An Exact Rescaling Velocity Method for some Kinetic Flocking Models

In this work, we discuss kinetic descriptions of flocking models, of the so-called CuckerSmale [4] and Motsch-Tadmor [10] 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 the recent work [6], 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.