Binary Interaction Algorithms for the Simulation of Flocking and Swarming Dynamics

Microscopic models of flocking and swarming take into account large numbers of interacting individuals. Numerical resolution of large flocks implies huge computational costs. Typically for $N$ interacting individuals we have a cost of $O(N^2)$. We tackle the problem numerically by considering approximated binary interaction dynamics described by kinetic equations and simulating such equations by suitable stochastic methods. This approach permits us to compute approximate solutions as functions of a small scaling parameter $\varepsilon$ at a reduced complexity of $O(N)$ operations. Several numerical results show the efficiency of the algorithms proposed.