In this talk, we present spatial-temporally second-order, energy stable numerical schemes for two classes of hydrodynamic phase field models of binary viscous fluid mixtures of different densities. One is quasi-incompressible while the other is incompressible. We introduce a novel invariant energy qudratization (IEQ) technique to arrive at fully discrete linear schemes, where in each time step only a linear system needs to be solved. These schemes are then proved to be unconditionally energy stable rigorously so that a large time step is plausible. Both spatial and temporal mesh refinements are conducted to illustrate the second order accuracy of the schemes. Several Numerical examples and conducted, and predictions by the two fluid mixture models are compared and discussed. As a conclusion, we believe the quasi-incompressible model is more reliable than the incompressible one.
This is a joint work with Yuezheng Gong (Nanjing University of Aeronautics and Astronautics, China), Xiaogang Yang (Wuhan Institute of Technology, China) and Qi Wang (Beijing Computational Science Research Center, China).

After obtained his Bachelor's degree in Nankai University, Dr. Jia Zhao went to University of South Carolina, where he received his PHD in 2015. Afterward, he held a postdoc position at University of North Carolina at Chapel Hill for two years. Currently, he holds a tenure-track assistant professor position at Utah State University. Dr. Zhao has a keen interest in interdisciplinary research, aiming to strike a balance among mathematical modeling, numerical analysis, and high-performance computations. His research is highly interdisciplinary, residing at the interface between applied mathematics, scientific computing, soft matter physics, and mathematical biology. His current research projects include numerical analysis of thermodynamically consistent hydrodynamic models, modeling and computation of multiphase complex fluids and complex biological systems (biofilms, cell motility, and liquid-liquid phase separation in intracellular dynamics).