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
N1 - Funding Information:
The authors thank Axel Voigt and Jean Taylor for illuminating and enjoyable discussions, and Sorin Mitran for making his BEARCLAW package available. SMW thanks Stephen Watson for pointing out several important references on the coarsening of faceted surfaces. The authors also gratefully acknowledge partial support from the University of Minnesota Office of Sponsored Projects, the National Science Foundation Division of Mathematical Sciences (DMS-0612878) and the Division of Materials Research (DMR-0606356 and DMR-0502737), and the European Community through grant STRP-016447.
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
U2 - 10.1016/j.jcp.2007.04.020
DO - 10.1016/j.jcp.2007.04.020
M3 - Article
AN - SCOPUS:34548460677
SN - 0021-9991
VL - 226
SP - 414
EP - 446
JO - Journal of Computational Physics
JF - Journal of Computational Physics
IS - 1
ER -