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

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.

Keywords:

• kinematic wave model
• continuous-time traffic flow model
• link transmission model
• differential algebraic equations
• well-posedness

Acknowledgments

The authors are grateful to the three reviewers for their constructive comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ke Han http://orcid.org/0000-0003-3529-6246

W.Y. Szeto http://orcid.org/0000-0001-7059-3532

Notes

1. In the mathematical modelling of a physical system, the term well-posedness refers to the property of having a unique solution, and the behaviour of that solution hardly changes when there is a slight change in the initial/boundary conditions.

2. In the case of a single conservation law (1(1) ), a solution cannot be defined in the classical sense because may be discontinuous due to the presence of shock waves. Instead, an alternative solution, called the weak solution, is defined through integrals. See Bressan (2000) and Evans (2010) for more details

3. Flow is the rate of change of volume. Their units are respectively vehicle per unit time, and vehicle.

4. We use the notation to represent a wave with state behind the wave and state in front of the wave.

Additional information

Funding

This work is jointly supported by the National Natural Science Foundation of China [71271183] and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [HKU 17207214E].