EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues ESAIM: M2AN, 40 4 (2006) 653-687 Article references

Articles citing this article

The Citing articles tool gives a list of articles citing the current article.
The citing articles come from EDP Sciences database, as well as other publishers participating in CrossRef Cited-by Linking Program. You can set up your personal account to receive an email alert each time this article is cited by a new article (see the menu on the right-hand side of the abstract page).

Cited article:

Auxiliary relaxation method to derive thermodynamically consistent phase field models with constraints and structure preserving numerical approximations

Qi Hong, Zengyan Zhang and Jia Zhao
Journal of Computational Physics 522 113598 (2025)
https://doi.org/10.1016/j.jcp.2024.113598

A free-energy based multiple-distribution-function lattice Boltzmann method for multi-component and multi-phase flows

Zhonghua Qiao, Xuguang Yang and Yuze Zhang
Applied Thermal Engineering 257 124241 (2024)
https://doi.org/10.1016/j.applthermaleng.2024.124241

Implementation of a ternary lattice Boltzmann model in LAMMPS

Gokul Raman Arumugam Kumar, James P. Andrews and Ulf D. Schiller
Computer Physics Communications 294 108898 (2024)
https://doi.org/10.1016/j.cpc.2023.108898

Linear relaxation method with regularized energy reformulation for phase field models

Jiansong Zhang, Xinxin Guo, Maosheng Jiang, Tao Zhou and Jia Zhao
Journal of Computational Physics 515 113225 (2024)
https://doi.org/10.1016/j.jcp.2024.113225

Three-dimensional lattice Boltzmann flux solver for three-phase/component flow

Da Zhang, Yan Li, Liang Gong, Chenlin Zhu and Chang Shu
Physics of Fluids 36 (8) (2024)
https://doi.org/10.1063/5.0224828

Fast and stable dimension splitting simulations for the hydrodynamically coupled three-component conserved Allen–Cahn phase field model

Yuna Yang, Yan Wang, Xufeng Xiao and Xinlong Feng
International Journal of Multiphase Flow 174 104765 (2024)
https://doi.org/10.1016/j.ijmultiphaseflow.2024.104765

Stability and error estimates of Strang splitting method for the nonlocal ternary conservative Allen–Cahn model

Zhifeng Weng, Shuying Zhai, Weizhong Dai, Yanfang Yang and Yuchang Mo
Journal of Computational and Applied Mathematics 441 115668 (2024)
https://doi.org/10.1016/j.cam.2023.115668

A reduction-consistent phase field model for non-isothermal multiphase flows of N immiscible incompressible fluids

Yao Xiao, Zhong Zeng, Liangqi Zhang, Jingzhu Wang, Yiwei Wang and Chenguang Huang
International Journal of Heat and Mass Transfer 228 125657 (2024)
https://doi.org/10.1016/j.ijheatmasstransfer.2024.125657

A second-order phase field-lattice Boltzmann model with equation of state inputting for two-phase flow containing soluble surfactants

Shi-Ting Zhang, Yang Hu, Qianping Li, De-Cai Li, Qiang He and Xiao-Dong Niu
Physics of Fluids 36 (2) (2024)
https://doi.org/10.1063/5.0191792

Patterning and folding of intestinal villi by active mesenchymal dewetting

Tyler R. Huycke, Teemu J. Häkkinen, Hikaru Miyazaki, Vasudha Srivastava, Emilie Barruet, Christopher S. McGinnis, Ali Kalantari, Jake Cornwall-Scoones, Dedeepya Vaka, Qin Zhu, Hyunil Jo, Roger Oria, Valerie M. Weaver, William F. DeGrado, Matt Thomson, Krishna Garikipati, Dario Boffelli, Ophir D. Klein and Zev J. Gartner
Cell 187 (12) 3072 (2024)
https://doi.org/10.1016/j.cell.2024.04.039

Wetting and Spreading Behavior of Axisymmetric Compound Droplets on Curved Solid Walls Using Conservative Phase Field Lattice Boltzmann Method

Yue Wang and Jun-Jie Huang
Entropy 26 (2) 172 (2024)
https://doi.org/10.3390/e26020172

Physically consistent modelling of surface tension forces in the Volume-of-Fluid method for three or more phases

