D. Theodosis, I. Karafyllis, G. Titakis, I. Papamichail and M. Papageorgiou, “A nonlinear heat equation arising from automated-vehicle traffic flow models,” J. Comput. Appl. Math., 2023, doi: 10.1016/j.cam.2023.115443.
https://doi.org/10.1016/j.cam.2023.115443
In this paper, a new nonlinear heat equation is studied that arises as a model of the collective behaviour of automated vehicles. The properties of the solutions of this equation are studied by introducing the appropriate notion of a weak solution that requires certain entropy-like conditions. To obtain an approximation of the solution of the nonlinear heat equation, a new conservative first-order finite difference scheme is proposed that respects the corresponding entropy conditions, and certain links between the weak solution and the numerical scheme are provided. Finally, a traffic simulation scenario and a comparison with the Lighthill–Whitham–Richards (LWR) model are provided, illustrating the benefits of the use of automated vehicles.