Out of equilibrium thermodynamics in multi-component exact models

Many body integrable systems allow the calculation of the equilibrium spectrum through a set of equations called "thermodynamic Bethe ansatz"(TBA). In particular, the macroscopic quantity encoding the informations about the spectrum is the so-called root density. A recent theoretical development is Generalized Hydrodynamics(GHD) which provides exact equations for the time evolution of the root density. The prototypical setup in which these equations are tested is the quantum quench, i.e. a sudden change of some parameter of the Hamiltonian (typically related to the external potential). The goal of the thesis is to develop numerical simulation for the time evolution of single-component and multi-component gases.