We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $ρ$ ahead. The averaging kernel is of exponential type: $w_\varepsilon(s)=\varepsilon^{-1} e^{-s/\varepsilon}$. For any decreasing velocity function $v$, we prove that, as $\varepsilon\to 0$, the limit of solutions to the nonlocal equation coincides with the unique entropy-admissible solution to the scalar conservation law $ρ_t + (ρv(ρ))_x=0$.
Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counter-examples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular l…
We study a class of traffic flow models with nonlocal look-ahead interactions. The global regularity of solutions depend on the initial data. We obtain sharp critical threshold conditions that distinguish the initial data into a trichotomy: subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams.
This paper focuses on the proof of the stability of entropy weak solutions of a nonlocal balance law modeling vehicular traffic flow on a road with on- and off-ramps. The stability is obtained with respect to a kernel function in the source term. We get an estimate of the L1-dependence of the solution with respect to the initial datum, the on-ramp rate, the off-ramp rate and the mentioned kernel function. We also show numerical experiments in order to perform an optimization problem in traffic flow with on-ramps.
Consider a nonlocal conservation where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (a…
We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $ρ$ ahead. The averaging kernel is of exponential type: $w_\varepsilon(s)=\varepsilon ^{-1} e^{-s/\varepsilon}$. By a transformation of coordinates, the problem can be reformulated as a $2\times 2$ hyperbolic system with relaxation. Uniform BV bounds on the solution are thus obtained, independent of the scaling parameter $\varepsilon $. Letting $\varepsilon\to 0$, the limit yields a weak solution to the corresponding conservation law $ρ_t + ( ρv(ρ))_x=0$. In…
We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model one considers an averaged downstream velocity. The road condition is piecewise constant with a jump at $x=0$. We study stationary traveling wave profiles cross $x=0$, for all possible cases. We show that, depending on the case, there could exit infinitely many profiles, a unique profile, or no profiles at all. Furthermore, some of the profiles are time asym…
We consider several non-local models for traffic flow, including both microscopic ODE models and macroscopic PDE models. The ODE models describe the movement of individual cars, where each driver adjusts the speed according to the road condition over an interval in the front of the car. These models are known as the FtLs (Follow-the-Leaders) models. The corresponding PDE models, describing the evolution for the density of cars, are conservation laws with non-local flux functions. For both types of models, we study stationary traveling wave profiles and stationary discrete traveling wave profil…
We investigate a class of nonlocal conservation laws with the nonlinear advection coupling both local and nonlocal mechanism, which arises in several applications such as the collective motion of cells and traffic flows. It is proved that the $C^1$ solution regularity of this class of conservation laws will persist at least for a short time. This persistency may continue as long as the solution gradient remains bounded. Based on this result, we further identify sub-thresholds for finite time shock formation in traffic flow models with Arrhenius look-ahead dynamics.
We present a network formulation for a traffic flow model with nonlocal velocity in the flux function. The modeling framework includes suitable coupling conditions at intersections to either ensure maximum flux or distribution parameters. Based on an upwind type numerical scheme, we prove the maximum principle and the existence of weak solutions on networks. We also investigate the limiting behavior of the proposed models when the nonlocal influence tends to infinity. Numerical examples show the difference between the proposed coupling conditions and a comparison to the Lighthill-Whitham-Richa…
We study a nonlocal traffic flow model with an Arrhenius type look-ahead interaction. We show a sharp critical threshold condition on the initial data which distinguishes the global smooth solutions and finite time wave break-down.
We study a nonlocal particle model describing traffic flow on rough roads. In the model, each driver adjusts the speed of the car according to the condition over an interval in the front, leading to a system of nonlocal ODEs which we refer to as the FtLs (follow-the-leaders) model. Assuming that the road condition is discontinuous at the origin, we seek stationary wave profiles (see Definition 1.1) for the system of ODEs across this discontinuity. We derive a non-local delay differential equation with discontinuous coefficient, satisfied by the profiles, together with conditions on the asympto…
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 traffic flow over the on-ramp and off-ramps. We approximate the problem using an upwind-type numerical scheme and we provide L^\infty and BV 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.
We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective isto minimise/maximise macroscopic quantities, such as traffic volume or average speed,controlling few agents, for example smart traffic lights and automated cars. The measuretheoretic approach allows to study in a same setting local and nonlocal drivers interactionsand to consider the control variables as additional measures interacting with the driversdistribution. We also propose a gradient descent adjoint-based optimization method, ob-…
