Entropy solutions of mildly singular non-local scalar conservation laws with congestion via deterministic particle method

We develop deterministic particle schemes to solve nonlocal scalar conservation laws with congestion. We show that the discrete approximations converge to the unique entropy solution under more general assumptions than the existing literature: the velocity fields are allowed to be timedependent (with no regularity in time), they are allowed to be less regular in space (in particular the interaction force can have a discontinuity at the origin), no prescribed attractive/repulsive regimes or symmetry are required, and the mobility can have unbounded support. We treat in a unified manner two different schemes, with sampled and integrated interaction, showing that they both converge to the entropy solution, albeit with different trade-offs between accuracy and computational effort. We complement our results with some numerical simulations, among which we show the applicability to the multispecies setting, for which the integrated scheme appears to be the better choice.