Year: 2013
Author: Craig Collins, Jie Shen, Steven M. Wise
Communications in Computational Physics, Vol. 13 (2013), Iss. 4 : pp. 929–957
Abstract
We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step size. Owing to energy stability, we show that the scheme is stable in the time and space discrete ℓ∞(0,T;H1h) and ℓ2(0,T;H2h) norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn-Hilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part – for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy- and boundary-driven flows.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.171211.130412a
Communications in Computational Physics, Vol. 13 (2013), Iss. 4 : pp. 929–957
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 29
Author Details
-
Second-Order Decoupled Linear Energy-Law Preserving gPAV Numerical Schemes for Two-Phase Flows in Superposed Free Flow and Porous Media
Gao, Yali | Han, DaozhiJournal of Scientific Computing, Vol. 100 (2024), Iss. 2
https://doi.org/10.1007/s10915-024-02576-4 [Citations: 0] -
New efficient and unconditionally energy stable schemes for the Cahn–Hilliard–Brinkman system
Zheng, Nan | Li, XiaoliApplied Mathematics Letters, Vol. 128 (2022), Iss. P.107918
https://doi.org/10.1016/j.aml.2022.107918 [Citations: 8] -
Second‐order, fully decoupled, linearized, and unconditionally stable scalar auxiliary variable schemes for Cahn–Hilliard–Darcy system
Gao, Yali | He, Xiaoming | Nie, YufengNumerical Methods for Partial Differential Equations, Vol. 38 (2022), Iss. 6 P.1658
https://doi.org/10.1002/num.22829 [Citations: 10] -
Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system: Primitive variable and projection-type schemes
Guo, Z. | Lin, P. | Lowengrub, J. | Wise, S.M.Computer Methods in Applied Mechanics and Engineering, Vol. 326 (2017), Iss. P.144
https://doi.org/10.1016/j.cma.2017.08.011 [Citations: 75] -
Convergence analysis of a positivity-preserving numerical scheme for the Cahn-Hilliard-Stokes system with Flory-Huggins energy potential
Guo, Yunzhuo | Wang, Cheng | Wise, Steven | Zhang, ZhengruMathematics of Computation, Vol. 93 (2023), Iss. 349 P.2185
https://doi.org/10.1090/mcom/3916 [Citations: 0] -
Optimal distributed control of the Cahn–Hilliard–Brinkman system with regular potential
You, Bo | Li, FangNonlinear Analysis, Vol. 182 (2019), Iss. P.226
https://doi.org/10.1016/j.na.2018.12.014 [Citations: 3] -
A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn–Hilliard–Navier–Stokes equation
Han, Daozhi | Wang, XiaomingJournal of Computational Physics, Vol. 290 (2015), Iss. P.139
https://doi.org/10.1016/j.jcp.2015.02.046 [Citations: 160] -
Decoupled energy-law preserving numerical schemes for the Cahn-Hilliard-Darcy system
Han, Daozhi | Wang, XiaomingNumerical Methods for Partial Differential Equations, Vol. 32 (2016), Iss. 3 P.936
https://doi.org/10.1002/num.22036 [Citations: 25] -
Semi-implicit spectral deferred correction methods for highly nonlinear partial differential equations
Guo, Ruihan | Xia, Yinhua | Xu, YanJournal of Computational Physics, Vol. 338 (2017), Iss. P.269
https://doi.org/10.1016/j.jcp.2017.02.059 [Citations: 24] -
Fully decoupled pressure projection scheme for the numerical solution of diffuse interface model of two-phase flow
