The Hamilton–Jacobi partial differential equation and the three representations of traffic flow

• New and existing first-order traffic flow models are formulated under the same H-J theory.
• New graphical and exact numerical solution methods including fixed and moving bottlenecks.
• Model for predicting cumulative counts that does not require memory.
• Models based on vehicle trajectories require comparison of two terms regardless of initial data.

This paper applies the theory of Hamilton–Jacobi partial differential equations to the case of first-order traffic flow models. The traffic flow surface is analyzed with respect to the three 2-dimensional coordinate systems arising in the space of vehicle number, time and distance. In each case, the solution to the initial and boundary value problems are presented. Explicit solution methods and examples are shown for the triangular flow-density diagram case. This unveils new models and shows how a number of existing models are cast as special cases.