__Numerical methods in general setting:__

- Low rank structure of hyperbolic systems

*Sparse+Low-energy decomposition for viscous conservation laws*

with Thomas Y. Hou and Hayden Schaeffer, J. Comput. Phys.

__Uncertainties__

- Elliptic type

*Exploring the locally low dimensional structure in solving random elliptic PDEs*

with Thomas Y. Hou and Pengchuan Zhang, submitted

*A sparse decomposition of low rank symmetric positive semi-definite matrices*

with Thomas Y. Hou and Pengchuan Zhang, submitted

__Multiscale analysis__

- Kinetic equation and fluid limit
- Asymptotic preserving methods

*A BGK-penalization based asymptotic-preserving scheme for the multi-species Boltzmann equation*

with: Shi Jin, Numer. Methods Partial Diﬀerential Equations

*Exponential Runge–Kutta for the inhomogeneous Boltzmann equations with high order of accuracy*

with Lorenzo Pareschi, J. Comput. Phys.

*Exponential Runge-Kutta based AP scheme for multi-species Boltzmann equation*

with Xu Yang, Commun. Comput. Phys.

*Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy*

with Jingwei Hu and Lorenzo Pareschi, J. Sci. Comput.

*Numerical methods for plasma physics in collisional regimes*

with Giacomo Dimarco, Lorenzo Pareschi and Bokai Yan, invited review, J. Plasma Physics

*Exponential asymptotic preserving scheme for the Fokker-Planck-Landau equation*

with Giacomo Dimarco, Lorenzo Pareschi and Bokai Yan, in preparation

*Implicit Asymptotic Preserving Method for Linear Transport Equations*

with Li Wang, submitted

- Boundary layer analysis

*A convergent method for linear half-space kinetic equations*

with Jianfeng Lu and Weiran Sun, submitted

*Half-space Kinetic Equations with General Boundary Conditions*

with Jianfeng Lu and Weiran Sun, to appear, Math Comp

*Diffusion approximations of linear transport equations: asymptotics and numerics*

with Jianfeng Lu and Weiran Sun, J. Comput. Phys.

*Validity and regularization of classical half-space equations*

with Jianfeng Lu and Weiran Sun, submitted

- Asymptotic preserving methods
- Quantum system with crossing energies

*Semiclassical models for Schrodinger equations with periodic potentials and band-crossings*

with Lihui Chai and Shi Jin, Kinet. Relat. Models

*Semiclassical model for the surface hopping problem, multiband Wigner approach and domain decomposition method*

with Lihui Chai, Shi Jin and Omar Morandi, SIAM-Multiscale Model. Simul.

__Thesis:__

*Modeling and Computational Methods for Multi-scale Quantum Dynamics and Kinetic Equations*