Nonlocal macroscopic models of multi-population pedestrian flows for walking facilities optimization

We propose a nonlocal macroscopic pedestrian flow model for two populations with different destinations trying to avoid each other in a confined environment, where the nonlocal term accounts for anisotropic interactions, mimicking the effect of different cones of view, and the presence of walls or other obstacles in the domain. In particular, obstacles can be incorporated in the density variable, thus avoiding to include them in the vector field of preferred directions. In order to efficiently compute the solution, we propose a Finite Difference scheme that couples high-order WENO approximations for spatial discretization, a multi-step TVD method for temporal discretization, and a high-order numerical derivative formula to approximate the derivatives of nonlocal terms, and in this way reducing consistently the amount of calculations. Numerical tests confirm that each population manages to evade both the presence of the obstacles and the other population.

Including obstacles in the nonlocal operator and having a computationally affordable simulation code allows to tackle the shape optimization of the walking domain as a classical PDE constrained optimization problem. In particular, we compute the optimal positions and sizes of obstacles that minimize the pedestrian evacuation time.