A Continuous-time Link-based Kinematic Wave Model for Dynamic Traffic Networks Ke Han a* Benedetto Piccoli b † Terry L. Friesz c ‡ Tao Yao c § a Department of Mathematics, Pennsylvania State University, PA 16802, USA b Department of Mathematics, Rutgers University, NJ 08102, USA c Department of Industrial and Manufacturing Engineering, Pennsylvania State University, PA 16803, USA Abstract This paper is concerned with an analytical model of traffic flow on networks consistent with the Lighthill-Whitham-Richards model [39, 47]. The model captures the formation, propagation and dissipation of physical queues as well as spillbacks on a network level. Similar models have been mainly studied either in discrete-time [12, 13, 51], or at the level of specific junctions [29, 30, 34]. In this paper, we take one major step towards an analytical framework for analyzing the network model from two approaches: (1) we de- rive a system of differential algebraic equations (DAEs) for the explicit representation of network dynamics. The DAE system is the continuous-time counterpart of the link trans- mission model proposed in [51]; (2) we conduct mathematical analysis of the continuous- time model and discuss solution existence, uniqueness as well as continuous dependence on initial/boundary conditions. The proposed work has a positive impact on analytical Dynamic Traffic Assignment models in that it not only provides theoretical guarantee for continuous-time models, it also brings insights to the analytical properties of discrete models such as the cell transmission model [12, 13] and the link transmission model [51]. 1 Introduction This paper focuses on the Lighthill-Witham-Richards model [39, 47] on a vehicular network. We model the traffic dynamics on a link level with the following first order hyperbolic partial differential equation (PDE) which describes the spatial-temporal evolution of density and flow: ∂ ∂t ρ(t, x)+ ∂ ∂x f ( ρ(t, x) ) =0 (1.1) where ρ(t, x) : [0, +∞) × [a, b] → [0,ρ j ] denotes density, f (ρ) : [0,ρ jam ] → [0,C ] denotes flow. ρ jam is the jam density, corresponding to a bumper-to-bumper situation, C is the flow * Corresponding author, e-mail: kxh323@psu.edu; † e-mail: piccoli@camden.rutgers.edu; ‡ e-mail: tfriesz@psu.edu; § e-mail: tyy1@engr.psu.edu; 1 arXiv:1208.5141v1 [math.AP] 25 Aug 2012