  We prove the non-linear stability of a class of travelling-wave solutions to
the extended Aw-Rascle system with a singular offset function, which is
formally equivalent to the compressible pressureless Navier-Stokes system with
a singular viscosity. These solutions encode the effect of congestion by
connecting a congested left state to an uncongested right state, and may also
be viewed as approximations of solutions to the 'hard-congestion model'. By
using carefully weighted energy estimates we are able to prove the non-linear
stability of viscous shock waves to the Aw-Rascle system under a small zero
integral perturbation, which in particular extends previous results that do not
handle the case where the viscosity is singular.
