A discontinuous Galerkin method on kinetic flocking models

TL;DR

This paper analyzes kinetic flocking models derived from agent-based systems, proving flocking behavior and introducing a high-order discontinuous Galerkin numerical scheme to handle singularities.

Contribution

It provides the first rigorous proof of flocking in kinetic models and develops a novel high-order DG method for these systems.

Findings

Proved flocking behavior in kinetic models

Developed a high-order positive preserving DG scheme

Successfully handled asymptotic singularities

Abstract

We study kinetic representations of flocking models. They arise from agent-based models for self-organized dynamics, such as Cucker-Smale and Motsch-Tadmor models. We prove flocking behavior for the kinetic descriptions of flocking systems, which indicates a concentration in velocity variable in infinite time. We propose a discontinuous Galerkin method to treat the asymptotic $\delta$-singularity, and construct high order positive preserving scheme to solve kinetic flocking systems.