We consider a semi-discrete car-following model and the
macroscopic Aw-Rascle model for traffic flow
given in Lagrangian form.
The solution of the car-following model converges to a weak
entropy solution of the system of hyperbolic
with Cauchy initial data.
For the homogeneous system, we allow vacuum in the initial
By using properties of the semi-discrete model, we show that this
solution of the hyperbolic system is stable in the $L^1$-norm.