Splitting scheme for a macroscopic crowd motion model with congestion for a two-typed population

We study the extension of the macroscopic crowd motion model with congestion to a population divided into two types. As the set of pairs of density whose sum is bounded is not geodesically convex in the product of Wasserstein spaces, the generic splitting scheme may be ill-posed. We thus analyze precisely the projection operator on the set of admissible densities, and link it to the projection on the set of measures of bounded density in the mono-type case. We then derive a numerical scheme to adapt the one-typed population splitting scheme.