In the last two decades, the number of traffic models which have emerged from engineering research is quite significant. Most of the initial models can be traced back to the famous Lighthill-Whitham-Richards (LWR) model, and numerous extensions of these models have been created, which include second order models, jamiton models, systems of hyperbolic conservation laws, and many others. While some of the first instantiations of numerical solutions for these models also go back to the 1950s – for example, the famous Godunov scheme (1957) – these schemes have taken some time to permeate and make their way into the engineering community.
Moreover, the models developed by traffic engineers are becoming so complex that the mathematical community has a hard time to keep up with modelers, to prove existence and uniqueness of the solutions which engineers produce analytically, numerically or heuristically. Some recent examples include the famous articles of Bardos, Leroux and Nedelec proving for the first time in 1979 the existence and uniqueness of specific conservation laws on bounded domains. More recently, the viscosity solution of Crandall and Lions gave a new meaning to integral forms of conservations laws (Hamilton-Jacobi equations) which appear naturally in traffic.
The goal of this workshop is precisely to bring together communities which can mutually benefit from each other: traffic engineering and mathematics. The mathematics community has historically provided the engineering community with the proper ways to scientifically derive results used in practice, and the engineering community has provided the mathematics community with a variety of interesting problems to study. In the 21st century, the amount of work to be done on the mathematical side to provide a sound basis for the current work in engineering is considerable. It is growing due to new sources of data (such as smartphones) that have generated even more complicated problems.
The workshop will be divided into three parts, each of which will investigate various aspects of these considerable challenges. The first subtopic, fundamental models, will assemble experts who have made initial models such as the LWR model progressively more complex because of the need to incorporate new data and paradigms. The second subtopic will assemble experts who have worked on integral forms of the LWR model, in particular the Hamilton-Jacobi model. In the third topic, extensions of traffic flow models to better fit reality will be discussed.
This workshop will include a poster session; a request for posters will be sent to registered participants in advance of the workshop.