Traffic Modelling and Some Inequalities in Banach Spaces

Modelling traffic flow has been around since the appearance of traffic jams. Ideally, if we can correctly predict the behavior of vehicle flow given an initial set of data, then adjusting the flow in crucial areas can maximize the overall throughput of traffic along a stretch of road. We consider a mathematical model for traffic flow on single land and without exits or entries. So, we are just observing what happens as time evolves if we fix at initial time (t = 0) some special distribution of cars (initial datum u_0). Because we do approximations, we need the notion of convergence and its corresponding topology. The numerical approximation of scalar conservation laws is carried out by using conservative methods such as the Lax-Friedrichs and the Lax-Wendroff schemes. The Lax-Friedrichs scheme gives regular numerical solutions even when the exact solution is discontinuous (shock waves). We say the scheme is diffusive meaning that the scheme is solving in fact an evolution equation of the form u_t+f(u)_x = epsilon u_xx, where epsilon is a small parameter depending on ∆x and ∆t. The Lax-Wendroff scheme is more precise than the Lax-Friedrichs scheme, and give the right position of the discontinuities for the shock waves. But it develop oscillations. We say the scheme is dispersive what means the scheme is solving approximatively an evolution equation of the form u_t + f(u)_x = delta u_xxx, where delta is a small parameter depending on ∆x and ∆t. An elaboration and an implementation of Lax-Friedrichs schemes and of Lax-Wendroff schemes even extended to second order provided numerical solutions to the problem of traffic flows on the road. Since along the roads the schemes present the same features as for conservation laws, the new and original aspect is given by the treatment of the solution at junctions. Our tests show the effectiveness of the approximations, revealing that Lax-Wendroff schemes is more accurate than Lax-Friedrichs schemes