Sohaib, Muhammad | Shah, AbdullahCommunications in Nonlinear Science and Numerical Simulation, Vol. 112 (2022), Iss. P.106547
https://doi.org/10.1016/j.cnsns.2022.106547 [Citations: 15] -
An efficient, unconditionally energy stable local discontinuous Galerkin scheme for the Cahn–Hilliard–Brinkman system
Guo, Ruihan | Xu, YanJournal of Computational Physics, Vol. 298 (2015), Iss. P.387
https://doi.org/10.1016/j.jcp.2015.06.007 [Citations: 5] -
Second order linear decoupled energy dissipation rate preserving schemes for the Cahn-Hilliard-extended-Darcy model
Li, Yakun | Yu, Wenkai | Zhao, Jia | Wang, QiJournal of Computational Physics, Vol. 444 (2021), Iss. P.110561
https://doi.org/10.1016/j.jcp.2021.110561 [Citations: 4] -
Nonlocal Cahn-Hilliard-Brinkman System with Regular Potential: Regularity and Optimal Control
Dharmatti, Sheetal | Perisetti, Lakshmi Naga MahendranathJournal of Dynamical and Control Systems, Vol. 27 (2021), Iss. 2 P.221
https://doi.org/10.1007/s10883-020-09490-6 [Citations: 2] -
Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis
Ebenbeck, Matthias | Garcke, HaraldJournal of Differential Equations, Vol. 266 (2019), Iss. 9 P.5998
https://doi.org/10.1016/j.jde.2018.10.045 [Citations: 44] -
Analysis of a Mixed Finite Element Method for a Cahn--Hilliard--Darcy--Stokes System
Diegel, Amanda E. | Feng, Xiaobing H. | Wise, Steven M.SIAM Journal on Numerical Analysis, Vol. 53 (2015), Iss. 1 P.127
https://doi.org/10.1137/130950628 [Citations: 93] -
An improved error analysis for a second-order numerical scheme for the Cahn–Hilliard equation
Guo, Jing | Wang, Cheng | Wise, Steven M. | Yue, XingyeJournal of Computational and Applied Mathematics, Vol. 388 (2021), Iss. P.113300
https://doi.org/10.1016/j.cam.2020.113300 [Citations: 17] -
A second order fully-discrete linear energy stable scheme for a binary compressible viscous fluid model
Zhao, Xueping | Wang, QiJournal of Computational Physics, Vol. 395 (2019), Iss. P.382
https://doi.org/10.1016/j.jcp.2019.06.030 [Citations: 17] -
Fully Discrete Second-Order Linear Schemes for Hydrodynamic Phase Field Models of Binary Viscous Fluid Flows with Variable Densities
Gong, Yuezheng | Zhao, Jia | Yang, Xiaogang | Wang, QiSIAM Journal on Scientific Computing, Vol. 40 (2018), Iss. 1 P.B138
https://doi.org/10.1137/17M1111759 [Citations: 67] -
Quasi-incompressible models for binary fluid flows in porous media
Li, Yakun | Wang, QiApplied Mathematics Letters, Vol. 136 (2023), Iss. P.108450
https://doi.org/10.1016/j.aml.2022.108450 [Citations: 0] -
On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems
Porta, Francesco Della | Grasselli, MaurizioCommunications on Pure and Applied Analysis, Vol. 15 (2016), Iss. 2 P.299
https://doi.org/10.3934/cpaa.2016.15.299 [Citations: 22] -
Finite-dimensional global attractor of the Cahn–Hilliard–Brinkman system
Li, Fang | Zhong, Chengkui | You, BoJournal of Mathematical Analysis and Applications, Vol. 434 (2016), Iss. 1 P.599
https://doi.org/10.1016/j.jmaa.2015.09.026 [Citations: 5] -
Error Analysis of a Mixed Finite Element Method for the Molecular Beam Epitaxy Model
Qiao, Zhonghua | Tang, Tao | Xie, HehuSIAM Journal on Numerical Analysis, Vol. 53 (2015), Iss. 1 P.184
https://doi.org/10.1137/120902410 [Citations: 15] -
