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A free-energy stable p-adaptive nodal discontinuous Galerkin for the Cahn-Hilliard equation

Gerasimos Ntoukas, Juan Manzanero, Gonzalo Rubio, Eusebio Valero, Esteban Ferrer
2020 Zenodo  
A novel free-energy stable discontinuous Galerkin method is developed for the Cahn-Hilliard equation with non-conforming elements.  ...  Lastly, we examine an unsteady case such as the spinodal decomposition and show that the same results for the free-energy are recovered with a 35% reduction of the degrees of freedom for the two-dimensional  ...  Conclusions In this work, we have established the theoretical framework for a free-energy stable discontinuous Galerkin scheme for the Cahn-Hilliard equation that targets p-non-conforming meshes.  ... 
doi:10.5281/zenodo.4008979 fatcat:g7mxgoxnvbbuvpfck2c2vetnl4

A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

Gerasimos Ntoukas, Juan Manzanero, Gonzalo Rubio, Eusebio Valero, Esteban Ferrer
2021 Journal of Computational Physics  
A novel free-energy stable discontinuous Galerkin method is developed for the Cahn-Hilliard equation with non-conforming elements.  ...  Lastly, we examine an unsteady case such as the spinodal decomposition and show that the same results for the free-energy are recovered, with a 35% reduction of the degrees of freedom for the two-dimensional  ...  A free-energy stable p-adaptive nodal discontinuous Galerkin for the Cahn-Hilliard equation Introduction A variety of different approaches and methodologies have been developed throughout the years to  ... 
doi:10.1016/j.jcp.2021.110409 fatcat:gv3sfch6zfh2zmjpxqiqfkjos4

A discontinuous Galerkin method for the Cahn–Hilliard equation

Garth N. Wells, Ellen Kuhl, Krishna Garikipati
2006 Journal of Computational Physics  
A discontinuous Galerkin finite element method has been developed to treat the high-order spatial derivatives appearing in the Cahn-Hilliard equation.  ...  The developed discontinuous Galerkin approach avoids the need for mixed finite element methods, coupled equations or interpolation functions with a high degree of continuity that have been employed in  ...  Garikipati was supported by a Presidential Early Career Award for Scientists and Engineers through the US Department of Energy, and the Alexander von Humboldt Foundation of Germany.  ... 
doi:10.1016/j.jcp.2006.03.010 fatcat:gvmzosu7pzcolpnzsb5e5bt62m

A High-Order Discontinuous Galerkin Solver for Multiphase Flows [chapter]

Juan Manzanero, Carlos Redondo, Gonzalo Rubio, Esteban Ferrer, Eusebio Valero, Susana Gómez-Álvarez, Ángel Rivero-Jiménez
2020 Lecture Notes in Computational Science and Engineering  
Specifically, we use a Discontinuous Galerkin Spectral Element Method (DGSEM) [2] that allows the generation of provably stable schemes [8] .  ...  The volume fraction is substituted by a phase-field parameter, which identifies each phase. In this work, the Cahn-Hilliard equation [4] is chosen to model the evolution of the phase-field parameter.  ...  The authors acknowledge the computer resources and technical assistance provided by the Centro de Supercomputación y Visualización de Madrid (CeSViMa).  ... 
doi:10.1007/978-3-030-39647-3_24 fatcat:xkxzyy6ianarldvhbwwshl2fxq

Natural element analysis of the Cahn–Hilliard phase-field model

Amirtham Rajagopal, Paul Fischer, Ellen Kuhl, Paul Steinmann
2010 Computational Mechanics  
We present a natural element method to treat higher-order spatial derivatives in the Cahn-Hilliard equation.  ...  The Cahn-Hilliard equation is a fourth-order nonlinear partial differential equation that allows to model phase separation in binary mixtures.  ...  Acknowledgments This project is part of the Baravia California Technology Center BaCaTeC Internationalization of High-Tech-Initiative and was initiated during a visit of Paul Steinmann as Timoshenko visiting  ... 
doi:10.1007/s00466-010-0490-4 fatcat:psqqp7rayvfcblr3hbg6m6lgwq

A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation [article]

Juan Manzanero, Gonzalo Rubio, David A. Kopriva, Esteban Ferrer, Eusebio Valero
2019 arXiv   pre-print
We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard equation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property.  ...  The latter permits us to show that the discrete free-energy is bounded, and as a result, the scheme is provably stable.  ...  This work was supported by a grant from the Simons Foundation (#426393, David Kopriva).  ... 
arXiv:1902.08089v1 fatcat:htlei64z5vdhpa5vhogyyba6ui

A projection-based, semi-implicit time-stepping approach for the Cahn-Hilliard Navier-Stokes equations on adaptive octree meshes [article]

Makrand A. Khanwale, Kumar Saurabh, Masado Ishii, Hari Sundar, James A. Rossmanith, Baskar-Ganapathysubramanian
2021 arXiv   pre-print
[A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes., arXiv:2009.06628 (2020)], to a block iterative  ...  We use a projection-based semi-implicit time discretization for the Navier-Stokes and a fully-implicit time discretization for the Cahn-Hilliard equation.  ...  Other related work: Han and Wang [29] used a block-iterative strategy with an energy-stable time scheme but with a non-linear scheme for Cahn-Hilliard.  ... 
arXiv:2107.05123v1 fatcat:3clzf6i2jngqta7t5ueoyuuyea

The Phase Field Method for Geometric Moving Interfaces and Their Numerical Approximations [article]

