Changhui Tan
I am a postdoctoral research associate in CSCAMM and Department of Mathematics, University of Maryland.
Gauss elimination
The purpose of this note is to clarify questions and statements in class, and a step-by-step way to find an LU decomposition by hand.
Polynomial interpolation
The purpose of this note is to clarify questions and statements in class, and to provide detailed procedures to find polynomial interpolations.
Carleson measure
This note is taken in the PDE discussion group in 2012, on the topic of important spaces in fluid dynamics.
Lecture 2: Carleson measure
An introduction to BMO Space
This note is taken in the PDE discussion group in 2012, on the topic of important spaces in fluid dynamics.
Lecture 1: An introduction to BMO Space
Ph.D. Defense
I have finished my Ph.D. defense today.
Committees
Prof. Eitan Tadmor (Chair/Advisor), Prof. Pierre-Emmanuel Jabin, Prof. Dave Levermore, Prof. Antoine Mellet and Prof. Howard Elman (Dean's representative).
My thesis title is Multi-scale problems on collective dynamics and image processing.
doi:10.13016/M2WG6T | |
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Critical thresholds in flocking hydrodynamics with nonlocal alignment
Eitan Tadmor, and Changhui Tan
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372.2028 (2014): 20130401.
Abstract
We study the large-time behavior of Eulerian systems augmented with non-local alignment. Such systems arise as hydrodynamic descriptions of agent-based models for self-organized dynamics, e.g. Cucker & Smale (2007 IEEE Trans. Autom. Control 52, 852–862. (doi:10.1109/TAC.2007.895842)) and Motsch & Tadmor (2011 J. Stat. Phys. 144, 923–947. (doi:10.1007/s10955-011-0285-9)) models. We prove that, in analogy with the agent-based models, the presence of non-local alignment enforces strong solutions to self-organize into a macroscopic flock. This then raises the question of existence of such strong solutions. We address this question in one- and two-dimensional set-ups, proving global regularity for subcritical initial data. Indeed, we show that there exist critical thresholds in the phase space of the initial configuration which dictate the global regularity versus a finite-time blow-up. In particular, we explore the regularity of non-local alignment in the presence of vacuum.
doi:10.1098/rsta.2013.0401 | |
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An exact rescaling velocity method for some kinetic flocking models.
Thomas Rey, and Changhui Tan
SIAM Journal on Numerical Analysis, Volume 54, No 2, pp. 641-664 (2016).
Abstract
In this work, we discuss kinetic descriptions of flocking models of the so-called Cucker–Smale [IEEE Trans. Automat. Control, 52 (2007), pp. 852–862] and Motsch–Tadmor [J. Statist. Phys., 144 (2011), pp. 923–947] types. These models are given by Vlasov-type equations where the interactions taken into account are only given long-range bi-particles interaction potentials. We introduce a new exact rescaling velocity method, inspired by the recent work [F. Filbet and T. Rey, J. Comput. Phys., 248 (2013) pp. 177–199], allowing us to observe numerically the flocking behavior of the solutions to these equations, without a need of remeshing or taking a very fine grid in the velocity space. To stabilize the exact method, we also introduce a modification of the classical upwind finite volume scheme which preserves the physical properties of the solution, such as momentum conservation.
doi:10.1137/140993430 | |
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Lovett instructor of Mathematics
I have recently accepted a 3-year Lovett instructor of Mathematics in Rice University. I will work with Professor Alex Kiselev.
Critical thresholds in 1D Euler equations with nonlocal forces
Jose A. Carrillo, Young-Pil Choi, Eitan Tadmor, and Changhui Tan
Mathematical Models and Methods in Applied Sciences, Volume 26, No 1, pp. 185-206 (2016).
Abstract
We study the critical thresholds for the compressible pressureless Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity in one dimension. We provide a complete description for the critical threshold to the system without interaction forces leading to a sharp dichotomy condition between global-in-time existence or finite-time blowup of strong solutions. When the interaction forces are considered, we also give a classification of the critical thresholds according to the different type of interaction forces. We also remark on global-in-time existence when the repulsion is modeled by the isothermal pressure law.
doi:10.1142/S0218202516500068 | |
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