Continuous-time link-based kinematic wave model: formulation, solution existence, and well-posedness

TL;DR

This paper introduces a continuous-time link-based kinematic wave model for traffic networks, deriving it from conservation laws and variational principles, and proves its well-posedness and solution existence.

Contribution

It formulates a novel continuous-time DAE system for traffic flow, establishing existence and stability, and demonstrates its numerical efficiency on various networks.

Findings

The model captures shock formation and queue spillback effectively.

The DAE system is shown to be well-posed with continuous dependence on initial data.

Numerical tests confirm the model's efficiency on different network sizes.

Abstract

We present a continuous-time link-based kinematic wave model (LKWM) for dynamic traffic networks based on the scalar conservation law model. 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. 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.