Volume 35, Issue 1
Energy Stable Numerical Method for the TDGL Equation with the Reticular Free Energy in Hydrogel

Dong Liao, Hui ZhangZhengru Zhang

J. Comp. Math., 35 (2017), pp. 37-51.

Published online: 2017-02

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  • Abstract

Here we focus on the numerical simulation of the phase separation about macromolecule microsphere composite (MMC) hydrogel. The model is based on time-dependent Ginzburg-Landau (TDGL) equation with the reticular free energy. An unconditionally energy stable difference scheme is proposed based on the convex splitting of the corresponding energy functional. In the numerical experiments, we observe that simulating the whole process of the phase separation requires a considerably long time. We also notice that the total free energy changes significantly in initial stage and varies slightly in the following time. Based on these properties, we apply the adaptive time stepping strategy to improve the computational efficiency. It is found that the application of time step adaptivity can not only resolve the dynamical changes of the solution accurately but also significantly save CPU time for the long time simulation.

  • Keywords

TDGL equation, Unconditionally energy stable scheme, Adaptive time-stepping method, Phase transition.

  • AMS Subject Headings

65M06, 65M12, 65Z05.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

7775024@163.com (Dong Liao)

hzhang@bnu.edu.cn (Hui Zhang)

zrzhang@bnu.edu.cn (Zhengru Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-37, author = {Liao , Dong and Zhang , Hui and Zhang , Zhengru}, title = {Energy Stable Numerical Method for the TDGL Equation with the Reticular Free Energy in Hydrogel}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {1}, pages = {37--51}, abstract = {

Here we focus on the numerical simulation of the phase separation about macromolecule microsphere composite (MMC) hydrogel. The model is based on time-dependent Ginzburg-Landau (TDGL) equation with the reticular free energy. An unconditionally energy stable difference scheme is proposed based on the convex splitting of the corresponding energy functional. In the numerical experiments, we observe that simulating the whole process of the phase separation requires a considerably long time. We also notice that the total free energy changes significantly in initial stage and varies slightly in the following time. Based on these properties, we apply the adaptive time stepping strategy to improve the computational efficiency. It is found that the application of time step adaptivity can not only resolve the dynamical changes of the solution accurately but also significantly save CPU time for the long time simulation.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1607-m2014-0109}, url = {http://global-sci.org/intro/article_detail/jcm/9762.html} }
TY - JOUR T1 - Energy Stable Numerical Method for the TDGL Equation with the Reticular Free Energy in Hydrogel AU - Liao , Dong AU - Zhang , Hui AU - Zhang , Zhengru JO - Journal of Computational Mathematics VL - 1 SP - 37 EP - 51 PY - 2017 DA - 2017/02 SN - 35 DO - http://doi.org/10.4208/jcm.1607-m2014-0109 UR - https://global-sci.org/intro/article_detail/jcm/9762.html KW - TDGL equation, Unconditionally energy stable scheme, Adaptive time-stepping method, Phase transition. AB -

Here we focus on the numerical simulation of the phase separation about macromolecule microsphere composite (MMC) hydrogel. The model is based on time-dependent Ginzburg-Landau (TDGL) equation with the reticular free energy. An unconditionally energy stable difference scheme is proposed based on the convex splitting of the corresponding energy functional. In the numerical experiments, we observe that simulating the whole process of the phase separation requires a considerably long time. We also notice that the total free energy changes significantly in initial stage and varies slightly in the following time. Based on these properties, we apply the adaptive time stepping strategy to improve the computational efficiency. It is found that the application of time step adaptivity can not only resolve the dynamical changes of the solution accurately but also significantly save CPU time for the long time simulation.

Dong Liao, Hui Zhang & Zhengru Zhang. (2020). Energy Stable Numerical Method for the TDGL Equation with the Reticular Free Energy in Hydrogel. Journal of Computational Mathematics. 35 (1). 37-51. doi:10.4208/jcm.1607-m2014-0109
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