Continuous-time link based kinematic wave model: Formulation, solution existence and well-Posedness

We present a continuous-time link based kinematic wave model (LKWM) for dynamic traffic networks based on the scalar conservation law model (Lighthill and Whitham, 1955; Richards, 1956). Derivation of the LKWM involves the variational principle for the Hamilton-Jacobi equation and junction models defined via the notions of demand and supply. We show that the proposed LKWM can be formulated as a system of differential algebraic equations (DAEs), which captures shock formation and propagation, as well as queue spillback. The DAE system, as we show in this paper, is the continuous-time counterpart of the link transmission model (Yperman et al., 2005). In addition, we present a solution existence theory for the continuous-time network model and investigate continuous dependence of the solution on the initial data, a property known as well-posedness. We test the DAE system extensively on several small and large networks and demonstrate its numerical efficiency.