Many interesting applications of hyperbolic systems of equations are stiff, and require the time step to satisfy restrictive stability conditions. One way to avoid small time steps is to use implicit time integration. Implicit integration is quite straightforward for first order schemes. High order schemes instead need also to control spurious oscillations, which requires limiting in space and time also in the implicit case. We propose a framework to simplify considerably the application of high order non oscillatory schemes through the introduction of a low order implicit predictor, which is used both to set up the nonlinear weights of a standard high order space reconstruction, and to achieve limiting in time.In this preliminary work, we concentrate on the case of a third order scheme, based on DIRK integration in time and CWENO reconstruction in space. The numerical tests involve linear and nonlinear scalar conservation laws.

We study a hierarchy of models based on kinetic equations for the descriptions of traffic flow in presence of automated and human-driven vehicles. For a sufficiently large penetration rate of autonomous vehicles stabilization of traffic phenomena, like stop-and-go waves, is observed. The influence of the penetration rate is investigated analytically and numerically. Even though the microscopic interaction rules of autonomous cars are simplistic, features observed in real world traffic can be recovered on a macroscopic modeling level.

We study kinetic models for traffic flow characterized by the property of producing backward propagating waves. These waves may be identified with the phenomenon of stop-and-go waves typically observed on highways. In particular, a refined modeling of the space of the microscopic speeds and of the interaction rate in the kinetic model allows to obtain weakly unstable backward propagating waves in dense traffic, without relying on non-local terms or multi-valued fundamental diagrams. A stability analysis of these waves is carried out using the Chapman-Enskog expansion. This leads to a BGK-type model derived as the mesoscopic limit of a Follow-The-Leader or Bando model, and its macroscopic limit belongs to the class of second-order Aw-Rascle and Zhang models.