# Sample Project: ODE and PDE Models for Traffic Flow.

We consider the first-order, following-the-leader model, for an infinitely long highway with N identical cars, and possible traffic lights at certain locations. The unknown variable x_i(t) denotes the position of the ith car at time t, with x_{i+1}(t)-x_i(t) \ge L for all t and i, where L is the vehicle length. Initially, the ith car is at position \bar x_i. During time intervals when the lights are green, we consider the following microscopic model:

\dot x_{N}(t) = V,\qquad \dot x_i(t) = \mathcal{V}\left((x_{i+1}-x_i)(t)\right)- e^{-t/\epsilon} \quad i=1,\cdots,N-1. (1)
The foremost car x_N has the empty highway ahead of it, thus it travels with a fixed speed V. All other cars travel with a speed that depends only on the distance between the current car and the one ahead, described by the monotone decreasing function \mathcal{V} \in \mathbf{C}^{0,1}([L,+\infty[;[0,V]) satisfying suitable assumptions. Here the parameter \epsilon >0 may be treated as a relaxation parameter.

With \epsilon =0, as N\to+\infty, (1) converges to the macroscopic Lighthill-Whitham model

\rho_t + (v(\rho) \rho)_x =0\,, \qquad \rho(0,x)=\bar\rho(x).. (2)
Here \rho(t,x) is the density of cars, and v(\rho) is the velocity of cars, assumed to be a decreasing function of the density. Equation (2) is simply the conservation of cars. One observes that the two models are related through particle paths, i.e. the trajectories of single cars in (2), \dot x = v(\rho(t,x(t)))withx(0)=\bar x.

The students will conduct numerical simulations for model (1) using ODE solvers, with the option of adding traffic lights, and model (2) using the Lax-Friedrichs method or front tracking. They will observe the behavior of (1) for large N, with \epsilon=0 and \epsilon>0, and observe the convergence to (2) and to a relaxation model. In order to reduce congestion on the highway, our goal is to maximize the flux \rho v at a chosen point X, by varying the parameters in the model. The students will run numerical simulations with different values of L, V, \epsilon, and different functions \mathcal{V}. Students may also include other factors into the model, then run simulations and evaluate the performance of these new models.