A second-order decoupled energy stable numerical scheme for Cahn-Hilliard-Hele-Shaw system
Gao, Yali | Li, Rui | Mei, Liquan | Lin, YanpingApplied Numerical Mathematics, Vol. 157 (2020), Iss. P.338
https://doi.org/10.1016/j.apnum.2020.06.010 [Citations: 17] -
Two‐phase flows in karstic geometry
Han, Daozhi | Sun, Dong | Wang, XiaomingMathematical Methods in the Applied Sciences, Vol. 37 (2014), Iss. 18 P.3048
https://doi.org/10.1002/mma.3043 [Citations: 46] -
A fully decoupled numerical method for Cahn–Hilliard–Navier–Stokes–Darcy equations based on auxiliary variable approaches
Gao, Yali | Li, Rui | He, Xiaoming | Lin, YanpingJournal of Computational and Applied Mathematics, Vol. 436 (2024), Iss. P.115363
https://doi.org/10.1016/j.cam.2023.115363 [Citations: 4] -
Well-posedness for the Brinkman–Cahn–Hilliard system with unmatched viscosities
Conti, Monica | Giorgini, AndreaJournal of Differential Equations, Vol. 268 (2020), Iss. 10 P.6350
https://doi.org/10.1016/j.jde.2019.11.049 [Citations: 22] -
On a SAV-MAC scheme for the Cahn–Hilliard–Navier–Stokes phase-field model and its error analysis for the corresponding Cahn–Hilliard–Stokes case
Li, Xiaoli | Shen, JieMathematical Models and Methods in Applied Sciences, Vol. 30 (2020), Iss. 12 P.2263
https://doi.org/10.1142/S0218202520500438 [Citations: 41] -
Efficient, adaptive energy stable schemes for the incompressible Cahn–Hilliard Navier–Stokes phase-field models
Chen, Ying | Shen, JieJournal of Computational Physics, Vol. 308 (2016), Iss. P.40
https://doi.org/10.1016/j.jcp.2015.12.006 [Citations: 100] -
Optimal rate convergence analysis of a second order scheme for a thin film model with slope selection
Wang, Shufen | Chen, Wenbin | Pan, Hanshuang | Wang, ChengJournal of Computational and Applied Mathematics, Vol. 377 (2020), Iss. P.112855
https://doi.org/10.1016/j.cam.2020.112855 [Citations: 8] -
Numerical Analysis of Second Order, Fully Discrete Energy Stable Schemes for Phase Field Models of Two-Phase Incompressible Flows
Han, Daozhi | Brylev, Alex | Yang, Xiaofeng | Tan, ZhijunJournal of Scientific Computing, Vol. 70 (2017), Iss. 3 P.965
https://doi.org/10.1007/s10915-016-0279-5 [Citations: 71] -
A fully-decoupled discontinuous Galerkin approximation of the Cahn–Hilliard–Brinkman–Ohta–Kawasaki tumor growth model
Zou, Guang-an | Wang, Bo | Yang, XiaofengESAIM: Mathematical Modelling and Numerical Analysis, Vol. 56 (2022), Iss. 6 P.2141
https://doi.org/10.1051/m2an/2022064 [Citations: 16] -
Entropy/energy stable schemes for evolutionary dispersal models
Liu, Hailiang | Yu, HuiJournal of Computational Physics, Vol. 256 (2014), Iss. P.656
https://doi.org/10.1016/j.jcp.2013.08.032 [Citations: 5] -
New efficient time-stepping schemes for the Navier–Stokes–Cahn–Hilliard equations
Li, Minghui | Xu, ChuanjuComputers & Fluids, Vol. 231 (2021), Iss. P.105174
https://doi.org/10.1016/j.compfluid.2021.105174 [Citations: 6] -
A stable second-order BDF scheme for the three-dimensional Cahn–Hilliard–Hele–Shaw system
Li, Yibao | Yu, Qian | Fang, Weiwei | Xia, Binhu | Kim, JunseokAdvances in Computational Mathematics, Vol. 47 (2021), Iss. 1
https://doi.org/10.1007/s10444-020-09835-6 [Citations: 10] -
Long Time Numerical Simulations for Phase-Field Problems Using p-Adaptive Spectral Deferred Correction Methods
