Hydrodynamic Cucker-Smale model with time delay and obstacle avoidance

We study a hydrodynamic Cucker-Smale-type model incorporating both time delays and obstacle potentials. The model governs the evolution of velocity and density fields of the system, where delayed interactions drive alignment and obstacle potentials account for responses to obstacles or predators. We further extend the framework to two-species systems. To numerically solve the model, we design a high-order finite volume method based on a Lax–Friedrichs numerical flux with fifth-order weighted essentially non-oscillatory reconstruction and third-order Runge–Kutta time discretization, ensuring numerical stability and high-order accuracy. Numerical experiments confirm the stability and accuracy of the proposed scheme and illustrate how time delays and obstacle potentials, under specific communication kernels and initial conditions, affect the emergence of flocking or non-flocking behavior.