, X ; Zhang, H. ( November 2017 , SIAM journal on scientific computing)

   In this paper, we consider the numerical approximations for a hydrodynamical model of smectic-A liquid crystals. The model, derived from the variational approach of the modified Oseen– Frank energy, is a highly nonlinear system that couples the incompressible Navier–Stokes equations and a constitutive equation for the layer variable. We develop two linear, second order time marching schemes based on the Invariant Energy Quadratization method for nonlinear terms in the constitutive equation, the projection method for the Navier–Stokes equations, and some subtle implicit-explicit treatments for the convective and stress terms. Moreover, we prove the well-posedness of the linear system and their unconditionally energy stabilities rigorously. Various numerical experiments are presented to demonstrate the stability and the accuracy of the numerical schemes in simulating the dynamics under shear flow and the magnetic field. 

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2. A novel Decoupled and stable scheme for an anisotropic phase-field dendritic crystal growth model

   Zhang J, Chen C ( April 2019 , Applied mathematics letters)

   We consider numerical approximations for a phase-field dendritic crystal growth model, which is a highly nonlinear system that couples the anisotropic Allen–Cahn type equation and the heat equation. By combining the stabilized-Invariant Energy Quadratization method with a novel decoupling technique, the scheme requires solving only a sequence of linear elliptic equations at each time step, making it the first, to the best of the author’s knowledge, totally decoupled, linear, unconditionally energy stable scheme for the model. We further prove the unconditional energy stability rigorously and present various numerical simulations to demonstrate the stability and accuracy. 

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3. On a novel full decoupling, linear, second‐order accurate, and unconditionally energy stable numerical scheme for the anisotropic phase‐field dendritic crystal growth model

   https://doi.org/10.1002/nme.6697  

   Yang, Xiaofeng ( April 2021 , International Journal for Numerical Methods in Engineering)

   The anisotropic phase‐field dendritic crystal growth model is a highly nonlinear system that couples the anisotropic Allen–Cahn equation and the thermal equation together. Due to the high anisotropy and nonlinear couplings in the system, how to develop an accurate and efficient, especially a fully decoupled scheme, has always been a challenging problem. To solve the challenge, in this article, we construct a novel fully decoupled numerical scheme which is also linear, energy stable, and second‐order time accurate. The key idea to realize the full decoupling structure is to introduce an ordinary differential equation to deal with the nonlinear coupling terms satisfying the so‐called “zero‐energy‐contribution” property. This scheme is very effective and easy to implement since only a few fully decoupled elliptic equations with constant coefficients need to be solved at each time step. We rigorously prove the solvability of each step and the unconditional energy stability, and perform a large number of numerical simulations in 2D and 3D to demonstrate its stability and accuracy numerically.

    

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4. Simplified conservative discretization of the Cahn-Hilliard-Navier-Stokes equations

   https://doi.org/10.1016/j.jcp.2024.113382  

   Goulding, Jason ; Ayazi, Mehrnaz ; Shinar, Tamar ; Schroeder, Craig ( December 2024 , Journal of Computational Physics)

   In this paper we construct a novel discretization of the Cahn-Hilliard equation coupled with the Navier-Stokes equations. The Cahn-Hilliard equation models the separation of a binary mixture. We construct a very simple time integration scheme for simulating the Cahn-Hilliard equation, which is based on splitting the fourth-order equation into two second-order Helmholtz equations. We combine the Cahn-Hilliard equation with the Navier-Stokes equations to simulate phase separation in a two-phase fluid flow in two dimensions. The scheme conserves mass and momentum and exhibits consistency between mass and momentum, allowing it to be used with large density ratios. We introduce a novel discretization of the surface tension force from the phase-field variable that has finite support around the transition region. The model has a parameter that allows it to transition from a smoothed continuum surface force to a fully sharp interface formulation. We show that our method achieves second-order accuracy, and we compare our method to previous work in a variety of experiments. 

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5. 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)

   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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