Feng, Xinlong | Tang, Tao | Yang, JiangSIAM Journal on Scientific Computing, Vol. 37 (2015), Iss. 1 P.A271
https://doi.org/10.1137/130928662 [Citations: 76] -
An Efficient Two-Grid Scheme for the Cahn-Hilliard Equation
Zhou, Jie | Chen, Long | Huang, Yunqing | Wang, WanshengCommunications in Computational Physics, Vol. 17 (2015), Iss. 1 P.127
https://doi.org/10.4208/cicp.231213.100714a [Citations: 22] -
Uniqueness and Regularity for the Navier--Stokes--Cahn--Hilliard System
Giorgini, Andrea | Miranville, Alain | Temam, RogerSIAM Journal on Mathematical Analysis, Vol. 51 (2019), Iss. 3 P.2535
https://doi.org/10.1137/18M1223459 [Citations: 65] -
An energy-stable finite-difference scheme for the binary fluid-surfactant system
Gu, Shuting | Zhang, Hui | Zhang, ZhengruJournal of Computational Physics, Vol. 270 (2014), Iss. P.416
https://doi.org/10.1016/j.jcp.2014.03.060 [Citations: 32] -
A divergence-free generalized moving least squares approximation with its application
Mohammadi, Vahid | Dehghan, MehdiApplied Numerical Mathematics, Vol. 162 (2021), Iss. P.374
https://doi.org/10.1016/j.apnum.2020.12.017 [Citations: 11] -
A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver
C. Aristotelous, Andreas | Karakashian, Ohannes | M. Wise, StevenDiscrete & Continuous Dynamical Systems - B, Vol. 18 (2013), Iss. 9 P.2211
https://doi.org/10.3934/dcdsb.2013.18.2211 [Citations: 30] -
Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation
Dong, Lixiu | Feng, Wenqiang | Wang, Cheng | Wise, Steven M. | Zhang, ZhengruComputers & Mathematics with Applications, Vol. 75 (2018), Iss. 6 P.1912
https://doi.org/10.1016/j.camwa.2017.07.012 [Citations: 45] -
Optimal \(\boldsymbol{{L^2}}\) Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn–Hilliard–Navier–Stokes System
Cai, Wentao | Sun, Weiwei | Wang, Jilu | Yang, ZongzeSIAM Journal on Numerical Analysis, Vol. 61 (2023), Iss. 3 P.1218
https://doi.org/10.1137/22M1486844 [Citations: 11] -
Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation
Gong, Yuezheng | Zhao, Jia | Wang, QiAdvances in Computational Mathematics, Vol. 44 (2018), Iss. 5 P.1573
https://doi.org/10.1007/s10444-018-9597-5 [Citations: 25] -
Uniquely solvable and energy stable decoupled numerical schemes for the Cahn–Hilliard–Stokes–Darcy system for two-phase flows in karstic geometry
Chen, Wenbin | Han, Daozhi | Wang, XiaomingNumerische Mathematik, Vol. 137 (2017), Iss. 1 P.229
https://doi.org/10.1007/s00211-017-0870-1 [Citations: 32] -
A Second-Order, Weakly Energy-Stable Pseudo-spectral Scheme for the Cahn–Hilliard Equation and Its Solution by the Homogeneous Linear Iteration Method
Cheng, Kelong | Wang, Cheng | Wise, Steven M. | Yue, XingyeJournal of Scientific Computing, Vol. 69 (2016), Iss. 3 P.1083
https://doi.org/10.1007/s10915-016-0228-3 [Citations: 98] -
On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions
Li, Fang | You, BoDiscrete & Continuous Dynamical Systems - B, Vol. 26 (2021), Iss. 12 P.6387
https://doi.org/10.3934/dcdsb.2021024 [Citations: 1] -
A Second Order in Time, Decoupled, Unconditionally Stable Numerical Scheme for the Cahn–Hilliard–Darcy System
Han, Daozhi | Wang, XiaomingJournal of Scientific Computing, Vol. 77 (2018), Iss. 2 P.1210
