Models for thermodynamic systems and hydrodynamic systems can be derived using a nonequilibrium thermodynamic paradigm known as the generalized Onsager principle [2]. The models developed under this framework possess energy dissipative properties and variational structures. In order to solve the partial differential equations in the models, numerical methods should be designed in such a way to respect energy dissipation as wel as the variational structure. Such methods are called energy stable schemes. We have developed a fairly general strategy to develop energy stable numerical schemes for models derived from the generalized Onsager principle. The strategy has been applied to various model systems ranging from multiphase fluid flows to multiphase phase field models.

When the system is subject to periodic boundary conditions, we have developed fully discretized energy stable schemes for multiphase and liquid crystal fluid flows. For physical boundary conditions, we have energy stable schemes for semi-discretized systems. In some of the works, a new method, which we call the energy quadratization technique, was introduced to arrive at linear energy stable schemes. Both first order and second order in time schemes are developed [1, 3-17]. These methods are shown to be more accurate and efficient compared with the non-energy stable schemes.

In [17], we compare the model prediction of an incompressible binary fluid mixture flow model with a more physically sound quasi-incompressible model. The incompressible model gives a prediction that is independent of the density ratio of the two fluid components, while the quasi-incompressible model shows the sensitivity to the density ratio. The deviation is drastic when the density ratio is large, which directly cast a doubt on the appropriateness of using the incompressible model for mixture fluid flows. Fig 1 depicts the drop

Figure 1. Drop dynamics subject to gravity. A heavy fluid drop is falling under the influence of gravity. The top panel is the model prediction using a quasi-incompressible model, while the bottom one is the one using an incompressible model.

Figure 2. Drop dynamics without the influence of gravity. The top three panels are the model prediction using the quasi-incompressible model with three different density ratios. The bottom two panels are the model prediction using the incompressible model with two differential density ratios, where the model does not discern the difference of the density ratio.

**References:**

1.Jia Zhao and Qi Wang, “Semi-Discrete Energy-Stable Schemes for a Tensor-Based Hydrodynamic Model of Nematic Liquid Crystal Flows.” **Journal of Scientific Computing**, 68(3), 2016, 1241-1266.

2.Xiaogang Yang, Jun Li, M. G. Forest, and Qi Wang, “Hydrodynamic Theories for Flows of Active Liquid Crystals and the Generalized Onsager Principle”, **Entropy**, 2016, 18, 202; doi:10.3390/e18060202.

3. Jia Zhao, Xiaofeng Yang, Jun Li and Qi Wang, “Energy stable numerical schemes for a hydrodynamic model of nematic liquid crystals.” **Siam J. Sci. Comp**., in press, 2016.

4. Yuezheng Gong, Xinfeng Liu, and Qi Wang, “Fully Discretized Energy Stable Schemes for Hydrodynamic Models of Two-phase Viscous Fluid Flows”, **Journal of Scientific Computing, **2016, DOI10.1007/s10915-016-0224-7**. **

5.Jia Zhao, Qi Wang, and Xiaofeng Yang, “Numerical Approximations to a New Phase Field Model for Immiscible Mixtures of Nematic Liquid Crystals and Viscous Fluids”, **Computer Methods in Applied Mechanics and Engineering**, in press 2016.

6.Jia Zhao, Huiyuan Li, Qi Wang, and Xiaofeng Yang, “A Linearly Decoupled Energy Stable Scheme for Phase Field Models of Three-phase Incompressible Viscous Fluid Flows”, **Journal of Scientific Computing**, in press, 2016.

7.Xiaofeng Yang, Jia Zhao, and Qi Wang, “Numerical Approximations for a phase field dendritic Growth Model Based on the Invariant Energy Quadratization Approach,” **International journal for Numerical Methods in Engineering**, in press, 2016.

8.Yuezheng Gong, Qi Wang, Yushun Wang, Jiaxiang Cai, “A conservative Fourier pseudospectral method for the nonlinear Schrodinger equation”, **Journal of Computational Physics**, in press, 2016.

9.Yuezheng Gong, Qi Wang, and Zhu Wang, Structure-Preserving Galerkin POD Reduced-Order Modeling of Hamiltonian Systems, **Computer Methods in Mechanics and Engineering,** 315, 2017, pp.780-798.

10.Xiaofeng Yang, Jia Zhao, and Qi Wang, “Linear and Unconditionally Energy Stable Schemes for Molecular Beam Epitaxial Growth Model Based on Invariant Energy Quadratization Methods,” **Journal of** **Computational Physics**, in press, 2016.

11. Jia Zhao, Xiaofeng Yang, Yuezheng Gong, and Qi Wang, A Novel Linear Second Order Unconditionally Energy-stable Scheme for a Hydrodynamic Q-tensor Model of Liquid Crystals, submitted to **Computer Methods in Applied Mechanics and Engineering**, 2016.

12. Yuezheng Gong, Jia Zhao, and Qi Wang, Linear Second Order in Time Energy Stable Schemes for Hydrodynamic Models of Binary Mixtures Based on a Spatially Pseudospectral Approximation，submitted to **Advances in Computational Mathematics**, 2016.

13. Xiaofeng Yang, Jia Zhao, and Qi Wang, Second Order Semi-discretized Numerical Schemes for Three-phase Fluid Mixtures, submitted to **Computer Methods in Applied Mechanics and Engineering**, 2016.

14. Yuezheng Gong, Jia Zhao, and Qi Wang, A Second Order Linear Energy Stable Scheme for the Hydrodynamic Model of Two-phase Quasi-incompressible Viscous Fluid Flows, submitted to **Journal of Scientific Computing**, 2016.

15. Xiaofeng Yang, Jia Zhao, Qi Wang, Jie Shen, Numerical Approximations for a three-component Cahn-Hilliard phase-field Model based on the Invariant Energy Quadratization method, submitted to **M3AS**, 2016.

16. Yuezheng Gong, Jia Zhao, and Qi Wang, An Energy Stable Algorithm for the Quasi-incompressible Hydrodynamic Model of Viscous Fluid Mixtures, submitted to **Computer Physics Communications**, 2016.

**17.** Yuezheng Gong, Xiaogang Yang, Jia Zhao, and Qi Wang, On Spatial-temporal Second-order Linear Schemes for Hydrodynamic Phase field Models of Viscous Fluid Flows with Variable Densities, to be submitted to **Mathematical Models and Methods in Applied Sciences, 2016.**