Élfego Ruiz-Gutiérrez, Josef Hasslberger, Markus Klein, Kenny Dalgarno and Nilanjan Chakraborty
Journal of Computational Physics 513 113149 (2024)
https://doi.org/10.1016/j.jcp.2024.113149

A Ternary Phase-Field Model for Separation of Water from a Water/Oil Mixed Drop Using Lamb Waves

Zhijie Zhang, Zichen Wang, Zhaodong Yang, Yang Wang and Wei Liang
Microgravity Science and Technology 36 (6) (2024)
https://doi.org/10.1007/s12217-024-10144-6

Mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces

Lulu Liu, Shijie Huang, Xufeng Xiao and Xinlong Feng
Journal of Computational Physics 513 113189 (2024)
https://doi.org/10.1016/j.jcp.2024.113189

A modified phase field method for the simulation of two-phase system in complex geometries

Yi Shi
Physics of Fluids 36 (8) (2024)
https://doi.org/10.1063/5.0220227

Efficiently and consistently energy-stable L2-phase-field method for the incompressible ternary fluid problems

Junxiang Yang, Huan Han, Shuhong Liu and Zhigang Zuo
Physica D: Nonlinear Phenomena 470 134346 (2024)
https://doi.org/10.1016/j.physd.2024.134346

Energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn model

Shuying Zhai, Zhifeng Weng, Yuchang Mo and Xinlong Feng
Computers & Mathematics with Applications 155 150 (2024)
https://doi.org/10.1016/j.camwa.2023.12.006

An Enhanced and Highly Efficient Semi‐Implicit Combined Lagrange Multiplier Approach Preserving Original Energy Law for Dissipative Systems

Zhengguang Liu, Nan Zheng and Xiaoli Li
International Journal for Numerical Methods in Engineering (2024)
https://doi.org/10.1002/nme.7619

A parametric analysis of the long term performance of a solid oxide fuel cell anode

Tomasz A. Prokop, Szymon Buchaniec, Janusz Szmyd and Grzegorz Brus
International Journal of Heat and Fluid Flow 110 109583 (2024)
https://doi.org/10.1016/j.ijheatfluidflow.2024.109583

Fluid dynamics alters liquid–liquid phase separation in confined aqueous two-phase systems

Eric W. Hester, Sean Carney, Vishwesh Shah, Alyssa Arnheim, Bena Patel, Dino Di Carlo and Andrea L. Bertozzi
Proceedings of the National Academy of Sciences 120 (49) (2023)
https://doi.org/10.1073/pnas.2306467120

A note on the summation relation in phase-field equations

Reza Haghani, Hamidreza Erfani, James E. McClure and Carl Fredrik Berg
Physics of Fluids 35 (9) (2023)
https://doi.org/10.1063/5.0164445

Axisymmetric lattice Boltzmann model for three-phase fluids and its application to the Rayleigh-Plateau instability

Cheng Liu and Hong Liang
Acta Physica Sinica 72 (4) 044701 (2023)
https://doi.org/10.7498/aps.72.20221967

Kelvin–Helmholtz Instability Augmented by von Kármán Vortex Shedding during an Oil Droplet Impact on a Water Pool

Joydip Chaudhuri, Tapas Kumar Mandal and Dipankar Bandyopadhyay
Langmuir 39 (2) 800 (2023)
https://doi.org/10.1021/acs.langmuir.2c02761

Phase-field modeling and consistent energy-stable simulation of binary creeping flows in contact with solid

Junxiang Yang, Jingwen Wu and Zhijun Tan
Computer Methods in Applied Mechanics and Engineering 414 116180 (2023)
https://doi.org/10.1016/j.cma.2023.116180

Numerical Analysis of Droplet Impacting on an Immiscible Liquid via Three-Phase Field Method

Qingming Hu, Fengshi Hu, Donghui Xu and Kailiang Zhang
Micromachines 14 (5) 951 (2023)
https://doi.org/10.3390/mi14050951

Bubble rising and interaction in ternary fluid flow: a phase field study

Mingguang Shen and Ben Q. Li
RSC Advances 13 (6) 3561 (2023)
https://doi.org/10.1039/D2RA06144A

An efficient time-dependent auxiliary variable approach for the three-phase conservative Allen–Cahn fluids

Zhijun Tan, Junxiang Yang, Jianjun Chen and Junseok Kim
Applied Mathematics and Computation 438 127599 (2023)
https://doi.org/10.1016/j.amc.2022.127599

