Abstract
In this paper, we consider the development of numerical schemes for mean-field equations describing the collective behavior of a large group of interacting agents. The schemes are based on a generalization of the classical Chang–Cooper approach and are capable to preserve the main structural properties of the systems, namely nonnegativity of the solution, physical conservation laws, entropy dissipation, and stationary solutions. In particular, the methods here derived are second order accurate in transient regimes, whereas they can reach arbitrary accuracy asymptotically for large times. Several examples are reported to show the generality of the approach.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
G. Albi, L. Pareschi, G. Toscani, M. Zanella, Recent advances in opinion modeling: control and social influence, in Active Particles, Volume 1. Modeling and Simulation in Science, Engineering and Technology, ed. by N. Bellomo, P. Degond, E. Tadmor (Birkhäuser, cham, 2017)
G. Albi, L. Pareschi, M. Zanella, Opinion dynamics over complex networks: kinetic modeling and numerical methods. Kinet. Relat. Models 10(1), 1–32 (2017)
A.B.T. Barbaro, P. Degond, Phase transition and diffusion among socially interacting self-propelled agents. Discret. Contin. Dyn. Syst. - Ser. B 19, 1249–1278 (2014)
F. Bolley, J.A. Carrillo, Stochastic mean-field limit: non-Lipschitz forces and swarming. Math. Models Methods Appl. Sci. 21(11), 2179 (2011)
N. Bellomo, G. Ajmone Marsan, A. Tosin, Complex Systems and Society. Modeling and Simulation, Springer Briefs in Mathematics (Springer, Berlin, 2013)
C. Buet, S. Dellacherie, On the Chang and Cooper numerical scheme applied to a linear Fokker-Planck equation. Commun. Math. Sci. 8(4), 1079–1090 (2010)
C. Buet, S. Cordier, V. Dos Santos, A conservative and entropy scheme for a simplified model of granular media. Transp. Theory Stat. Phys. 33(2), 125–155 (2004)
J.A. Carrillo, M. Fornasier, J. Rosado, G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42(1), 218–236 (2010)
J.A. Carrillo, M. Fornasier, G. Toscani, F. Vecil, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences. Particle, Kinetic and Hydrodynamic Models of Swarming (Birkhuser, Boston, 2010), pp. 297–336
J.A. Carrillo, A. Chertock, Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. Commun. Comput. Phys. 17, 233–258 (2015)
J.S. Chang, G. Cooper, A practical difference scheme for Fokker-Planck equations. J. Comput. Phys. 6(1), 1–16 (1970)
S. Cordier, L. Pareschi, G. Toscani, On a kinetic model for a simple market economy. J. Stat. Phys. 120(1–2), 253–277 (2005)
F. Cucker, S. Smale, Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862 (2007)
M.R. D’Orsogna, Y.L. Chuang, A.L. Bertozzi, L. Chayes, Self-propelled particles with soft-core interactions: patterns, stability and collapse. Phys. Rev. Lett. 96, 104302 (2006)
G. Furioli, A. Pulvirenti, E. Terraneo, G. Toscani, Fokker-Planck equations in the modelling of socio-economic phenomena. Math. Models Methods Appl. Sci. 27(1), 115–158 (2017)
L. Gosse, Computing qualitatively correct approximations of Balance Laws. Exponential-Fit, Well-Balanced and Asymptotic-Preserving, SEMA SIMAI Springer Series (Springer, Berlin, 2013)
S. Gottlieb, C.W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)
E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equation I: Nonstiff Problems, vol. 8, Springer Series in Comput. Mathematics (Springer, Berlin, 1987). Second revised edition 1993
E.W. Larsen, C.D. Levermore, G.C. Pomraning, J.G. Sanderson, Discretization methods for one-dimensional Fokker-Planck operators. J. Comput. Phys. 61(3), 359–390 (1985)
L. Pareschi, G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods (Oxford University Press, Oxford, 2013)
L. Pareschi, G. Toscani, Wealth distribution and collective knowledge: a Boltzmann approach. Philos. Trans. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci. 372(2028), 20130396 (2014)
L. Pareschi, M. Zanella, Structure preserving schemes for nonlinear Fokker-Planck equations and applications. J. Sci. Comput. 74(3), 1575–1600 (2018)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Pareschi, L., Zanella, M. (2018). Structure Preserving Schemes for Mean-Field Equations of Collective Behavior. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 237. Springer, Cham. https://doi.org/10.1007/978-3-319-91548-7_31
Download citation
DOI: https://doi.org/10.1007/978-3-319-91548-7_31
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91547-0
Online ISBN: 978-3-319-91548-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)