Go back to Carlos Daganzo's Publications PageThis paper shows that if the kinematic wave model of freeway traffic flow in its general form is approximated by a particular type of finite difference equation, the finite difference results converge to the kinematic wave solution despite the existence of shocks in the latter. This result, which applies to initial and boundary condition problems with and without discontinuous data, is shown not to hold for other commonly used finite difference schemes.
In the proposed approximation, the flow between two neighboring lattice points is the minimum of the two values returned by: (i) a "sending" function evaluated at the density prevailing at the upstream lattice point and (ii) a "receiving" function evaluated at the downstream lattice point. The sending and receiving functions correspond to the increasing and decreasing branches of the freeway's flow-density curve.
The paper presents an asymptotic formula for the errors
introduced by the proposed finite difference approximation, and
describes quantitatively the finite difference's shock-capturing
behavior. Errors are shown to be approximately proportional to
the mesh spacing with a coefficient of proportionality that
depends on the wave speed, on its rate of change with density,
and on the slope and curvature of the initial density profile.
The asymptotic errors are smaller than those of Lax's first-
order, centered difference method which is also convergent. More
importantly though, the proposed procedure never yields negative
flows, and this makes it attractive in practical engineering
applications when the mesh cannot be made arbitrarily small.
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