We prove the well-posedness of entropy weak solutions of a scalar conservation law with non-local flux arising in traffic flow modeling. The result is obtained providing accurate L ∞ , BV and L 1 estimates for the sequence of approximate solutions constructed by an adapted Lax-Friedrichs scheme.

This article is motivated by the practical problem of highway traffic estimation using velocity measurements from GPS enabled mobile devices such as cell phones. In order to simplify the estimation procedure, a velocity model for highway traffic is constructed, which results in a dynamical system in which the observation operator is linear. This article presents a new scalar hyperbolic partial differential equation (PDE) model for traffic velocity evolution on highways, based on the seminal Lighthill-Whitham-Richards (LWR) PDE for density. Equivalence of the solution of the new velocity PDE and the solution of the LWR PDE is shown for quadratic flux functions. Because this equivalence does not hold for general flux functions, a discretized model of velocity evolution based on the Godunov scheme applied to the LWR PDE is proposed. Using an explicit instantiation of the weak boundary conditions of the PDE, the discrete velocity evolution model is generalized to a network, thus making the model applicable to arbitrary highway networks. The resulting velocity model is a nonlinear and nondifferentiable discrete time dynamical system with a linear observation operator, for which a Monte Carlo based ensemble Kalman filtering data

An extension of the Colombo phase transition model is proposed. The congestion phase is described by a two-dimensional zone defined around a standard fundamental diagram. General criteria for building such a set-valued fundamental diagram are enumerated and instantiated on several standard fluxes with different concavity properties. The solution to the Riemann problem in the presence of phase transitions is obtained through the design of a Riemann solver, which enables the construction of the solution of the Cauchy problem using wavefront tracking. The free-flow phase is described using a Newell-Daganzo fundamental diagram, which allows for a more tractable definition of phase transition compared to the original Colombo phase transition model. The accuracy of the numerical solution obtained by a modified Godunov scheme is assessed on benchmark scenarios for the different flux functions constructed. Introduction.First order scalar models of traffic. Hydrodynamic models of traffic go back to the 1950s with the work of Lighthill and Whitham [31] and Richards [38], who built the first model of the evolution of vehicle density on the highway using a first order scalar hyperbolic partial differential equation (PDE) referred to as the LWR PDE. Their model relies on the knowledge of an empirically measured flux function, also called the fundamental diagram in transportation engineering, for which measurements go back to 1935 with the pioneering work of Greenshields [22]. Numerous other flux functions have since been proposed in the hope of capturing effects of congestion more accurately, in particular, Greenberg [21], Underwood [44], Newell [34], Daganzo [10], and Papageorgiou [47]. The existence and uniqueness of an entropy solution to the Cauchy problem [39] for the class of scalar conservation laws to which the LWR PDE belongs go back to the work of Oleinik [35] and Kruzhkov [27] (see also the seminal article of Glimm [18]), which was extended later to the initial-boundary value problem [2] and specifically instantiated for the scalar case with a concave flux function in [29], in particular for traffic in [40]. Numerical solutions of the LWR PDE go back to the seminal Godunov scheme [20,30], which was shown to converge to the entropy solution of the first order hyperbolic PDE (in particular, the LWR

Traffic state estimation is a challenging problem for the transportation community due to the limited deployment of sensing infrastructure. However, recent trends in the mobile phone industry suggest that GPS equipped devices will become standard in the next few years. Leveraging these GPS equipped devices as traffic sensors will fundamentally change the type and the quality of traffic data collected on large scales in the near future. New traffic models and data assimilation algorithms must be developed to efficiently transform this data into usable traffic information.In this work, we introduce a new partial differential equation (PDE) based on the Lighthill-Whitham-Richards PDE, which serves as a flow model for velocity. We formulate a Godunov discretization scheme to cast the PDE into a Velocity Cell Transmission Model (CTM-v), which is a nonlinear dynamical system with a time varying observation matrix. The Ensemble Kalman Filtering (EnKF) technique is applied to the CTMv to estimate the velocity field on the highway using data obtained from GPS devices, and the method is illustrated in microsimulation on a fully calibrated model of I880 in California. Experimental validation is performed through the unprecedented 100-vehicle Mobile Century experiment, which used a novel privacy-preserving traffic monitoring system to collect GPS cell phone data specifically for this research.

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