Research Interests

My main research interests are computational and mathematical biology, and numerical methods for PDEs on structured and unstructured meshes. My current research interests in the computational and mathematical biology are modeling and computational analysis of morphogen gradient formation in developmental biology. One is to study the dorsal-ventral patterning in Drosophila and Zebrafish embryos. A morphogen is a substance whose non-uniform distribution in a field of cells differentially determines the fate and phenotype of those cells. During the embryo development of both vertebrates and invertebrates, the bone morphogenetic protein (BMP) binding with cell receptors acts as a morphogen to induce the dorsal-ventral patterning. The BMP activity gradient needs the interaction of a multi-protein network to regulate and create its unusual shape. The central questions are: how those ligands cooperate to produce the desired pattern in the Drosophila and Zebrafish embryos; how the BMP gradient achieves its robustness. The other study is on the formation of the skeletal pattern in a growing embryonic vertebrate limb. Computational challenges are due to the complex high dimensional geometry of the embryos and the stiff reaction-diffusion systems, with a moving boundary.

Efficient, high accuracy and easy implementation numerical methods are essential for the computational analysis of complex biological and other scientific problems. I mainly work on designing high accuracy and efficient numerical methods for both time-dependent and steady states of reaction-advection-diffusion equations and convection dominated PDEs, on both structured and unstructured meshes. The numerical schemes I am currently working on include Discontinuous Galerkin (DG) finite element methods, high order Weighted ENO methods, fast sweeping methods and integrating factor methods.

Selected Publications

• Y.-T. Zhang and C.-W. Shu, High order WENO schemes for Hamilton-Jacobi equations on triangular meshes, SIAM Journal on Scientific Computing, v24 ((2003), pp. 1005-1030.
• Y.-T. Zhang, J. Shi, C.-W. Shu and Y. Zhou, Numerical viscosity and resolution of high-order weighted essentially nonoscillatory schemes for compressible flows with high Reynolds numbers, Physical Review E, v68 (2003), 046709.
• J. Shi, Y.-T. Zhang and C.-W. Shu, Resolution of high order WENO schemes for complicated flow structures, Journal of Computational Physics, v186 (2003), pp. 690-696.
• S. Zhang, Y.-T. Zhang and C.-W. Shu, Multi-stage interaction of a shock wave and a strong vortex, Physics of Fluids, to appear, (2005).
• Y.-T. Zhang, H.-K. Zhao and J. Qian, High order fast sweeping methods for static Hamilton-Jacobi equations, Journal of Scientific Computing, to appear, (2005).