Model of Vehicular Traffic by Hilliges and Weidlich Revisited

Driving of cars on a highway is a complex process which can be described by different means using continuous or discrete basic equations. It always leads to equations of motion with asymmetric interaction. In 1994 Martin Hilliges and Wolfgang Weidlich from University of Stuttgart developed a phenomenological modeling for dynamic traffic flow in networks, published in Hilliges and Weidlich (Transpn Res B 29:407–431, 1995). The authors Hilliges and Weidlich introduce the model in its discrete formulation, carry out a continuous approximation and investigate stationary solutions with respect to stability analytically. Here we consider the Hilliges–Weidlich–Model once again using our optimal velocity function already introduced in previous papers by Mahnke and Pieret (Phys Rev E 56:2666–2671, 1997). We solve the equations of motion given by two coupled partial differential equations numerically and discuss the moving staedy state profiles of density as well as speed. As a first result we present the long-time behaviour. The investitations are still going on and comparisons to related research (Sugiyama Y, Masuoka K, Ishida T (2009) Dynamics of dissipative system with asymmetric interaction and N–body problem for the emergence of moving cluster. In: Appert-Rolland C, Chevoir F, Gondret P, Lassarre S, Lebacque J-P, Schreckenberg M (eds) Traffic and Granular Flow’07. Springer, Berlin, pp 556–563; Wagner, Phil Trans R Soc A 368:4481–4495, 2010) are in preparartion.