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

The interpolation between two opposite configurations of spheres along generalized geodesics. In particular, for $t = 0.45$ the pair $(\alpha_1^t, \alpha_2^t)$ is not in $K_2$

The cell of the mesh in position $(k, l)$. The densities are defined inside the cell, whereas the velocities are defined at the edges

The distribution of the image of the cell $(2, 2)$ by T. We first compute the image of the center, then draw a box of size $h$. The lower left part is lifted on the cell $(3, 4)$, the upper left part on $(3, 5)$, the upper right part on $(4, 5)$ and the lower right part on $(4, 4)$

Distribution of the 2-Wasserstein distances between pairs of estimated projections of the density in example 3

The motion of two crossing discs. The first column represents the sum of the two densities, and the other two the separated densities. The total time of the simulation is $T = 0.5$ for a timestep $dt = 0.01$ and a mesh size $N = 100$. The random projection step is averaged on $50$ experiments

The motion of two crossing discs in the presence of chemoattraction. We chose the same parameters that in the previous simulation, with $\kappa = 10$

Aggregation of a composite crowd driven by chemoattraction and short-range interactions. For $t = 0.50$, we see small numerical diffusion due to the stochastic projection on $K_2$. The parameters here are $N = 150$, $\kappa = 8$, $\alpha = 0.2$, $\eta = 0.1$, $R = 0.04$. The random projection step is averaged on $100$ runs and the mesh size is $N = 150$. The time parameters are still $T = 0.5$ and $dt = 0.01$

In solid line, the function $g_5$. In dashed, its approximation by a strictly convex smooth function $f_5$. In dotted line is displayed the common limit to $f_n$ and $g_n$