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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I have recently accepted a 3-year Lovett instructor of Mathematics in Rice University. I will work with Professor Alex Kiselev. 

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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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.

	doi:10.1142/S0218202517400139
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