The simulation model proposed in [M. Hilliges and W. Weidlich. A phenomenological model for dynamic traffic flow in networks. Transportation Research Part B: Methodological, 29 (6): 407–431, 1995] can be understood as a simple method for approximating solutions of scalar conservation laws whose flux is of density times velocity type, where the density and velocity factors are evaluated on neighboring cells. The resulting scheme is monotone and converges to the unique entropy solution of the underlying problem. The same idea is applied to devise a numerical scheme for a class of one-dimensional scalar conservation laws with nonlocal flux and initial and boundary conditions. Uniqueness of entropy solutions to the nonlocal model follows from the Lipschitz continuous dependence of a solution on initial and boundary data. By various uniform estimates, namely a maximum principle and bounded variation estimates, along with a discrete entropy inequality, the sequence of approximate solutions is shown to converge to an entropy weak solution of the nonlocal problem. The improved accuracy of the proposed scheme in comparison to schemes based on the Lax-Friedrichs flux is illustrated by numerical examples. A second-order scheme based on MUSCL methods is presented.

We present a non-local version of a scalar balance law modeling traffic flow with on-ramps and off-ramps. The source term is used to describe the inflow and output flow over the on-ramp and off-ramps respectively. We approximate the problem using an upwind-type numerical scheme and we provide ${\mathbf{L}}^{\mathrm{\infty }}$ and $\mathbf{B}\mathbf{V}$ estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar balance laws. Some numerical simulations illustrate the behaviour of solutions in sample cases.