Structure Preserving Schemes for Nonlinear Fokker–Planck Equations and Applications
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.
KeywordsStructure preserving methods Finite difference schemes Fokker–Planck equations Emerging collective behavior
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”.
- 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
- 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
- 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