Prediction-Correction algorithm is used to handle the congestion in some pedestrian flow model (of second order). The idea consists in transporting the population according to a spontaneous velocity field at a first step, and then straighten up the density using some kind of projection. It was used first in a pioneering work by B. Maury and al. where the projection is taken in \(W_2\)-Wasserstein space. In this talk, we’ll give an overview on this algorithm and we propose a new manner to handle this kind of algorithms showing how these algorithms are closely connected to nonlinear diffusion diffusion-transport equation of Hele-Shaw type. We focus the conference on two types of strategy and we present the main theoretical and numerical results that we obtain on these models.