This paper is concerned with the development of continuum traffic flow models. The paper starts with a brief discussion of the need for such models and of some of the problems associated with such models. It then focuses on the development and evolving understanding of the kinematic wave model. The paper focuses with the significant theoretic developments. It starts with a formulation of the model as originally developed by Lighthill and Whitman and Richards. The two basic premises of the model are presented: conservation of traffic and the existence of an equation of state relating traffic flow and traffic density. After these have been established the approach of Gordon Newell using the accumulated flow A (t, x) is reviewed. The important advance here is the observation that, given initial conditions, the accumulated flow at a point (t, x) can be found by intergrating along characteristics through that point. If the characteristics do not intersect the resulting function A (t, x) is well defined. However in some regions the characteristics will intersect. This is associated with the occurrence of shock waves. Newell's formulation makes finding the shock wave straightforward. The shock wave is the curve along which the accumulated flows as evaluated along the intersecting characteristics are equal. The recent variational formulation due to Daganzo is described. This shows that the accumulated flow A (t, x) is known. The usefulness of this approach is shown by the solution methods that Daganzo has also developed. In particular the possibility of solutions with known bounded errors. Finally the paper raises various theoretical problems that need to be addressed for further development of continuum models. (A) Reprinted with permission from Elsevier. For the covering abstract see ITRD E134766.
Samenvatting