Research Interests of Zhiliang Xu

My main research interests are in computational fluid dynamics, numerical methods for PDEs and interface methods. My current research interests in the computational biofluids are modeling and computational analysis of platelet aggregation in blood vessels. One approach to study the platelet aggregation is to use the continuum modeling. The other approach is to use the multiscale modeling. In either approach, computational challenges are due to the complex vessel geometry, the moving boundary formed by the aggregated platelets and the chemical reaction and diffusion process at different length scales.

My other research projects include to develop a uniformly high order accuracy interface tracking method for free surface flows with applications to a variety of hydro processes. High order accuracy numerical methods are essential for obtaining numerical solutions at long time. I mainly work on designing high order accuracy conservative front tracking method for both time-dependent Euler equations and incompressible Navier-Stokes equations. The front tracking method represents interfaces as lower dimensional meshes moving through a uniform Cartesian grid. We are interested in using high order accuracy methods including Discontinuous Galerkin (DG) finite element methods, ENO/WENO methods to obtain Cartesian grid solution conservatively.

Pictures from publications:

Fig 1.

Spike amplitude in the RM instability simulations, as functions of time.

The conservative tracked amplitude for a coarse grid is in approximate agreement with the non-conservative tracked amplitude for a fine grid.

From: James Glimm, Xiaolin Li, Yingjie Liu, Zhiliang Xu and Ning Zhao, Conservative Front Tracking with Improved Accuracy, Siam J. of Numerical Analysis, 41, No.5, 2003.

Fig 2.

Density and grid plots for the spherical RM simulation at time = 0.3.

The left frame shows the density plot from AMR front tracking, and the middle frame shows the density plot from uniform grid front tracking.

The right frame shows the AMR grid plots.

From: James Glimm Xiaolin Li and Zhiliang Xu, Front Tracking Algorithm Using Adaptively Refined Meshes. Proceedings of the 2003 Chicago Workshop on Adaptive Mesh Refinement Methods, Adaptive Mesh Refinement - Theory and Applications, the Lecture Notes in Computational Science and Engineering, ISSN: 1439-7358

Fig 3.

Plot of jet interface at late time with different bubble spacings

From: Zhiliang Xu, M. Kim, T.S. Lu and etc, al, Discrete Bubble Modeling of Unsteady Cavitating Flow, International J. forMultiscale Computational Engineering, accepted for publication (2006).