An Exact Rescaling Velocity Method for some Kinetic Flocking Models

TL;DR

This paper introduces an exact rescaling velocity method for kinetic flocking models like Cucker-Smale and Motsch-Tadmor, enabling efficient numerical simulation of flocking behavior without fine velocity grids.

Contribution

A novel exact rescaling velocity method is developed for kinetic flocking models, improving numerical observation of flocking without remeshing or fine velocity discretization.

Findings

The method accurately captures flocking behavior.

It preserves physical properties like momentum.

It reduces computational complexity in velocity space.

Abstract

In this work, we discuss kinetic descriptions of flocking models, of the so-called Cucker-Smale and Motsch-Tadmor types. These models are given by Vlasov-type equations where the interactions taken into account are only given long-range bi-particles interaction potential. We introduce a new exact rescaling velocity method, inspired by a recent work of Filbet and Rey, allowing to observe numerically the flocking behavior of the solutions to these equations, without a need of remeshing or taking a very fine grid in the velocity space. To stabilize the exact method, we also introduce a modification of the classical upwind finite volume scheme which preserves the physical properties of the solution, such as momentum conservation.