
Research
Here are the latest updates for Changhui Tan's research profile.
Here is the Curriculum Vitae and List of Publications.
First-order aggregation models with alignment
Razvan C. Fetecau, Weiran Sun, and Changhui Tan
Physica D: Nonlinear Phenomena, Volume 325, pp. 146-163 (2016).
Abstract
We include alignment interactions in a well-studied first-order attractive–repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the population density, becomes implicit, and can have non-unique solutions. We investigate the well-posedness of the model and show rigorously how it can be obtained as a macroscopic limit of a second-order kinetic equation. We work within the space of probability measures with compact support and use mass transportation ideas and the characteristic method as essential tools in the analysis. A discretization procedure that parallels the analysis is formulated and implemented numerically in one and two dimensions.
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doi:10.1016/j.physd.2016.03.011 |
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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.
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doi:10.1142/S0218202516500068 |
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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.
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doi:10.1137/140993430 |
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A discontinuous Galerkin method on kinetic flocking models
Changhui Tan
Mathematical Models and Methods in Applied Sciences, Volume 27, No 7, pp. 1199-1221 (2017).
Abstract
We study kinetic representations of flocking models. They arise from agent-based models for self-organized dynamics, such as Cucker–Smale [Emergent behaviors in flocks, IEEE Trans. Autom. Control. 52 (2007) 852–862] and Motsch–Tadmor [A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys. 144 (2011) 923– 947] models. We first establish a well-posedness theory and large-time flocking behavior for the kinetic systems, which indicates a concentration in velocity variable in infinite time. We then apply a discontinuous Galerkin method to treat the asymptotic \(\delta\)-singularity, and construct high-order positive-preserving schemes to solve kinetic flocking systems.
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doi:10.1142/S0218202517400139 |
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