A shock-capturing numerical scheme for a non-conservative self-organized hydrodynamics model

This work focuses on the Self-Organized Hydrodynamics (SOH) model, which is the macroscopic limit of the well-known individual based Vicsek model. The SOH model is a hyperbolic non-conservative PDE system with a geometric constraint, which causes issues for both its theoretical and numerical resolutions. In this work, we first reformulate the model without its constraint for some well-prepared initial conditions. We then focus on shock wave solutions of the SOH model and we define generalized Rankine-Hugoniot conditions. Moreover, a Godunov-type scheme is designed and a viscous correction is added in order to numerically recover shock wave solutions. Finally, some exact solutions to the Riemann problem are computed thanks to the generalized Rankine-Hugoniot conditions. Thus the shock-capturing scheme is compared to an usual splitting method and the exact solutions on some test cases. These simulations confirm the relevance of the viscous Godunov-type scheme to simulate shock waves of the SOH model.