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We propose a family of compactly supported parametric interaction functionsin the general Cucker-Smale flocking dynamics such that the mean-fieldmacroscopic system of mass and momentum balance equations with non-localdamping terms can be converted from a system of partial integro-differentialequations to an augmented system of partial differential equations in a compactset. We treat the interaction functions as Green's functions for an operatorcorresponding to a semi-linear Poisson equation and compute the density andmomentum in a translating reference frame, i.e. one that is taken in referenceto the flock's centroid. This allows us to consider the dynamics in a fixed,flock-centered compact set without loss of generality. We approach thecomputation of the non-local damping using the standard finite differencetreatment of the chosen differential operator, resulting in a tridiagonalsystem which can be solved quickly.

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