Coalescing particle systems and applications to nonlinear Fokker-Planck equations
This work provides a computationally efficient method for simulating blow-up in chemotaxis models, relevant for researchers studying pattern formation and singularities in PDEs.
The authors develop a numerical method for a stochastic particle system with singular interactions that avoids pairwise computations, and prove its hydrodynamic limit converges to nonlinear Fokker-Planck equations like the Patlak-Keller-Segel model. They simulate finite-time singularities and post-blow-up dynamics, and observe novel scaling behavior in two-species Keller-Segel blow-up.
We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.