This is a review based on the presentation done at the seminar Laurent Schwartz in December 2021. It is announcing results in the forthcoming [Menegaki-Mouhot-Marahrens'22]. This work presents a new simple quantitative method for proving the hydrodynamic limit of a class of interacting particle systems on lattices. We present here this method in a simplified setting, for the zero-range process and the Ginzburg-Landau process with Kawasaki dynamics, in the parabolic scaling and in dimension $1$. The rate of convergence is quantitative and uniform in time. The proof relies on a consistence-stability approach in Wasserstein distance, and it avoids the use of the ``block estimates''.