A variational formulation of kinematic waves: basic theory and complex boundary conditions

Abstract

This paper proves that the solution of every well-posed kinematic wave (KW) traffic problem with a concave flow-density relation is a set of least-cost (shortest) paths in space-time with a special metric. The equi-cost contours are the vehicle trajectories. If the flow-density relation is strictly concave the set of shortest paths is unique and matches the set of waves. Shocks, if they arise, are curves in the solution region where the shortest paths end. The new formulation extends the range of applications of kinematic wave theory and simplifies it considerably. For example, moving restrictions such as slow buses, which cannot be treated easily with existing methods, can be modeled as shortcuts in space-time. These shortcuts affect the nature of the solution but not the complexity of the solution process. Hybrid models of traffic flow where discrete vehicles (e.g., trucks) interact with a continuum KW stream can now be easily implemented.