A Large Time-Stepping Mixed Finite Method of the Modified Cahn–Hilliard Equation

2020 ◽  
Vol 46 (6) ◽  
pp. 1551-1569
Author(s):  
Hongen Jia ◽  
Huanhuan Hu ◽  
Lingxiong Meng
Keyword(s):  
Large Time ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
Finite Method

On large time-stepping methods for the Cahn–Hilliard equation

2007 ◽  
Vol 57 (5-7) ◽  
pp. 616-628 ◽  
Author(s):  
Yinnian He ◽  
Yunxian Liu ◽  
Tao Tang
Keyword(s):  
Large Time ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

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.

Parameter-Free Time Adaptivity Based on Energy Evolution for the Cahn-Hilliard Equation

2016 ◽  
Vol 19 (5) ◽  
pp. 1542-1563 ◽  
Author(s):  
Fuesheng Luo ◽  
Tao Tang ◽  
Hehu Xie
Keyword(s):  
Large Time ◽  
Main Idea ◽  
Energy Evolution ◽  
Time Step ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Phase Field Models ◽  
Adaptive Time ◽  
Cahn Hilliard Equation ◽  
Adaptive Time Stepping

AbstractIt is known that large time-stepping method are useful for simulating phase field models. In this work, an adaptive time-stepping strategy is proposed based on numerical energy stability and equi-distribution principle. The main idea is to use the energy variation as an indicator to update the time step, so that the resulting algorithm is free of user-defined parameters, which is different from several existing approaches. Some numerical experiments are presented to illustrate the effectiveness of the algorithms.

scholarly journals Towards Infinite Tilings with Symmetric Boundaries

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 444
Author(s):  
Florian Stenger ◽  
Axel Voigt
Keyword(s):  
Finite Elements ◽  
Large Time ◽  
Time Discretization ◽  
Numerical Approach ◽  
Computational Effort ◽  
Special Boundaries ◽  
Hilliard Equation ◽  
Self Similar ◽  
Energy Stable ◽  
Cahn Hilliard Equation

Large-time coarsening and the associated scaling and statistically self-similar properties are used to construct infinite tilings. This is realized using a Cahn–Hilliard equation and special boundaries on each tile. Within a compromise between computational effort and the goal to reduce recurrences, an infinite tiling has been created and software which zooms in and out evolve forward and backward in time as well as traverse the infinite tiling horizontally and vertically. We also analyze the scaling behavior and the statistically self-similar properties and describe the numerical approach, which is based on finite elements and an energy-stable time discretization.

Unconditionally Gradient Stable Time Marching the Cahn-Hilliard Equation

1998 ◽  
Vol 529 ◽  
Author(s):  
David J. Eyre
Keyword(s):  
Numerical Methods ◽  
Linear Equations ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Time Marching ◽  
Cahn Hilliard Equation

AbstractNumerical methods for time stepping the Cahn-Hilliard equation are given and discussed. The methods are unconditionally gradient stable, and are uniquely solvable for all time steps. The schemes require the solution of ill-conditioned linear equations, and numerical methods to accurately solve these equations are also discussed.

scholarly journals On the sharp interface limit of a model for phase separation on biological membranes

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Abels ◽  
Johannes Kampmann
Keyword(s):  
Lipid Raft ◽  
Cell Membranes ◽  
System Modeling ◽  
Bulk Diffusion ◽  
Type Equation ◽  
Sharp Interface ◽  
Interface Problem ◽  
Hilliard Equation ◽  
System A ◽  
Cahn Hilliard Equation

AbstractWe rigorously prove the convergence of weak solutions to a model for lipid raft formation in cell membranes which was recently proposed in [H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci. 26 2016, 6, 1149–1189] to weak (varifold) solutions of the corresponding sharp-interface problem for a suitable subsequence. In the system a Cahn–Hilliard type equation on the boundary of a domain is coupled to a diffusion equation inside the domain. The proof builds on techniques developed in [X. Chen, Global asymptotic limit of solutions of the Cahn–Hilliard equation, J. Differential Geom. 44 1996, 2, 262–311] for the corresponding result for the Cahn–Hilliard equation.

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