Discontinuous Galerkin method for a nonlocal hydrodynamic model of flocking dynamics

In this paper we devise an efficient and robust numerical method for a nonlocal nonlinear model of flocking dynamics. The governing equations are a hydrodynamic limit of the model of Cucker and Smale which consists of the compressible Euler equations with added nonlinear nonlocal interaction terms. The numerical scheme is based on the discontinuous Galerkin method. A semi-implicit scheme is used in the time discretization which requires only the solution of one linear system per time level while retaining the stability of an implicit scheme. A crucial point is the construction of a suitable linearization of the nonlocal terms which does not result in fill-in of the system matrices. Element-wise and inter-element artificial diffusion is added to the scheme along with a postprocessing procedure to deal with near-vacuum states that typically arise in the solution. We demonstrate the efficiency and robustness of the scheme on numerical experiments in 1D and 2D.