A General Strategy for Numerical Approximations of Non-Equilibrium Models – Part I: Thermodynamical Systems

Authors

  • Jia Zhao Department of Mathematics & Statistics, Utah State University, Logan, UT, 84322, USA
  • Xiaofeng Yang Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
  • Yuezheng Gong College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P. R. China
  • Xueping Zhao Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
  • Xiaogang Yang School of Science, Wuhan Institute of Technology, Wuhan 430205, Hubei Province, P. R. China
  • Jun Li Tianjin Normal University, Tianjin, 100093, P R China
  • Qi Wang Beijing Computational Science Research Center, Beijing, 100193, P R China; Department of Mathematics, University of South Carolina, Columbia, SC 29028, USA

Keywords:

Energy stable schemes, nonequilibrium models, thermodynamically consistent models, energy quadratization.

Abstract

We present a general approach to deriving energy stable numerical approximations for thermodynamically 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.

Published

2018-08-15

Issue

Section

Articles