https://doi.org/10.1007/s10915-018-0748-0 [Citations: 36] -
Second order linear thermodynamically consistent approximations to nonlocal phase field porous media models
Yu, Wenkai | Li, Yakun | Zhao, Jia | Wang, QiComputer Methods in Applied Mechanics and Engineering, Vol. 386 (2021), Iss. P.114089
https://doi.org/10.1016/j.cma.2021.114089 [Citations: 2] -
Numerical simulation of endocytosis: Viscous flow driven by membranes with non-uniformly distributed curvature-inducing molecules
Lowengrub, John | Allard, Jun | Aland, SebastianJournal of Computational Physics, Vol. 309 (2016), Iss. P.112
https://doi.org/10.1016/j.jcp.2015.12.055 [Citations: 23] -
A Second Order BDF Numerical Scheme with Variable Steps for the Cahn--Hilliard Equation
Chen, Wenbin | Wang, Xiaoming | Yan, Yue | Zhang, ZhuyingSIAM Journal on Numerical Analysis, Vol. 57 (2019), Iss. 1 P.495
https://doi.org/10.1137/18M1206084 [Citations: 98] -
Numerical analysis for the Cahn-Hilliard-Hele-Shaw system with variable mobility and logarithmic Flory-Huggins potential
Guo, Yayu | Jia, Hongen | Li, Jichun | Li, MingApplied Numerical Mathematics, Vol. 150 (2020), Iss. P.206
https://doi.org/10.1016/j.apnum.2019.09.014 [Citations: 12] -
A numerical method for the quasi-incompressible Cahn–Hilliard–Navier–Stokes equations for variable density flows with a discrete energy law
Guo, Z. | Lin, P. | Lowengrub, J.S.Journal of Computational Physics, Vol. 276 (2014), Iss. P.486
https://doi.org/10.1016/j.jcp.2014.07.038 [Citations: 86] -
A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids
Feng, Wenqiang | Guo, Zhenlin | Lowengrub, John S. | Wise, Steven M.Journal of Computational Physics, Vol. 352 (2018), Iss. P.463
https://doi.org/10.1016/j.jcp.2017.09.065 [Citations: 19] -
Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system
Liu, Yuan | Chen, Wenbin | Wang, Cheng | Wise, Steven M.Numerische Mathematik, Vol. 135 (2017), Iss. 3 P.679
https://doi.org/10.1007/s00211-016-0813-2 [Citations: 79] -
Well-posedness of stochastic Cahn-Hilliard-Brinkman system with regular potential
Dai, Haoran | You, Bo | Li, FangStochastic Analysis and Applications, Vol. 43 (2025), Iss. 2 P.181
https://doi.org/10.1080/07362994.2024.2447787 [Citations: 0] -
A Decoupled Unconditionally Stable Numerical Scheme for the Cahn–Hilliard–Hele-Shaw System
Han, Daozhi
Journal of Scientific Computing, Vol. 66 (2016), Iss. 3 P.1102
https://doi.org/10.1007/s10915-015-0055-y [Citations: 29] -
Highly efficient, robust and unconditionally energy stable second order schemes for approximating the Cahn-Hilliard-Brinkman system
Jiang, Peng | Jia, Hongen | Liu, Liang | Zhang, Chenhui | Wang, DanxiaApplied Numerical Mathematics, Vol. 201 (2024), Iss. P.175
https://doi.org/10.1016/j.apnum.2024.03.001 [Citations: 0] -
An efficient fully-discrete local discontinuous Galerkin method for the Cahn–Hilliard–Hele–Shaw system
Guo, Ruihan | Xia, Yinhua | Xu, YanJournal of Computational Physics, Vol. 264 (2014), Iss. P.23
https://doi.org/10.1016/j.jcp.2014.01.037 [Citations: 28] -
Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation
Feng, Xinlong | Song, Huailing | Tang, Tao | Yang, JiangInverse Problems & Imaging, Vol. 7 (2013), Iss. 3 P.679
https://doi.org/10.3934/ipi.2013.7.679 [Citations: 69]