Energy stable and large time-stepping methods for the Cahn–Hilliard equation

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

Applied Numerical Mathematics ◽

10.1016/j.apnum.2006.07.026 ◽

2007 ◽

Vol 57 (5-7) ◽

pp. 616-628 ◽

Cited By ~ 121

Author(s):

Yinnian He ◽

Yunxian Liu ◽

Tao Tang

Keyword(s):

Large Time ◽

Time Stepping ◽

Hilliard Equation ◽

Cahn Hilliard Equation

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Towards Infinite Tilings with Symmetric Boundaries

Symmetry ◽

10.3390/sym11040444 ◽

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.

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

Communications in Computational Physics ◽

10.4208/cicp.scpde14.45s ◽

2016 ◽

Vol 19 (5) ◽

pp. 1542-1563 ◽

Cited By ~ 6

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.

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