Qiang Du, Xiaobing Feng
2019 arXiv   pre-print
This paper surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs).  ...  Instead of describing technical details of various numerical methods and their analyses, the paper presents a holistic overview about the main ideas of phase field modeling, its mathematical foundation  ...  For example it is well-known [215, 11, 97] that the Cahn-Hilliard equations given in Example 6 is the H −1 -gradient flow of the Cahn-Hilliard free energy J := εJ .  ... 
arXiv:1902.04924v2 fatcat:tqpsb3quufa4bkelzhxcxqfsbe

Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation

Yan Xu, Chi-Wang Shu
2011 Communications in Computational Physics  
The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L2stability for general solutions.  ...  AbstractIn this paper, we develop, analyze and test local discontinuous Galerkin (LDG) methods for solving the Degasperis-Procesi equation which contains nonlinear high order derivatives, and possibly  ...  Figs. 3 and 4 are some of the simulation results of chaotic solutions in [99]. 3.3.3 The Cahn-Hilliard equation LDG methods have been designed in [93] for solving the Cahn-Hilliard equation in a bounded  ... 
doi:10.4208/cicp.2009.09.023 fatcat:jwudzqy6w5e3jcm6bpjjy4shey

Benchmark Problems for the Numerical Discretization of the Cahn–Hilliard Equation with a Source Term

Sungha Yoon, Hyun Geun Lee, Yibao Li, Chaeyoung Lee, Jintae Park, Sangkwon Kim, Hyundong Kim, Junseok Kim, Nikos I. Karachalios
2021 Discrete Dynamics in Nature and Society  
In this paper, we present benchmark problems for the numerical discretization of the CahnHilliard equation with a source term.  ...  To test numerical discretization, we present two benchmark problems: one is the growth of a disk or a sphere and the other is the growth of a rotated ellipse or a rotated ellipsoid.  ...  Wise, “A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver,” Discrete & Contin- uous Dynamical Systems-B  ... 
doi:10.1155/2021/1290895 fatcat:cqw6fvauyfheve5tz2srpvvdui

A pilot study of a phenomenological model of adipogenesis in maturing adipocytes using Cahn–Hilliard theory

F. J. Vermolen, A. Segal, A. Gefen
2011 Medical and Biological Engineering and Computing  
The process is modeled by the use of the Cahn-Hilliard equation, which is massconserving and allows the formation of secondary phases in the context of spinodal decomposition.  ...  It turns out that the Cahn-Hilliard theory can model many of the features during adipogenesis qualitatively. keywords Adipose Á Adipogenesis Á Spinodal decomposition Á Finite-element method  ...  Numerical procedure The Cahn-Hilliard equation To solve the Cahn-Hilliard equation, we use a standard Galerkin finite-element method with order reduction, that is the Cahn-Hilliard equation is split  ... 
doi:10.1007/s11517-011-0802-7 pmid:21761246 pmcid:PMC3223594 fatcat:6vchscgnmnbqhc7zrqkihhkari

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

Markus Kästner, Philipp Metsch, René de Borst
2016 Journal of Computational Physics  
This ensures the functionality of an adaptive time stepping scheme which is required for the efficient numerical solution of the Cahn-Hilliard equation.  ...  Using a manufactured solution, a mixed formulation of the Cahn-Hilliard equation and the direct discretisation of the weak form, which requires a C 1 -continuous approximation, are compared in terms of  ...  Kästner acknowledges financial support of the German  ... 
doi:10.1016/j.jcp.2015.10.047 fatcat:sq5y34b5tvgzhoayykmdcrdyjq

Entropy-stable discontinuous Galerkin approximation with summation-by-parts property for the incompressible Navier-Stokes/Cahn-Hilliard system [article]

Juan Manzanero, Gonzalo Rubio, David A. Kopriva, Esteban Ferrer, Eusebio Valero
2019 arXiv   pre-print
We develop an entropy stable two-phase incompressible Navier--Stokes/Cahn--Hilliard discontinuous Galerkin (DG) flow solver method.  ...  The model poses the Cahn-Hilliard equation as the phase field method, a skew-symmetric form of the momentum equation, and an artificial compressibility method to compute the pressure.  ...  This work was supported by a grant from the Simons Foundation (#426393, David Kopriva).  ... 
arXiv:1910.11252v2 fatcat:bsrtdnouojgcxduw6t3l2z6a34

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

Matthieu Brachet, Philippe Parnaudeau, Morgan Pierre
2022 Discrete and Continuous Dynamical Systems. Series S  
<p style='text-indent:20px;'>We review space and time discretizations of the Cahn-Hilliard equation which are energy stable.  ...  In many cases, we prove that a solution converges to a steady state as time goes to infinity. The proof is based on Lyapunov theory and on a Lojasiewicz type inequality.  ...  The authors are thankful to M. A. Jendoubi for helpful comments. The authors also thank the anonymous referee for his/her relevant questions which helped improve the quality of the manuscript.  ... 
doi:10.3934/dcdss.2022110 fatcat:7sfe7flgrjaqxhiaroiubcppi4

Finite element solution of nonlocal Cahn-Hilliard equations with feedback control time step size adaptivity [article]

Gabriel F. Barros, Adriano M. A. Côrtes, Alvaro L. G. A. Coutinho
2020 arXiv   pre-print
In this study, we evaluate the performance of feedback control-based time step adaptivity schemes for the nonlocal Cahn-Hilliard equation derived from the Ohta-Kawasaki free energy functional.  ...  We assess the performance of the adaptive schemes for the nonlocal Cahn-Hilliard equation in terms of the number of time steps required for the complete simulation and the computational effort measured  ...  Acknowledgements This research was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001.  ... 
arXiv:2009.14739v1 fatcat:e6mmkhpc6zgdbf3vlvlasjijaa
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