Application of relaxation schemes in the microscopic theory of hydrodynamics

We consider stochastic evolution of particles moving on $\mathbb Z$ with opposite speeds. This model of interacting exclusions admits a hyperbolic (Euler) scaling, and its hydrodynamic limit results in the Leroux system of PDE theory. The basic model can be modified by introducing a spin-flip, or a creation-annihilation mechanism. In a regime of shock waves the method of compensated compactness is applied. We are going to discuss usefulness of another tool of the theory of conservation laws, the technique of relaxation schemes is extended to microscopic systems.