Spectral method for a kinetic swarming model
Abstract
Here we present the first numerical method for a kinetic description of the Vicsek swarming model. The kinetic model poses a unique challenge, as there is a distribution dependent collision invariant to satisfy when computing the interaction term. We use a spectral representation linked with a discrete constrained optimization to compute these interactions. To test the numerical scheme we investigate the kinetic model at different scales and compare the solution with the microscopic and macroscopic descriptions of the Vicsek model. Lastly, we observe that the kinetic model captures key features such as vortex formation and traveling waves.
- Authors:
-
[3]
- Univ. of Texas at Austin, Austin, TX (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Arizona State Univ., Tempe, AZ (United States)
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1254839
- Report Number(s):
- LA-UR-15-20570
Journal ID: ISSN 0021-9991; PII: S0021999115002892
- Grant/Contract Number:
- AC52-06NA25396; 11-07444; 11-07465
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 297; Journal Issue: C; Journal ID: ISSN 0021-9991
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Spectral method; Finite Volume method; Kinetic equation; Vicsek model; Hyperbolic systems
Citation Formats
Gamba, Irene M., Haack, Jeffrey R., and Motsch, Sebastien. Spectral method for a kinetic swarming model. United States: N. p., 2015.
Web. doi:10.1016/j.jcp.2015.04.033.
Gamba, Irene M., Haack, Jeffrey R., & Motsch, Sebastien. Spectral method for a kinetic swarming model. United States. https://doi.org/10.1016/j.jcp.2015.04.033
Gamba, Irene M., Haack, Jeffrey R., and Motsch, Sebastien. Tue .
"Spectral method for a kinetic swarming model". United States. https://doi.org/10.1016/j.jcp.2015.04.033. https://www.osti.gov/servlets/purl/1254839.
@article{osti_1254839,
title = {Spectral method for a kinetic swarming model},
author = {Gamba, Irene M. and Haack, Jeffrey R. and Motsch, Sebastien},
abstractNote = {Here we present the first numerical method for a kinetic description of the Vicsek swarming model. The kinetic model poses a unique challenge, as there is a distribution dependent collision invariant to satisfy when computing the interaction term. We use a spectral representation linked with a discrete constrained optimization to compute these interactions. To test the numerical scheme we investigate the kinetic model at different scales and compare the solution with the microscopic and macroscopic descriptions of the Vicsek model. Lastly, we observe that the kinetic model captures key features such as vortex formation and traveling waves.},
doi = {10.1016/j.jcp.2015.04.033},
journal = {Journal of Computational Physics},
number = C,
volume = 297,
place = {United States},
year = {Tue Apr 28 00:00:00 EDT 2015},
month = {Tue Apr 28 00:00:00 EDT 2015}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 11 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
A Nonlocal Continuum Model for Biological Aggregation
journal, July 2006
- Topaz, Chad M.; Bertozzi, Andrea L.; Lewis, Mark A.
- Bulletin of Mathematical Biology, Vol. 68, Issue 7
A Well-Posedness Theory in Measures for some Kinetic Models of Collective Motion
journal, March 2011
- CaÑIzo, J. A.; Carrillo, J. A.; Rosado, J.
- Mathematical Models and Methods in Applied Sciences, Vol. 21, Issue 03
Numerical Simulations of a Nonconservative Hyperbolic System with Geometric Constraints Describing Swarming Behavior
journal, July 2011
- Motsch, Sebastien; Navoret, Laurent
- Multiscale Modeling & Simulation, Vol. 9, Issue 3
Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator
journal, January 2000
- Pareschi, Lorenzo; Russo, Giovanni
- SIAM Journal on Numerical Analysis, Vol. 37, Issue 4
Numerical methods for kinetic equations
journal, May 2014
- Dimarco, G.; Pareschi, L.
- Acta Numerica, Vol. 23
Fast deterministic method of solving the Boltzmann equation for hard spheres
journal, September 1999
- Bobylev, A. V.; Rjasanow, S.
- European Journal of Mechanics - B/Fluids, Vol. 18, Issue 5
Flocks, herds, and schools: A quantitative theory of flocking
journal, October 1998
- Toner, John; Tu, Yuhai
- Physical Review E, Vol. 58, Issue 4
Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states
journal, April 2009
- Gamba, Irene M.; Tharkabhushanam, Sri Harsha
- Journal of Computational Physics, Vol. 228, Issue 6
A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit
journal, August 2014
- Gamba, Irene M.; Haack, Jeffrey R.
- Journal of Computational Physics, Vol. 270
Diffusion in a Continuum Model of Self-Propelled Particles with Alignment Interaction
journal, September 2010
- Degond, Pierre; Yang, Tong
- Mathematical Models and Methods in Applied Sciences, Vol. 20, Issue supp01
Collective Memory and Spatial Sorting in Animal Groups
journal, September 2002
- Couzin, Iain D.; Krause, Jens; James, Richard
- Journal of Theoretical Biology, Vol. 218, Issue 1
Study of rarefied shear flow by the discrete velocity method
journal, July 1964
- Broadwell, James E.
