A variational formulation of kinematic waves: basic theory and complex boundary conditions
Section snippets
Foreword
The problems addressed in this paper are related to Hamilton–Jacobi theory. Developments in this theory with the “viscosity solution” approach (see Evans, 2002, Chapter 10) do not yet address all the peculiar boundary conditions that arise in traffic flow applications. Because complex traffic problems were the motivation for this study, the derivations in this paper build on prior traffic work rather than the viscosity approach. This approach is physically more meaningful and turns out to be
A variational principle
We consider a solution of (4), N(t,x), which is a continuous solution of , , and start by looking for the functional .
Where N is differentiable it must satisfy , , , . In connection with these equations, it will be convenient to abbreviate Qk by uand N′ by r so that (5c) becomes,The scalar r is the rate at which cars overtake an observer moving with the wave.
Since , holds where N is differentiable, q can be eliminated from (8) and r becomes:
Well-posedness tests
With the new formulation, determining if a meaningful solution exists is very easy. For a solution to exist there must be a shortest path with finite cost to every point in the solution domain, and the shortest path to every point on the boundary should emanate from itself. (Otherwise, N(t,x) would be discontinuous at the point in question and a proper solution to the problem would not exist.) Note that a proper solution always exists for the initial value problem, since valid paths cannot
Acknowledgements
Prof. L.C. Evans of U.C. Berkeley (Mathematics Department) read an earlier draft of this manuscript and clued me in to the viscosity results of Hamilton–Jacobi theory. J. Laval and A. Lago provided many comments; their help is appreciated. The comments of an anonymous referee are also gratefully acknowledged.
References (15)
A simplified theory of kinematic waves in highway traffic. I: General theory. II: Queuing at Freeway Bottlenecks. III: Multi-destination flows
Trans. Res.
(1993)- et al.
Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations
(1997) A simple traffic analysis procedure
Networks Spatial Econom.
(2001)- Daganzo, C.F., 2003a. “A variational formulation for a class of first-order PDE's”, Institute of Transportation Studies...
- Daganzo, C.F., 2003b. “A variational formulation of kinematic waves: Solution methods”, Institute of Transportation...
Partial Differential Equations
(1964)Partial differential equations
Cited by (292)
Modelling the dual dynamic traffic flow evolution with information perception differences between human-driven vehicles and connected autonomous vehicles
2024, Physica A: Statistical Mechanics and its ApplicationsThe Traffic Reaction Model: A kinetic compartmental approach to road traffic modeling
2024, Transportation Research Part C: Emerging TechnologiesAn MFD approach to route guidance with consideration of fairness
2023, Transportation Research Part C: Emerging TechnologiesNon-unimodal and non-concave relationships in the network Macroscopic Fundamental Diagram caused by hierarchical streets
2023, Transportation Research Part B: MethodologicalSelf-organized criticality of traffic flow: Implications for congestion management technologies
2023, Transportation Research Part C: Emerging Technologies