Structure Preserving Schemes for Nonlinear Fokker–Planck Equations and Applications

Article
  • 37 Downloads

Abstract

In this paper we focus on the construction of numerical schemes for nonlinear Fokker–Planck equations that preserve the structural properties, like non negativity of the solution, entropy dissipation and large time behavior. The methods here developed are second order accurate, they do not require any restriction on the mesh size and are capable to capture the asymptotic steady states with arbitrary accuracy. These properties are essential for a correct description of the underlying physical problem. Applications of the schemes to several nonlinear Fokker–Planck equations with nonlocal terms describing emerging collective behavior in socio-economic and life sciences are presented.

Keywords

Structure preserving methods Finite difference schemes Fokker–Planck equations Emerging collective behavior 

Notes

Acknowledgements

The research that led to the present paper was partially supported by the research grant Numerical methods for uncertainty quantification in hyperbolic and kinetic equations of the group GNCS of INdAM. MZ acknowledges “Compagnia di San Paolo”.

References

  1. 1.
    Albi, G., Pareschi, L.: Binary interaction algorithms for the simulation of flocking and swarming dynamics. Multiscale Model. Simul. 11(1), 1–29 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Albi, G., Pareschi, L., Toscani, G., Zanella, M.: Recent advances in opinion modeling: control and social influence. In: Bellomo, N., Degond, P., Tadmor, E. (eds.) Active Particles Vol.1: Advances in Theory, Models, and Applications. Birkhäuser–Springer, Basel (2017)Google Scholar
  3. 3.
    Albi, G., Pareschi, L., Zanella, M.: Opinion dynamics over complex networks: Kinetic modelling and numerical methods. Kinetic Relat. Models 10(1), 1–32 (2017)MathSciNetMATHGoogle Scholar
  4. 4.
    Barbaro, A.B.T., Degond, P.: Phase transition and diffusion among socially interacting self-propelled agents. Discrete Contin. Dyn. Syst. Ser. B 19, 1249–1278 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barbaro, A.B.T., Cañizo, J.A., Carrillo, J.A., Degond, P.: Phase transitions in a kinetic model of Cucker–Smale type. Multiscale Model. Simul. 14(3), 1063–1088 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bessemoulin-Chatard, M., Filbet, F.: A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34, 559–583 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Boscarino, S., Filbet, F., Russo, G.: High order semi-implicit schemes for time dependent partial differential equations. J. Sci. Comput. 68, 975–1001 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Buet, C., Cordier, S., Dos Santos, V.: A conservative and entropy scheme for a simplified model of granular media. Transp. Theory Stat. Phys. 33(2), 125–155 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Buet, C., Dellacherie, S.: On the Chang and Cooper numerical scheme applied to a linear Fokker–Planck equation. Commun. Math. Sci. 8(4), 1079–1090 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Burger, M., Carrillo, J.A., Wolfram, M.-T.: A mixed finite element method for nonlinear diffusion equations. Kinet. Relat. Models 3, 59–83 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Carrillo, J.A., Chertock, A., Huang, Y.: A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. Commun. Comput. Phys. 17, 233–258 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Carrillo, J.A., Choi, Y.-P., Hauray, M.: The derivation of swarming models: mean-field limit and Wasserstein distances. In: Collective Dynamics from Bacteria to Crowds, CISM Courses and Lect., vol. 553, pp. 1–46. Springer, Vienna (2014)Google Scholar
  13. 13.
    Carrillo, J.A., Fornasier, M., Rosado, J., Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker–Smale model. SIAM J. Math. Anal. 42(1), 218–236 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Carrillo, J.A., McCann, R.J., Villani, C.: Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Revista Matemática Iberoamericana 19, 971–1018 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Chainais-Hillairet, C., Jüngel, A., Schuchnigg, S.: Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities. ESAIM Math. Model. Numer. Anal. 50(1), 135–162 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Chang, J.S., Cooper, G.: A practical difference scheme for Fokker–Planck equations. J. Comput. Phys. 6(1), 1–16 (1970)CrossRefMATHGoogle Scholar
  17. 17.
    Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dimarco, G., Pareschi, L.: Numerical methods for kinetic equations. Acta Numerica 23, 369–520 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    D’Orsogna, M.R., Chuang, Y.L., Bertozzi, A.L., Chayes, L.: Self-propelled particles with soft-core interactions: patterns, stability and collapse. Phys. Rev. Lett. 96(10), 104302 (2006)Google Scholar
  20. 20.
    Furioli, G., Pulvirenti, A., Terraneo, E., Toscani, G.: Fokker–Planck equations in the modeling of socio-economic phenomena. Math. Models Methods Appl. Sci. 27(1), 115–158 (2017)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Gosse, L.: Computing Qualitatively Correct Approximations of Balance Laws. Exponential-Fit, Well-Balanced and Asymptotic-Preserving. SEMA SIMAI Springer Series. Springer, Berlin (2013)CrossRefMATHGoogle Scholar
  22. 22.
    Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics. SIAM Publications, Philadelphia (1995)CrossRefMATHGoogle Scholar
  24. 24.
    Larsen, E.W., Levermore, C.D., Pomraning, G.C., Sanderson, J.G.: Discretization methods for one-dimensional Fokker–Planck operators. J. Comput. Phys. 61(3), 359–390 (1985)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mohammadi, M., Borzì, A.: Analysis of the Chang–Cooper discretization scheme for a class of Fokker–Planck equations. J. Numer. Math. 23(3), 271–288 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Naldi, G., Pareschi, L., Toscani, G.: Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences. Birkhauser, Boston (2010)CrossRefMATHGoogle Scholar
  27. 27.
    Pareschi, L., Toscani, G.: Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods. Oxford University Press, Oxford (2013)MATHGoogle Scholar
  28. 28.
    Pareschi, L., Russo, G.: Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxations. J. Sci. Comput. 25, 129–155 (2005)MathSciNetMATHGoogle Scholar
  29. 29.
    Scharfetter, H.L., Gummel, H.K.: Large signal analysis of a silicon Read diode oscillator. IEEE Trans. Electr. Dev. 16, 64–77 (1969)CrossRefGoogle Scholar
  30. 30.
    Toscani, G.: Kinetic models of opinion formation. Commun. Math. Sci. 4(3), 481–496 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of FerraraFerraraItaly
  2. 2.Department of Mathematical SciencesPolitecnico di TorinoTurinItaly

Personalised recommendations