Improving long-term solid oxide fuel cell stack performance prediction accuracy using a microstructure evolution model

Tomasz A. Prokop and Grzegorz Brus
Journal of Power Sources 580 233368 (2023)
https://doi.org/10.1016/j.jpowsour.2023.233368

Comparison between phase-field model and coarse-grained model for characterizing cell-resolved morphological and mechanical properties in a multicellular system

Guoye Guan, Xiangyu Kuang, Chao Tang and Lei Zhang
Communications in Nonlinear Science and Numerical Simulation 117 106966 (2023)
https://doi.org/10.1016/j.cnsns.2022.106966

A consistent and conservative Phase-Field method for compressible multiphase flows with shocks

Ziyang Huang and Eric Johnsen
Journal of Computational Physics 488 112195 (2023)
https://doi.org/10.1016/j.jcp.2023.112195

A free-energy-based and interfacially consistent phase-field model for solid-state sintering without artificial void generation

Qingcheng Yang, Yongxin Gao, Arkadz Kirshtein, Qiang Zhen and Chun Liu
Computational Materials Science 229 112387 (2023)
https://doi.org/10.1016/j.commatsci.2023.112387

Efficient IMEX and consistently energy-stable methods of diffuse-interface models for incompressible three-component flows

Junxiang Yang, Jian Wang, Zhijun Tan and Junseok Kim
Computer Physics Communications 282 108558 (2023)
https://doi.org/10.1016/j.cpc.2022.108558

Unveiling the spatiotemporal evolution of liquid-lens coalescence: Self-similarity, vortex quadrupoles, and turbulence in a three-phase fluid system

Nadia Bihari Padhan and Rahul Pandit
Physics of Fluids 35 (11) (2023)
https://doi.org/10.1063/5.0172631

A multiphase Cahn–Hilliard system with mobilities and the numerical simulation of dewetting

Elie Bretin, Roland Denis, Simon Masnou, Arnaud Sengers and Garry Terii
ESAIM: Mathematical Modelling and Numerical Analysis 57 (3) 1473 (2023)
https://doi.org/10.1051/m2an/2023023

High-order supplementary variable methods for thermodynamically consistent partial differential equations

Qi Hong, Qi Wang and Yuezheng Gong
Computer Methods in Applied Mechanics and Engineering 416 116306 (2023)
https://doi.org/10.1016/j.cma.2023.116306

Numerical study on the engulfing behavior between immiscible droplets in a confined shear flow

Wankun Liu and Jang Min Park
Chemical Engineering Science 266 118265 (2023)
https://doi.org/10.1016/j.ces.2022.118265

Comparison study of phase-field and level-set method for three-phase systems including two minerals

Mathis Kelm, Stephan Gärttner, Carina Bringedal, et al.
Computational Geosciences 26 (3) 545 (2022)
https://doi.org/10.1007/s10596-022-10142-w

A unified finite volume framework for phase‐field simulations of an arbitrary number of fluid phases

Milad Bagheri, Bastian Stumpf, Ilia V. Roisman, Abdolrahman Dadvand, Martin Wörner and Holger Marschall
The Canadian Journal of Chemical Engineering 100 (9) 2291 (2022)
https://doi.org/10.1002/cjce.24510

The collision of immiscible droplets in three-phase liquid systems: A numerical study using phase-field lattice Boltzmann method

Adel Ebadi and S.M. Hosseinalipour
Chemical Engineering Research and Design 178 289 (2022)
https://doi.org/10.1016/j.cherd.2021.12.019

A simulation study on hydrogel performance for enhanced oil recovery using phase-field method

Seyed Hosein Hayatolgheibi, Forough Ameli and Mohammad Reza Moghbeli
Scientific Reports 12 (1) (2022)
https://doi.org/10.1038/s41598-022-06388-0

High-order discontinuous Galerkin approximation for a three-phase incompressible Navier–Stokes/Cahn–Hilliard model

Juan Manzanero, Carlos Redondo, Miguel Chávez-Módena, Gonzalo Rubio, Eusebio Valero, Susana Gómez-Álvarez and Ángel Rivero-Jiménez
Computers & Fluids 244 105545 (2022)
https://doi.org/10.1016/j.compfluid.2022.105545

