Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity
Résumé
We consider an extension of the traffic flow model proposed by Lighthill, Whitham and Richards, in which the mean velocity depends on a weighted mean of the downstream traffic density. We prove well-posedness and a regularity result for entropy weak solutions of the corresponding Cauchy problem, and use a finite volume central scheme to compute approximate solutions. We perform numerical tests to illustrate the theoretical results and to investigate the limit as the convolution kernel tends to a Dirac delta function. 1. Introduction. Macroscopic traffic flow models provide nowadays a validated and powerful approach to simulate and manage traffic on road networks (we refer the interested reader to [25] for an overview of modelling approaches and practical applications and to [11] for a detailed review of the mathematical theory of macroscopic models on networks). These models are based on hyperbolic equations derived from fluid dynamics, and describe the spatio-temporal evolution of macroscopic quantities like vehicle density and mean velocity. One of the seminal macroscopic models was introduced in the mid 1950s by Lighthill and Whitham [21] and Richards [23], who proposed to complete the one-dimensional mass conservation equation ∂ t ρ(t, x) + ∂ x f (t, x) = 0 with a closure relation between speed and density, leading to the fundamental diagram f (t, x) = f (ρ(t, x)) = ρ(t, x)v(ρ(t, x)), which can be derived from empirical speed-density or flow-density data. The classical LWR model therefore reads ∂ t ρ(t, x) + ∂ x (ρ(t, x)v(ρ(t, x))) = 0, x ∈ R, t > 0,
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