Speaker
Description
Kinetic models obtained as the mean field limit of interacting particle systems provide an effective framework for describing collective dynamics in a wide range of applications, including vehicular traffic modeling, coordinated animal motion, biological population dynamics, and the control of robotic swarms. Compared to direct multi agent simulations, the continuous formulation based on Fokker–Planck type equations drastically reduces the computational cost as the number of agents increases, while preserving an accurate description of the macroscopic behavior.
In this work, we present a numerical method for the simulation of a class of nonlocal kinetic equations with nonlinear drift, derived from an underlying particle interaction model. The strategy relies on an operator splitting approach that separates the transport term from the drift–diffusion dynamics. Transport is discretized using an explicit Lax–Wendroff scheme, while the velocity drift–diffusion part is treated with a Chang–Cooper method, ensuring stability, mass conservation, and positivity of the solution. The combination of these two components yields a robust and accurate scheme.
The method is validated by comparing the PDE solution with that of the original particle model, showing excellent agreement between the two descriptions and confirming the correctness of the mean field limit. We also analyze the computational cost of both approaches, highlighting how the kinetic formulation becomes significantly more efficient beyond a certain threshold in the number of agents.
These results demonstrate that the kinetic formulation, combined with accurate numerical schemes, provides an effective tool for the simulation of complex systems driven by collective interactions.