Elasto-capillary fluid–structure interaction with compound droplets

Sthavishtha R. Bhopalam, Jesus Bueno and Hector Gomez
Computer Methods in Applied Mechanics and Engineering 400 115507 (2022)
https://doi.org/10.1016/j.cma.2022.115507

Optimal rate convergence analysis of a numerical scheme for the ternary Cahn–Hilliard system with a Flory–Huggins–deGennes energy potential

Lixiu Dong, Cheng Wang, Steven M. Wise and Zhengru Zhang
Journal of Computational and Applied Mathematics 415 114474 (2022)
https://doi.org/10.1016/j.cam.2022.114474

Numerical simulation of droplet evaporation in three-component multiphase flows using lattice Boltzmann method

Navid Latifiyan, Mohammad Hassan Rahimian, Reza Haghani-Hassan-Abadi, Mostafa Ashna and Azadeh Jafari
Acta Mechanica 233 (11) 4817 (2022)
https://doi.org/10.1007/s00707-022-03307-2

Single and double toroid formation during oil droplet impact on an air–water interface at low Reynolds number

Joydip Chaudhuri, Tapas Kumar Mandal and Dipankar Bandyopadhyay
Physics of Fluids 34 (1) (2022)
https://doi.org/10.1063/5.0077745

A consistent and conservative Phase-Field method for multiphase incompressible flows

Ziyang Huang, Guang Lin and Arezoo M. Ardekani
Journal of Computational and Applied Mathematics 408 114116 (2022)
https://doi.org/10.1016/j.cam.2022.114116

A phase field model for partially saturated geomaterials describing fluid–fluid displacements. Part I: The model and one-dimensional analysis

Siddhartha H. Ommi, Giulio Sciarra and Panagiotis Kotronis
Advances in Water Resources 164 104170 (2022)
https://doi.org/10.1016/j.advwatres.2022.104170

Phase-field-based lattice Boltzmann model for ternary fluid flows considering the wettability effect

Yajin Zhang, Bo Dong, Xiang An, et al.
Applied Mathematical Modelling 103 195 (2022)
https://doi.org/10.1016/j.apm.2021.10.012

Influence of the pre-impact shape of an oil droplet on the post-impact flow dynamics at air–water interface

Joydip Chaudhuri, Tapas Kumar Mandal and Dipankar Bandyopadhyay
Soft Matter 18 (21) 4102 (2022)
https://doi.org/10.1039/D2SM00219A

A phase-field-based lattice Boltzmann model for multiphase flows involving N immiscible incompressible fluids

Xiaolei Yuan, Baochang Shi, Chengjie Zhan and Zhenhua Chai
Physics of Fluids 34 (2) (2022)
https://doi.org/10.1063/5.0078507

A wettability pattern-mediated trapped bubble removal from a horizontal liquid–liquid interface

Imdad Uddin Chowdhury, Pallab Sinha Mahapatra and Ashis Kumar Sen
Physics of Fluids 34 (4) (2022)
https://doi.org/10.1063/5.0086149

Numerical investigations of electric field-mediated active droplet transfer in a ternary phase system

Satarupa Dutta, Partho Sarathi Gooh Pattader and Dipankar Bandyopadhyay
Physics of Fluids 34 (10) (2022)
https://doi.org/10.1063/5.0104912

A fully-discrete decoupled finite element method for the conserved Allen–Cahn type phase-field model of three-phase fluid flow system

Xiaofeng Yang and Xiaoming He
Computer Methods in Applied Mechanics and Engineering 389 114376 (2022)
https://doi.org/10.1016/j.cma.2021.114376

Upscaling of a Cahn–Hilliard Navier–Stokes model with precipitation and dissolution in a thin strip

Lars von Wolff and Iuliu Sorin Pop
Journal of Fluid Mechanics 941 (2022)
https://doi.org/10.1017/jfm.2022.308

Reduction in vacuum phenomenon for the triple junction in the ternary Cahn–Hilliard model

Junxiang Yang, Chaeyoung Lee and Junseok Kim
Acta Mechanica 232 (11) 4485 (2021)
https://doi.org/10.1007/s00707-021-03072-8

Ternary phase-field simplified multiphase lattice Boltzmann method and its application to compound droplet dynamics on solid surface in shear flow

Z. Chen, C. Shu, Y. Y. Liu and L. Q. Zhang
Physical Review Fluids 6 (9) (2021)
https://doi.org/10.1103/PhysRevFluids.6.094304

