Pingwen Zhang received his Ph.D. in mathematics in 1992 under the supervision of Professor Long-An Ying at Peking University. He then joined Peking University's School of Mathematical Sciences as a faculty member. Since 1999, he serves as the Director of its Department of Scientific Computing and Engineering. In 2001, he became the Executive Deputy Director of the Center of Computational Science and Engineering, an institute dedicated to research in large-scale computation with the goal of developing a parallel computing platform.

Throughout his career, Pingwen Zhang has benefited from the work of many excellent collaborators, post-docs and graduate students. His research on vortex methods resulted in a book co-authored with his advisor. Zhenhuan Teng and Zhang advanced both the theoretical and numerical analysis of conservation laws. Applying the boundary integral methods, Zhang and Thomas Hou demonstrated the well-posedness for the linearized motion of 3D water waves and proved the convergence of their new algorithm. Together with Ruo Li and Tao Tang, Zhang developed a new moving mesh method based on a harmonic mapping and successfully applied it to many physical problems. Zhang, Weinan E and Pingbing Ming proved that the error in HMM simulations of the elliptic and parabolic homogenization problems is controlled by the standard error plus a new term called e(HMM). Collaborators Weinan E and Tiejun Li, Post-Doctoral Candidates Hui Zhang and Guanghua Ji and many excellent students have worked with Zhang on the multiscale modeling of complex fluids. This extensive effort examines many aspects of the theoretical and numerical analysis of phase separation, transition and nucleation of polymers, liquid crystal polymers, and membranes. Together these colleagues have written more than 70 scientific journal articles over the last 15 years.

Pingwen Zhang also actively serves the mathematical community as editor for several leading journals in applied and computational mathematics, including the SIAM Journal on Numerical Analysis, Journal of Computational Mathematics, Communications in Computational Physics and Communications in Mathematical Sciences.

Multiscale modeling of complex fluids predicts macroscopic properties and behavior from fundamental molecular processes by bridging size and time scales and linking computational processes. The difficulty of such simulations reflects the wide variation in size of polymeric structures, from nanometers to centimeters, and the time scale of dynamic process completion, from femtoseconds to hours for large scale ordering processes. No single model or simulation algorithm can handle such a range of size and time scales. Pingwen Zhang and his collaborators proposed a new kinetic model of liquid crystalline polymers (LCP), including nonhomogeneous extensions of Doi's kinetic theory. Using multiscale simulations based on their model and moment tensor models of LCP, they tried to understand the dynamics of defects, microstructure and pattern formulations induced by flow. By applying liquid crystal theories, they derived full dynamic equations for membranes, incorporating elastic and viscous effects in both the presence and absence of bulk fluids. Nucleation is an important topic in the physics of phase transitions of first order. Zhang and his collaborators developed a numerical method based on the string method to compute the size and shape of the critical droplet and the free-energy barrier between two ordered phases. They also studied the phenomena of visco-elastic phase separation in polymer systems and the dynamic model coupling crystallization and phase separation. The scientific contributions of their project are:

- By studying the structure of stationary solutions to the Doi-Onsager equation with Maier-Saupe potential on the sphere, they demonstrated that the stationary solutions were a set of axially symmetric functions. They also developed a complete classification of phase transition parameters and presented explicit formulas for all stationary solutions.
- They proved local well-posedness for the FENE models coupling fluids and polymers.
- Using the extended Doi model for nematic LCPs in planar shear flow (inhomogeneous in shear direction), they studied the formation of microstructure, pattern formulations and the dynamics of defects. They also proved numerically that the Bingham closure model, which maintains energy dissipation, is reliable. Leslie coefficients and Ericksen stress are two important parameters in the Eriksen-Leslie equations. M. Doi and N. Kuzuu found that the Leslie coefficients can be expressed in terms of the molecular parameters. Pingwen Zhang and his collaborators performed an asymptotic analysis systemically for the Doi model at the small Deborah case. They found that the Ericksen stress can be expressed in terms of a body force.

The adaptive mesh technique has proven very powerful in large scale scientific and engineering computing. Ruo Li, Tao Tang and Pingwen Zhang developed a moving mesh strategy based on a harmonic mapping, which makes the coordinate transformation from the logical to the physical domain. Their moving mesh algorithm has been successfully applied in many computational problems, for example, solving PDEs on non-Euclidean manifolds, simulating incompressible and compressible fluid flow, and solving multi-phase flow problems. The significant contributions of this technique are:

- Their new algorithm updates interior and boundary grids simultaneously and maintains mesh quality, effectively avoiding mesh entanglement. As a result, it can handle physical problems which involve large solution gradients or singularities.
- It also preserves the physical conservation properties at each mesh redistribution stage. Although the algorithm is simple, it has proven very useful. Many researchers are now using it to accurately simulate many physical problems involving mass conservation, energy conservation, divergence free properties, etc.
- Previous methods used explicit coordinate transformations, and strongly coupled the PDEs with the moving mesh equation. These methods cannot easily simulate practical problems in higher dimensional spaces. By separating the PDE solver from the mesh redistribution, Zhang and his collaborators have been able to implement the moving mesh algorithm as a distributable package. This facilitates its use by other scientists.

The heterogeneous multiscale method (HMM) for elliptic and parabolic homogenization problems is a general methodology for designing sublinear algorithms by exploiting scale separation. Pingwen Zhang and his collaborators proved that the error between the numerical solution of HMM and the real solution to the elliptic and parabolic homogenization problems are controlled by the standard error plus a new term called e(HMM). They obtained an estimate of e(HMM) for periodic and random homogenization problems. This overall strategy should prove useful when applying other multiscale methods across a wide variety of problems.

Pingwen Zhang and his collaborators have developed a new algorithm to simulate the boundary integral equations applicable to 3D water wave problems. Used to study the fluid dynamical instabilities associated with free interface problems, the boundary integral methods contain a kernel with a non-removable branch point singularity in the 3D case. This makes it more difficult to prove the stability of this case relative to the corresponding 2D problem. The well-posedness of 3D water waves relies on the subtle balance among the spectral properties of various singular integral operators. As a consequence, slight variations in this balance make it susceptible to numerical instability. Thomas Hou and Pingwen Zhang demonstrated the well-posedness for the linearized motion of 3D water waves, and proved the convergence of their new algorithm. A similar approach was used to analyze the singularity formulation of 3D vortex sheets.

Zhenhuan Teng and Pingwen Zhang proved some crucial results on the theoretical analysis and practical computation of conservation laws. They established the optimal error bounds for both the viscosity method and the monotone difference scheme satisfying the entropy conditions for the initial-value problem of scalar conservation laws. This was proven for the initial data with a finite number of piecewise constants. Tao Tang has extended this work to piecewise smooth initial data. The results are improvements over the half-order rates of the L1-convergence.

Pingwen Zhang's Ph.D. dissertation focused on numerical analysis of vortex methods. The vortex method can be traced back to 1931 when Rosenhead evaluated two-dimensional flow with a point vortex method. Chorin's 1973 paper on numerical studies of slightly viscous flow stimulated a large scientific effort. Pingwen Zhang and his collaborators derived a general formulation of the variable-elliptic-vortex method for the incompressible Euler equations and proved its consistency, stability and convergence. Moreover, they proved the fully discrete convergence for vortex methods in bounded domains. A comprehensive mathematical study of vortex methods can be found in Long-An Ying and Pingwen Zhang's book "Vortex Methods".