We have developed a general approach to deriving energy stable numerical approximations for thermodynamically consistent models for nonequilibrium phenomena. Thermodynamically consistent models are the ones that satisfy the thermodynamical laws, especially, the second law. These models include many well-known ones in materials science, physics, fluid dynamics and rheology etc.

The central idea behind the systematic numerical approximation is the energy quadratization (EQ) strategy, where the system’s free energy (or entropy) is transformed into a quadratic form by introducing new intermediate (or auxiliary) variables. This process is known as the EQ-reformulation. It in fact embeds the thermodynamical model into a higher dimensional variable space where the energy functional (or the entropy functional) is a quadratic functional, and in the meantime, the dissipative structure of the model system is fully retained.

For the equivalent thermodynamically consistent model with a quadratic energy, one can develop linear, high order semi-discrete schemes in time that preserve the energy dissipation property of the original thermodynamically consistent model equations in the semi-discrete sense. When coupled with an appropriate spatial discretization, for instance, finite element methods, spectral methods and finite difference methods that respect the discrete summation-by-parts rule, a fully discrete, high order, linear scheme can be developed to warrant the energy dissipation property of the fully discrete scheme. A host of energy stable numerical schemes for phase field models and other thermodynamically consistent models have been published in the literature which are merely examples of the general approach.

**References:**

Jia Zhao, Xiaofeng Yang, Yuezheng Gong, Xueping Zhao, Jun Li, Xiaogang Yang and Qi Wang, “A General Strategy for Numerical Approximations of Thermodynamically Consistent Nonequilibrium Models--Part I: Thermodynamical Systems”, *International Journal of Numerical Analysis and Modeling*, 15(16) (2018), pp 884-918.