Volume 15, Issue 6
A General Strategy for Numerical Approximations of Non-Equilibrium Models – Part I: Thermodynamical Systems

Jia Zhao, Xiaofeng Yang, Yuezheng Gong, Xueping Zhao, Xiaogang Yang, Jun LiQi Wang

Int. J. Numer. Anal. Mod., 15 (2018), pp. 884-918.

Published online: 2018-08

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  • Abstract

We present a general approach to deriving energy stable numerical approximations for thermodynamical consistent models for nonequilibrium phenomena. The central idea behind the systematic numerical approximation is the energy quadratization (EQ) strategy, where the system's free energy is transformed into a quadratic form by introducing new intermediate variables. By applying the EQ strategy, one can develop linear, high order semi-discrete schemes in time that preserve the energy dissipation property of the original thermodynamically consistent model equations. The EQ method is developed for time discretization primarily. When coupled with an appropriate spatial discretization, a fully discrete, high order, linear scheme can be developed to warrant the energy dissipation property of the fully discrete scheme. A host of examples for phase field models are presented to illustrate the effectiveness of the general strategy.

  • Keywords

Energy stable schemes nonequilibirum models thermodynamic consistent models energy quadratization.

  • AMS Subject Headings

65N30.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

jia.zhao@usu.edu (Jia Zhao)

xfyang@math.sc.edu (Xiaofeng Yang)

gongyuezheng@nuaa.edu.cn (Yuezheng Gong)

xpzhao@math.sc.edu (Xueping Zhao)

xgyang@wit.edu.cn (Xiaogang Yang)

nkjunli@hotmail.com (Jun Li)

qwang@math.sc.edu (Qi Wang)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-15-884, author = {Zhao , Jia and Yang , Xiaofeng and Gong , Yuezheng and Zhao , Xueping and Yang , Xiaogang and Li , Jun and Wang , Qi}, title = {A General Strategy for Numerical Approximations of Non-Equilibrium Models – Part I: Thermodynamical Systems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {15}, number = {6}, pages = {884--918}, abstract = {

We present a general approach to deriving energy stable numerical approximations for thermodynamical consistent models for nonequilibrium phenomena. The central idea behind the systematic numerical approximation is the energy quadratization (EQ) strategy, where the system's free energy is transformed into a quadratic form by introducing new intermediate variables. By applying the EQ strategy, one can develop linear, high order semi-discrete schemes in time that preserve the energy dissipation property of the original thermodynamically consistent model equations. The EQ method is developed for time discretization primarily. When coupled with an appropriate spatial discretization, a fully discrete, high order, linear scheme can be developed to warrant the energy dissipation property of the fully discrete scheme. A host of examples for phase field models are presented to illustrate the effectiveness of the general strategy.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12613.html} }
TY - JOUR T1 - A General Strategy for Numerical Approximations of Non-Equilibrium Models – Part I: Thermodynamical Systems AU - Zhao , Jia AU - Yang , Xiaofeng AU - Gong , Yuezheng AU - Zhao , Xueping AU - Yang , Xiaogang AU - Li , Jun AU - Wang , Qi JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 884 EP - 918 PY - 2018 DA - 2018/08 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12613.html KW - Energy stable schemes KW - nonequilibirum models KW - thermodynamic consistent models KW - energy quadratization. AB -

We present a general approach to deriving energy stable numerical approximations for thermodynamical consistent models for nonequilibrium phenomena. The central idea behind the systematic numerical approximation is the energy quadratization (EQ) strategy, where the system's free energy is transformed into a quadratic form by introducing new intermediate variables. By applying the EQ strategy, one can develop linear, high order semi-discrete schemes in time that preserve the energy dissipation property of the original thermodynamically consistent model equations. The EQ method is developed for time discretization primarily. When coupled with an appropriate spatial discretization, a fully discrete, high order, linear scheme can be developed to warrant the energy dissipation property of the fully discrete scheme. A host of examples for phase field models are presented to illustrate the effectiveness of the general strategy.

Jia Zhao, Xiaofeng Yang, Yuezheng Gong, Xueping Zhao, Xiaogang Yang, Jun Li & Qi Wang. (2020). A General Strategy for Numerical Approximations of Non-Equilibrium Models – Part I: Thermodynamical Systems. International Journal of Numerical Analysis and Modeling. 15 (6). 884-918. doi:
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