- Journal of Fluid Mechanics, Vol. 19, Issue 3
A simulation study on the schooling mechanism in fish.
journal, January 1982
- Aoki, Ichiro
- NIPPON SUISAN GAKKAISHI, Vol. 48, Issue 8
Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study
journal, January 2008
- Ballerini, M.; Cabibbo, N.; Candelier, R.
- Proceedings of the National Academy of Sciences, Vol. 105, Issue 4
Shock and Boundary Structure Formation by Spectral-Lagrangian Methods for the Inhomogeneous Boltzmann Transport Equation
journal, June 2010
- Tharkabhushanam, Irene M. Gamba and Sri Harsha
- Journal of Computational Mathematics, Vol. 28, Issue 4
State transitions and the continuum limit for a 2D interacting, self-propelled particle system
journal, August 2007
- Chuang, Yao-li; D’Orsogna, Maria R.; Marthaler, Daniel
- Physica D: Nonlinear Phenomena, Vol. 232, Issue 1
Self-Organized Fish Schools: An Examination of Emergent Properties
journal, June 2002
- Parrish, Julia K.; Viscido, Steven V.; Grünbaum, Daniel
- The Biological Bulletin, Vol. 202, Issue 3
Analysis of spectral methods for the homogeneous Boltzmann equation
journal, April 2011
- Filbet, Francis; Mouhot, Clément
- Transactions of the American Mathematical Society, Vol. 363, Issue 04
Flocks, herds and schools: A distributed behavioral model
journal, August 1987
- Reynolds, Craig W.
- ACM SIGGRAPH Computer Graphics, Vol. 21, Issue 4
Continuum Limit of Self-Driven Particles with Orientation Interaction
journal, August 2008
- Degond, Pierre; Motsch, SÉBastien
- Mathematical Models and Methods in Applied Sciences, Vol. 18, Issue supp01
Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming
journal, September 2013
- Bostan, Mihai; Carrillo, Jose Antonio
- Mathematical Models and Methods in Applied Sciences, Vol. 23, Issue 13
A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics
journal, January 1996
- Buet, C.
- Transport Theory and Statistical Physics, Vol. 25, Issue 1
A Fourier spectral method for homogeneous boltzmann equations
journal, April 1996
- Pareschi, Lorenzo; Perthame, Benoit
- Transport Theory and Statistical Physics, Vol. 25, Issue 3-5
A non-local model for a swarm
journal, June 1999
- Mogilner, Alexander; Edelstein-Keshet, Leah
- Journal of Mathematical Biology, Vol. 38, Issue 6
From particle to kinetic and hydrodynamic descriptions of flocking
journal, August 2008
- Tadmor, Eitan; Ha, Seung-Yeal
- Kinetic and Related Models, Vol. 1, Issue 3
Numerical study of discrete‐velocity gases
journal, December 1990
- Inamuro, Takaji; Sturtevant, Bradford
- Physics of Fluids A: Fluid Dynamics, Vol. 2, Issue 12
Fast Spectral Methods for the Fokker–Planck–Landau Collision Operator
journal, November 2000
- Pareschi, L.; Russo, G.; Toscani, G.
- Journal of Computational Physics, Vol. 165, Issue 1
A Numerical Method for the Accurate Solution of the Fokker–Planck–Landau Equation in the Nonhomogeneous Case
journal, June 2002
- Filbet, Francis; Pareschi, Lorenzo
- Journal of Computational Physics, Vol. 179, Issue 1
Monte Carlo solution of the Boltzmann equation via a discrete velocity model
journal, February 2011
- Morris, A. B.; Varghese, P. L.; Goldstein, D. B.
- Journal of Computational Physics, Vol. 230, Issue 4
A direct method for solving the Boltzmann equation
journal, January 1994
- Rogier, Francois; Schneider, Jacques
- Transport Theory and Statistical Physics, Vol. 23, Issue 1-3
Mean-field limit for the stochastic Vicsek model
journal, March 2012
- Bolley, François; Cañizo, José A.; Carrillo, José A.
- Applied Mathematics Letters, Vol. 25, Issue 3
Hydrodynamic models of self-organized dynamics: Derivation and existence theory
journal, January 2013
- Degond, Pierre; Liu, Jian-Guo; Motsch, Sebastien
- Methods and Applications of Analysis, Vol. 20, Issue 2
Works referencing / citing this record:
Global Weak Solutions for Kolmogorov–Vicsek Type Equations with Orientational Interactions
journal, April 2016
- Gamba, Irene M.; Kang, Moon-Jin
- Archive for Rational Mechanics and Analysis, Vol. 222, Issue 1
Kinetic Models for Topological Nearest-Neighbor Interactions
journal, October 2017
- Blanchet, Adrien; Degond, Pierre
- Journal of Statistical Physics, Vol. 169, Issue 5
Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives
journal, September 2019
- Albi, G.; Bellomo, N.; Fermo, L.
- Mathematical Models and Methods in Applied Sciences, Vol. 29, Issue 10