Semi-implicit DGM Applied to a Model of Flocking
We present the numerical solution of a hydrodynamics model of flocking using a suitable modified semi-implicit discontinuous Galerkin method. The investigated model describing the dynamics of flocks of birds or other individual entities forming herds or swarms was introduced by Fornasier et al. (Physica D 240(1):21–31, 2011). The main idea of this model comes from the well known Cucker-Smale model. The resulting equations consist of the Euler equations for compressible flow with an additional non-local non-linear source term. The model is discretized by the semi-implicit discontinuous Galerkin method for the compressible Euler equations of Feistauer and Kučera (J Comput Phys 224(1):208–221, 2007). We show that with a suitable treatment of the source term we can use this method for models like the model of flocking and find a numerical solution very efficiently.
The research of V. Kučera is supported by the Grant No. P201/13/00522S of the Czech Science Foundation. He is currently a Fulbright visiting scholar at Brown University, Providence, RI, USA, supported by the J. William Fulbright Commission in the Czech Republic. The research of A. Živčáková is supported by the Charles University in Prague, project GA UK No. 758214.