Hyperbolic systems with supercharacteristic relaxations and roll waves

We study a distinguished limit for general 2 x 2 hyperbolic systems with re laxation, which is valid in both the subcharacteristic and supercharacteris tic cases. This is a weakly nonlinear limit, which leads the underlying rel axation systems into a Burgers equation with a source term; the sign of the source term depends on the characteristic interleaving condition. In the s upercharacteristic case, the problem admits a periodic solution known as th e roll wave, generated by a small perturbation around equilibrium constants . Such a limit is justified in the presence of artificial viscosity, using the energy method.