# 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) |
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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) |
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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.