Second order linear decoupled energy dissipation rate preserving schemes for the Cahn-Hilliard-extended-Darcy model

Yakun Li, Wenkai Yu, Jia Zhao and Qi Wang
Journal of Computational Physics 444 110561 (2021)
https://doi.org/10.1016/j.jcp.2021.110561

Phase-field modeling with the TAF-ID of incipient melting and oxygen transport in nuclear fuel during power transients

C. Introïni, J. Sercombe, I. Ramière and R. Le Tellier
Journal of Nuclear Materials 556 153173 (2021)
https://doi.org/10.1016/j.jnucmat.2021.153173

On the lattice Boltzmann method and its application to turbulent, multiphase flows of various fluids including cryogens: A review

K. J. Petersen and J. R. Brinkerhoff
Physics of Fluids 33 (4) (2021)
https://doi.org/10.1063/5.0046938

Phase-field modeling of wetting and balling dynamics in powder bed fusion process

Lu Li, Ji-Qin Li and Tai-Hsi Fan
Physics of Fluids 33 (4) (2021)
https://doi.org/10.1063/5.0046771

A new efficient fully-decoupled and second-order time-accurate scheme for Cahn–Hilliard phase-field model of three-phase incompressible flow

Xiaofeng Yang
Computer Methods in Applied Mechanics and Engineering 376 113589 (2021)
https://doi.org/10.1016/j.cma.2020.113589

Hybrid lattice-Boltzmann finite-difference simulation of ternary fluids near immersed solid objects of general shapes

Jun-Jie Huang
Physics of Fluids 33 (7) (2021)
https://doi.org/10.1063/5.0056598

A discontinuous Galerkin approximation for a wall–bounded consistent three–component Cahn–Hilliard flow model

Juan Manzanero, Carlos Redondo, Gonzalo Rubio, Esteban Ferrer and Ángel Rivero–Jiménez
Computers & Fluids 225 104971 (2021)
https://doi.org/10.1016/j.compfluid.2021.104971

Improving the staircase approximation for wettability implementation of phase-field model: Part 1 – Static contact angle

Amin Zarareh, Sorush Khajepor, Stephen B Burnside and Baixin Chen
Computers & Mathematics with Applications 98 218 (2021)
https://doi.org/10.1016/j.camwa.2021.07.013

A positivity-preserving, energy stable scheme for a ternary Cahn-Hilliard system with the singular interfacial parameters

Lixiu Dong, Cheng Wang, Steven M. Wise and Zhengru Zhang
Journal of Computational Physics 442 110451 (2021)
https://doi.org/10.1016/j.jcp.2021.110451

Derivation of generalized Cahn-Hilliard equation for two-phase flow in porous media using hybrid mixture theory

Lynn Schreyer and Zachary Hilliard
Advances in Water Resources 149 103839 (2021)
https://doi.org/10.1016/j.advwatres.2020.103839

A consistent and conservative volume distribution algorithm and its applications to multiphase flows using Phase-Field models

Ziyang Huang, Guang Lin and Arezoo M. Ardekani
International Journal of Multiphase Flow 142 103727 (2021)
https://doi.org/10.1016/j.ijmultiphaseflow.2021.103727

Phase-field modeling of selective laser brazing of diamond grits

Lu Li, Shuai Li, Bi Zhang and Tai-Hsi Fan
Physics of Fluids 33 (5) (2021)
https://doi.org/10.1063/5.0049096

How Do Degenerate Mobilities Determine Singularity Formation in Cahn--Hilliard Equations?

Catalina Pesce and Andreas Muench
Multiscale Modeling & Simulation 19 (3) 1143 (2021)
https://doi.org/10.1137/21M1391249

A consistent and conservative model and its scheme for N-phase-M-component incompressible flows

Ziyang Huang, Guang Lin and Arezoo M. Ardekani
Journal of Computational Physics 434 110229 (2021)
https://doi.org/10.1016/j.jcp.2021.110229

Reduction-consistent multiple-relaxation-time lattice Boltzmann equation method for wall bounded N immiscible incompressible fluids

Lin Zheng, Song Zheng and Qinglan Zhai
Computers & Fluids 221 104896 (2021)
https://doi.org/10.1016/j.compfluid.2021.104896

Efficient and energy stable scheme for the hydrodynamically coupled three components Cahn-Hilliard phase-field model using the stabilized-Invariant Energy Quadratization (S-IEQ) Approach

