  We consider solutions to nonlinear hyperbolic systems of balance laws with
stiff relaxation and formally derive a parabolic-type effective system
describing the late-time asymptotics of these solutions. We show that many
examples from continuous physics fall into our framework, including the Euler
equations with (possibly nonlinear) friction. We then turn our attention to the
discretization of these stiff problems and introduce a new finite volume scheme
which preserves the late-time asymptotic regime. Importantly, our scheme
requires only the classical CFL (Courant, Friedrichs, Lewy) condition
associated with the hyperbolic system under consideration, rather than the more
restrictive, parabolic-type stability condition.
