Semi-linear Poisson-mediated Flocking in a Cucker-Smale Model*

Title : Semi-linear Poisson-mediated Flocking in a Cucker-Smale Model*
Authors :
Conference : 24th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2020) IFAC PapersOnLine 54-9 (2021), pp. 404-409

We propose a family of compactly supported parametric interaction functions in the general Cucker-Smale flocking dynamics such that the mean-field macroscopic system of mass and momentum balance equations with non-local damping terms can be converted from a system of partial integrodifferential equations to an augmented system of partial differential equations in a compact set. We treat the interaction functions as Green’s functions for an operator corresponding to a semi-linear Poisson equation and compute the density and momentum in a translating reference frame, i.e. one that is taken in reference to the flock’s centroid.  This allows us to consider the dynamics in a fixed, flock-centered compact set without loss of generality. We approach the computation of the non-local damping using the standard finite difference treatment of the chosen differential operator, resulting in a tridiagonal system which can be solved quickly.

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