Xiaofeng Yang
Journal of Computational Physics 438 110342 (2021)
https://doi.org/10.1016/j.jcp.2021.110342

Efficient, second-order in time, and energy stable scheme for a new hydrodynamically coupled three components volume-conserved Allen–Cahn phase-field model

Xiaofeng Yang
Mathematical Models and Methods in Applied Sciences 31 (04) 753 (2021)
https://doi.org/10.1142/S0218202521500184

A ternary Cahn–Hilliard–Navier–Stokes model for two-phase flow with precipitation and dissolution

Christian Rohde and Lars von Wolff
Mathematical Models and Methods in Applied Sciences 31 (01) 1 (2021)
https://doi.org/10.1142/S0218202521500019

Second order linear thermodynamically consistent approximations to nonlocal phase field porous media models

Wenkai Yu, Yakun Li, Jia Zhao and Qi Wang
Computer Methods in Applied Mechanics and Engineering 386 114089 (2021)
https://doi.org/10.1016/j.cma.2021.114089

Microstructure Evolution in a Solid Oxide Fuel Cell Stack Quantified with Interfacial Free Energy

Tomasz A. Prokop, Grzegorz Brus and Janusz S. Szmyd
Energies 14 (12) 3476 (2021)
https://doi.org/10.3390/en14123476

A conservative level set method for N-phase flows with a free-energy-based surface tension model

Amanda A. Howard and Alexandre M. Tartakovsky
Journal of Computational Physics 426 109955 (2021)
https://doi.org/10.1016/j.jcp.2020.109955

An Energy Stable Finite Element Scheme for the Three-Component Cahn–Hilliard-Type Model for Macromolecular Microsphere Composite Hydrogels

Maoqin Yuan, Wenbin Chen, Cheng Wang, Steven M. Wise and Zhengru Zhang
Journal of Scientific Computing 87 (3) (2021)
https://doi.org/10.1007/s10915-021-01508-w

Numerical study of the ternary Cahn–Hilliard fluids by using an efficient modified scalar auxiliary variable approach

Junxiang Yang and Junseok Kim
Communications in Nonlinear Science and Numerical Simulation 102 105923 (2021)
https://doi.org/10.1016/j.cnsns.2021.105923

CFD–based erosion and corrosion modeling in pipelines using a high–order discontinuous Galerkin multiphase solver

C. Redondo, M. Chávez–Modena, J. Manzanero, et al.
Wear 478-479 203882 (2021)
https://doi.org/10.1016/j.wear.2021.203882

Reduction-consistent Cahn–Hilliard theory based lattice Boltzmann equation method for N immiscible incompressible fluids

Lin Zheng, Song Zheng and Qinglan Zhai
Physica A: Statistical Mechanics and its Applications 574 126015 (2021)
https://doi.org/10.1016/j.physa.2021.126015

Phase-field-based lattice Boltzmann model for immiscible incompressible N -phase flows

Xiaolei Yuan, Hong Liang, Zhenhua Chai and Baochang Shi
Physical Review E 101 (6) (2020)
https://doi.org/10.1103/PhysRevE.101.063310

Decoupled, non-iterative, and unconditionally energy stable large time stepping method for the three-phase Cahn-Hilliard phase-field model

Jun Zhang and Xiaofeng Yang
Journal of Computational Physics 404 109115 (2020)
https://doi.org/10.1016/j.jcp.2019.109115

Consistent and conservative scheme for incompressible two-phase flows using the conservative Allen-Cahn model

Ziyang Huang, Guang Lin and Arezoo M. Ardekani
Journal of Computational Physics 420 109718 (2020)
https://doi.org/10.1016/j.jcp.2020.109718

Multiphase flows of N immiscible incompressible fluids: Conservative Allen-Cahn equation and lattice Boltzmann equation method

Lin Zheng, Song Zheng and Qinglan Zhai
Physical Review E 101 (1) (2020)
https://doi.org/10.1103/PhysRevE.101.013305

Unconditionally energy stable large time stepping method for the L2-gradient flow based ternary phase-field model with precise nonlocal volume conservation

Jun Zhang and Xiaofeng Yang
Computer Methods in Applied Mechanics and Engineering 361 112743 (2020)
https://doi.org/10.1016/j.cma.2019.112743