Volume 11, Issue 4
An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

Zhengru Zhang & Zhonghua Qiao

Commun. Comput. Phys., 11 (2012), pp. 1261-1278.

Published online: 2012-04

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  • Abstract

This 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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@Article{CiCP-11-1261, author = {}, title = {An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation}, journal = {Communications in Computational Physics}, year = {2012}, volume = {11}, number = {4}, pages = {1261--1278}, abstract = {

This 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.300810.140411s}, url = {http://global-sci.org/intro/article_detail/cicp/7410.html} }
TY - JOUR T1 - An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation JO - Communications in Computational Physics VL - 4 SP - 1261 EP - 1278 PY - 2012 DA - 2012/04 SN - 11 DO - http://dor.org/10.4208/cicp.300810.140411s UR - https://global-sci.org/intro/article_detail/cicp/7410.html KW - AB -

This 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.

Zhengru Zhang & Zhonghua Qiao. (2020). An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation. Communications in Computational Physics. 11 (4). 1261-1278. doi:10.4208/cicp.300810.140411s
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