Abstract: We study a distinguished limit for general $2 \times 2$ hyperbolic systems with relaxation, which is valid in both the subcharacteristic and supercharacteristic cases. This is a weakly nonlinear limit, which leads the underlying relaxation systems into Burgers equation with a source term; the sign of the source term depends on the characteristic interleaving condition. In the supercharacteristic case, the problem admits a periodic solution known as the 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.
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