Changhui Tan
I am a postdoctoral research associate in CSCAMM and Department of Mathematics, University of Maryland.
An asymptotic preserving scheme for kinetic models with singular limit
Alina Chertock, Changhui Tan, and Bokai Yan
Kinetic and Related Models, Volume 11, No 4, pp. 735-756 (2018).
Abstract
We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors, and prove that the rescaled solution does not have a singular limit, under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We set up a few numerical experiments to demonstrate the accuracy, stability, efficiency and asymptotic preserving property of the schemes.
doi:10.3934/krm.2018030 | |
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Global regularity for the fractional Euler alignment system
Tam Do, Alexander Kiselev, Lenya Ryzhik, and Changhui Tan
Archive for Rational Mechanics and Analysis, Volume 228, No 1, pp. 1-37 (2018).
Abstract
We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker–Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian \((-\partial_{xx})^{\alpha/2}, \alpha\in(0,1)\). The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all \(\alpha\in(0,1)\). To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.
doi:10.1007/s00205-017-1184-2 | |
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Finite time blow up in the hyperbolic Boussinesq system
Alexander Kiselev, and Changhui Tan
Advances in Mathematics, Volume 325, pp. 34-55 (2018).
Abstract
In recent work of Luo and Hou, a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a two dimensional model that we call “hyperbolic Boussinesq system”. This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot–Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of initial data.
doi:10.1016/j.aim.2017.11.019 | |
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Application of natural neighbor interpolation method in three-dimensional geological model
Yan-jun Guo, Mao Pan, Fei Yan, Zhe Wang, Changhui Tan, and Tiao Lu
Journal of PLA University of Science and Technology (Natural Science Edition), Volume 26, No 1, pp. 185-206 (2016).
Abstract
To enhance the accuracy of three-dimensional geological model, emphasize the high local relevance characteristics of the complex geological bodies, and avoid complicated calculation and dependence on human experience in traditional interpolation methods, the natural neighbor interpolation (NNI) method was used for three-dimensional discrete data interpolation in the process of modeling. But the existing NNI method could not be applied to the boundary interpolation of finite fields, which was the most difficult problem of its application in three-dimensional geological modeling. Based on the geometry of Voronoi Cells and Delaunay Triangles, the shape function was constructed using non-Sibsonian (Laplace) interpolation method. The continuity of the boundary in NNI method was proven, the boundary interpolation was implemented and the computational complexity was reduced. The accuracy and validity of the method were proven by building the city geological model.
doi:10.3969/j.issn.1009-3443.2009.06.026 | |
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The paper is related to the undergraduate thesis: Numerical analysis and algorithm design in natural neighbor method.
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Hierarchical construction of bounded solutions to divU = F
Eitan Tadmor, and Changhui Tan
"Nonlinear Partial Differential Equations", Proceedings of the 2010 Abel Symposium held in Oslo, Sep. 2010 (H. Holden & K. Karlsen eds.), Abel Symposia 7, Springer 2011, 255-269.
Abstract
We implement the hierarchical decomposition introduced in [Ta15], to construct uniformly bounded solutions of the problem \(\nabla\cdot U = F\), where the two-dimensional data is in the critical regularity space, \(F\in L^2_{\#}(\mathbb{T}^2)\). Criticality in this context, manifests itself by the lack of linear mapping, \(F\in L^2_{\#}(\mathbb{T}^2)\to U\in L^{\infty}(\mathbb{T}^2,\mathbb{R}^2)\) [BB03]. Thus, the intriguing aspect here is that although the problem is linear, the construction of its uniformly bounded solutions is not.
doi:10.1007/978-3-642-25361-4_14 | |
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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.
doi:10.1016/j.physd.2016.03.011 | |
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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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