  This paper studies steady-state traffic flow on a ring road with up- and
down- slopes using a semi-discrete model. By exploiting the relations between
the semi-discrete and the continuum models, a steady-state solution is uniquely
determined for a given total number of vehicles on the ring road. The solution
is exact and always stable with respect to the first-order continuum model,
whereas it is a good approximation with respect to the semi-discrete model
provided that the involved equilibrium constant states are linearly stable. In
an otherwise case, the instability of one or more equilibria could trigger
stop-and-go waves propagating in certain road sections or throughout the ring
road. The indicated results are reasonable and thus physically significant for
a better understanding of real traffic flow on an inhomogeneous road.
