Approximations of kinetic equations of swarm formation: Convergence and exact solutions

In the present paper we study Euler-type approximations along characteristics for a class of kinetic equations that describe swarm formations in the case when the interactions rate is variable. The proposed numerical schemes preserve essential properties of the kinetic equations and in particular preserve the probabilistic measure and are able to approximate the solution almost to the appearance of blow-ups. The blow-ups are referred here to the self-organization swarm behavior. Moreover we define a class of exact solutions traveling wave-type equilibrium solutions that we called TWES. (C) 2021 The Authors. Published by Elsevier Inc.

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In this paper, Euler-type approximations along characteristics were studied for a class of kinetic equations describing swarm formations with variable interaction rates. The proposed numerical schemes preserve essential properties of the kinetic equations and approximate the solution almost to the point of blow-ups, which indicate self-organization swarm behavior. Additionally, a class of exact solutions known as TWES, traveling wave-type equilibrium solutions, were defined.