Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method

Steven Wise, Junseok Kim, John Lowengrub

Research output: Contribution to journalArticle

127 Citations (Scopus)

Abstract

We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn-Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently strong, there are missing orientations in the equilibrium level curves of the diffuse interface solutions, corresponding to those missing from the sharp interface Wulff shape, and the anisotropic Cahn-Hilliard equation becomes ill-posed. To regularize the equation, a higher-order derivative term is added to the energy. This leads to a sixth-order, nonlinear parabolic equation for the order parameter. An implicit time discretization is used to remove the high-order time step stability constraints. Dynamic block-structured Cartesian mesh refinement is used to highly resolve narrow interfacial layers. A multilevel, nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level. One of the keys to the success of the method is the treatment of the anisotropic term. This term is discretized in conservation form in space and is discretized fully implicitly in time. Numerical simulations are presented that confirm the accuracy, efficiency and stability of the scheme. We study the dynamics of interfaces under strong anisotropy and compare near-equilibrium diffuse interface solutions to the sharp interface Wulff shapes in 2D and 3D. We also simulate large-scale coarsening of a corrugated surface (in 3D) evolving by anisotropic surface diffusion. We show the emergence of long-range order during coarsening and an interesting mechanism of ordered coarsening.

Original languageEnglish
Pages (from-to)414-446
Number of pages33
JournalJournal of Computational Physics
Volume226
Issue number1
DOIs
Publication statusPublished - 2007 Sep 10
Externally publishedYes

Fingerprint

multigrid methods
Coarsening
Anisotropy
Surface diffusion
Interfacial energy
Nonlinear equations
Finite difference method
Conservation
anisotropy
Derivatives
surface diffusion
Computer simulation
nonlinear equations
surface energy
conservation
curves
simulation

Keywords

  • Adaptive mesh refinement
  • Cahn-Hilliard equation
  • Cartesian grid methods
  • Nonlinear multigrid methods
  • Regularization
  • Strong anisotropy

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy(all)

Cite this

Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method. / Wise, Steven; Kim, Junseok; Lowengrub, John.

In: Journal of Computational Physics, Vol. 226, No. 1, 10.09.2007, p. 414-446.

Research output: Contribution to journalArticle

@article{bd1c483ef6ba44f8b030d7c2e8dc56cc,
title = "Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method",
abstract = "We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn-Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently strong, there are missing orientations in the equilibrium level curves of the diffuse interface solutions, corresponding to those missing from the sharp interface Wulff shape, and the anisotropic Cahn-Hilliard equation becomes ill-posed. To regularize the equation, a higher-order derivative term is added to the energy. This leads to a sixth-order, nonlinear parabolic equation for the order parameter. An implicit time discretization is used to remove the high-order time step stability constraints. Dynamic block-structured Cartesian mesh refinement is used to highly resolve narrow interfacial layers. A multilevel, nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level. One of the keys to the success of the method is the treatment of the anisotropic term. This term is discretized in conservation form in space and is discretized fully implicitly in time. Numerical simulations are presented that confirm the accuracy, efficiency and stability of the scheme. We study the dynamics of interfaces under strong anisotropy and compare near-equilibrium diffuse interface solutions to the sharp interface Wulff shapes in 2D and 3D. We also simulate large-scale coarsening of a corrugated surface (in 3D) evolving by anisotropic surface diffusion. We show the emergence of long-range order during coarsening and an interesting mechanism of ordered coarsening.",
keywords = "Adaptive mesh refinement, Cahn-Hilliard equation, Cartesian grid methods, Nonlinear multigrid methods, Regularization, Strong anisotropy",
author = "Steven Wise and Junseok Kim and John Lowengrub",
year = "2007",
month = "9",
day = "10",
doi = "10.1016/j.jcp.2007.04.020",
language = "English",
volume = "226",
pages = "414--446",
journal = "Journal of Computational Physics",
issn = "0021-9991",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method

AU - Wise, Steven

AU - Kim, Junseok

AU - Lowengrub, John

PY - 2007/9/10

Y1 - 2007/9/10

N2 - We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn-Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently strong, there are missing orientations in the equilibrium level curves of the diffuse interface solutions, corresponding to those missing from the sharp interface Wulff shape, and the anisotropic Cahn-Hilliard equation becomes ill-posed. To regularize the equation, a higher-order derivative term is added to the energy. This leads to a sixth-order, nonlinear parabolic equation for the order parameter. An implicit time discretization is used to remove the high-order time step stability constraints. Dynamic block-structured Cartesian mesh refinement is used to highly resolve narrow interfacial layers. A multilevel, nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level. One of the keys to the success of the method is the treatment of the anisotropic term. This term is discretized in conservation form in space and is discretized fully implicitly in time. Numerical simulations are presented that confirm the accuracy, efficiency and stability of the scheme. We study the dynamics of interfaces under strong anisotropy and compare near-equilibrium diffuse interface solutions to the sharp interface Wulff shapes in 2D and 3D. We also simulate large-scale coarsening of a corrugated surface (in 3D) evolving by anisotropic surface diffusion. We show the emergence of long-range order during coarsening and an interesting mechanism of ordered coarsening.

AB - We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn-Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently strong, there are missing orientations in the equilibrium level curves of the diffuse interface solutions, corresponding to those missing from the sharp interface Wulff shape, and the anisotropic Cahn-Hilliard equation becomes ill-posed. To regularize the equation, a higher-order derivative term is added to the energy. This leads to a sixth-order, nonlinear parabolic equation for the order parameter. An implicit time discretization is used to remove the high-order time step stability constraints. Dynamic block-structured Cartesian mesh refinement is used to highly resolve narrow interfacial layers. A multilevel, nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level. One of the keys to the success of the method is the treatment of the anisotropic term. This term is discretized in conservation form in space and is discretized fully implicitly in time. Numerical simulations are presented that confirm the accuracy, efficiency and stability of the scheme. We study the dynamics of interfaces under strong anisotropy and compare near-equilibrium diffuse interface solutions to the sharp interface Wulff shapes in 2D and 3D. We also simulate large-scale coarsening of a corrugated surface (in 3D) evolving by anisotropic surface diffusion. We show the emergence of long-range order during coarsening and an interesting mechanism of ordered coarsening.

KW - Adaptive mesh refinement

KW - Cahn-Hilliard equation

KW - Cartesian grid methods

KW - Nonlinear multigrid methods

KW - Regularization

KW - Strong anisotropy

UR - http://www.scopus.com/inward/record.url?scp=34548460677&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548460677&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2007.04.020

DO - 10.1016/j.jcp.2007.04.020

M3 - Article

AN - SCOPUS:34548460677

VL - 226

SP - 414

EP - 446

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -