Numerical simulations of a nonconservative hyperbolic system with geometric constraints describing swarming behavior

The Vicsek model is a very popular individual based model which describes collective behavior among animal societies. A large-scale limit of the Vicsek model has been derived in [Math. Models Methods Appl. Sci., 18 (2008), pp. 1193-1215], leading to a macroscopic version of the model. In this work, we want to numerically validate this macroscopic Vicsek (MV) model. However, there is no standard theory to study analytically or numerically the MV model since it is a nonconservative hyperbolic system with a geometric constraint. Different formulations of the MV model are presented and lead to several nonequivalent numerical schemes. In particular, we derive a numerical scheme, denoted by the splitting method, based on a relaxation of the geometric constraint. To test the veracity of these schemes, we compare the simulations of the macroscopic and microscopic models with each other. The numerical simulations reveal that the microscopic and macroscopic models are in good agreement, provided that we use the splitting method to simulate theMVmodel. This result confirms the relevance of the macroscopic model, but it also calls for a better theoretical understanding of this type of equation.