A conservative nonlinear difference scheme for the viscous Cahn-Hilliard equation

2004 ◽  
Vol 16 (1-2) ◽  
pp. 53-68 ◽  
Author(s):  
S. M. Choo ◽  
S. K. Chung
Keyword(s):  
Difference Scheme ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
Nonlinear Difference Scheme

scholarly journals The Cahn–Hilliard Equation with Generalized Mobilities in Complex Geometries

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Jaemin Shin ◽  
Yongho Choi ◽  
Junseok Kim
Keyword(s):  
Growth Rate ◽  
Finite Difference ◽  
Difference Scheme ◽  
Pore Size ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Complex Geometries ◽  
Hilliard Equation ◽  
Controlled Porosity ◽  
Cahn Hilliard Equation

In this study, we apply a finite difference scheme to solve the Cahn–Hilliard equation with generalized mobilities in complex geometries. This method is conservative and unconditionally gradient stable for all positive variable mobility functions and complex geometries. Herein, we present some numerical experiments to demonstrate the performance of this method. In particular, using the fact that variable mobility changes the growth rate of the phases, we employ space-dependent mobility to design a cylindrical biomedical scaffold with controlled porosity and pore size.

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

2012 ◽  
Vol 11 (4) ◽  
pp. 1261-1278 ◽  
Author(s):  
Zhengru Zhang ◽  
Zhonghua Qiao
Keyword(s):  
Difference Scheme ◽  
Numerical Simulations ◽  
Large Time ◽  
Second Order ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Adaptive Time ◽  
Energy Stable ◽  
Cahn Hilliard Equation ◽  
Adaptive Time Stepping

AbstractThis paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the Cahn-Hilliard model needs very long time to reach the steady state, and therefore large time-stepping methods become useful. The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations. The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative. An error estimate for the numerical solution is also obtained with second order in both space and time. By using this energy stable scheme, an adaptive time-stepping strategy is proposed, which selects time steps adaptively based on the variation of the free energy against time. The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

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