A finite difference approximation of the kinematic wave model of traffic flow

This article 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: (a) a ''sending'' function evaluated at the density prevailing at the upstream lattice point and (b) 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 article 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.