Time: Mar
24 , 2023

This part is selected from the work of the Department of Scientific and Engineering Computing (DSEC) which consists of twelve faculty members. Their research interests include the modeling, analysis and simulation of soft matter, model reduction of kinetic equation, numerical methods in fluid mechanics, finite element methods and fast solvers (such as multigrid) for discretized partial differential equations, adaptive mesh method, stochastic modeling and algorithms, numerical optimization and its application, numerical linear algebra, computational material sciences, image processing and image reconstruction. Over the past decade, PKU has seen some rather impressive progresses made in the above research fields. Some of them are listed as follows.

**1. Computable Modeling of Complex Fluids (ZHANG Pingwen)**

Complex fluids are common in nature and have important applications in everyday life and industry. The study of complex fluids is a rapidly expanding field and requires interdisciplinary approaches including experiments, physics and mathematics. Pingwen Zhang and his colleagues used mathematical modeling and numerical simulation to study static and dynamical properties of complex fluids. They focused on developing models for liquid crystals and polymers, understanding their mathematical properties and implementing efficient algorithms to numerically solve them.

**1) A Modeling Framework Bridging the Gap between Microscopic and Macroscopic**

Liquid crystals is a model system for complex fluids. On one hand, the Doi-Onsager framework gives an elegant description of the rod-like liquid crystals in the molecular level. On the other hand, phenomelogical models for liquid crystals are widely used by physicists and chemists. Examples include the Oseen-Frank model and Ericksen-Leslie model based on vector order parameter, and the Landau-de Gennes model based on tensor order parameter. The link between the microscopic level and macroscopic level has not been mathematically justified. To establish the connections between microscopic and macroscopic theories of liquid crystals, Zhang and his colleagues proposed a systematic modeling approach. Using analytical tools, they showed that the macroscopic model can be derived from molecular theory based on simple competing mechanism between inter-molecule interaction and thermal noises. By applying this approach to rod-like liquid crystals, important relationship between different order parameters and models has been revealed. Indeed, they proved axil-symmetry of equilibrium solutions to Onsager model with Maier-Saupe potential, which gives a mathematical justification of the assumption of axial-symmetry which is central to many macroscopic models for liquid crystals. Furthermore they gave a rigorous derivation for the famous Ericksen-Leslie theory starting from Doi-Onsager theory and derived a new tensor model based on Onsager's molecular model which can describe isotropic phase, nematic phase and smectic phase uniformly and can also recover both the Oseen-Frank and Ericksen-Leslie model. More details can be found in [Liu-Zhang- Zhang,Commun. Math. Sci.(2005); Wang-Zhang-Zhang,Comm. Pure Appl. Math.(2015); Han-Luo-Wang-Zhang,Arch. Ration. Mech. Anal.(2015)].