Thesis:
Modeling and Computational Methods for Multi-scale Quantum Dynamics and Kinetic Equations
Project finished:
BGK-penalization based asymptotic preserving scheme for multi-species Boltzmann equation
with: Shi Jin
We extended the idea proposed by F. Filbert and S. Jin in 2009 about using BGK operator to penalize Boltzmann collision term to multi-species system. The new difficulty comes from designing appropriate Maxwellian for BGK. It can be either defined by macro quantities averaged out of the entire system, or defined by those for each single species. For the second method, unlike one-species system, collision operator does not conserve momentum and energy for each species, so we need to solve the corresponding Euler system with the stiff source.
High order asymptotic accurate method for Boltzmann equation
with Lorenzo Pareschi
We consider the development of exponential methods for the robust time discretization of space inhomogeneous Boltzmann equations in stiff regimes. Compared to the space homo-geneous case, or more in general to the case of splitting based methods, studied in Dimarco Pareschi a major difficulty is that the local Maxwellian equilibrium state is not constant in a time step and thus needs a proper numerical treatment. We show how to derive asymptotic preserving (AP) schemes of arbitrary order and in particular using the Shu-Osher representation of Runge-Kutta methods we explore the monotonicity properties of such schemes, like strong stability preserving (SSP) and positivity preserving. Several numerical results confirm our analysis.
Semiclassical Models for Schrodinger Equations with Periodic Potentials and Band-crossings
with Lihui Chai and Shi Jin
We study the linear Schr\"odinger equation with a periodic potential in the semi-classical limit. When the so called Bloch band gap is small, the inter-band transition is significant but can not be described by the classical transport model. We derive a semi-classical Liouville system to capture the inter-band transition phenomena. This system can be seen as a first order approximation to the Wigner equation. A semi-classical hybrid model and the corresponding domain decomposition method are presented to solve this type of system efficiently.
Exponential Runge-Kutta based AP scheme for multi-species Boltzmann equation
with Xu Yang
We extend exponential RK method to multispecies Boltzmann equation. One needs to find an appropriate way to define the Maxwellian, and transform the original Boltzmann equation into an exponential form.
Paper in preparation:
Knudsen layer and boundary fluxes for multispecies Euler system
with Lihui Chai
The Knudsen layer of the Euler equation for multispecies system is studied. We address two issues, 1. the well-posedness of the half-space problem for multispecies linear Boltzmann equation and the numerical method to compute it; 2. coupling of the half-space linear Boltzmann equation and linear Euler equation. Based on these, a method to compute Euler equation with physical boundary that is imposed on phase space is carried out.
Semiclassical model for the surface hopping problem, multiband Wigner approach and domain decomposition method
with Lihui Chai, Shi Jin and Omar Morandi
Wigner transformation and Weyl quantization is used to address the surface hopping problem in quantum molecular dynamics. A domain decomposition method is naturally derived to handle the separation of the adiabatic and non-adiabatic approximation.
Before coming to UW-Madison
I played around different areas in applied math: I've done 4 months research with Dr. Xian-Sheng Hua in Microsoft Research Asia on advancing image searching engine by adopting and improving image sorting algorithms. I also spent one year working with Prof. Lei Guo on studying evolutionary network for bachelor thesis. I introduced a boolean evolution network to model a biological system, and analyzed the equilibrim state. I also did some numerical experiments corresponding to the analysis. Beyond that, I attended The Mathematical Contest in Modeling, in which, my collaborators and I modeled the ice melting process in antartica due to the increased global temperature, and measured its effects on land. We won Meritorious Prize.