On an implicit and stable resolution scheme for the Payne–Whitham model

The objective of this paper is to propose a new resolution scheme for the second-order macroscopic dynamic model of traffic flow, generally known as the Payne–Whitham model. Even though this model is normally used in a discrete form, it is important to consider that it is obtained by discretizing both in time and space a set of differential (continuous) equations. The discretization is of course necessary in order to numerically solve the differential equations with finite difference approximations, but the choice of the sampling intervals is crucial for the numerical stability of the scheme. In a previous work, we found some numerical constraints for the definition of stable freeway traffic simulations obtained by using the classic discretization, i.e. by applying an explicit finite difference numerical solution scheme. In this paper we propose a new discretization method in order to avoid numerical instability phenomena. In particular, we introduce an implicit discretization scheme which shows analogous results to the classic one in case of stable simulations and does not show any unstable behaviour in the cases in which the classic model fails. In this paper, the two methods (the classic explicit method and the new implicit one) are compared through an extensive simulative campaign.