Integrating Burger's Equation into a higher-order traffic flow model for delay propagation

To better model public transit vehicle flow, we modify an Aw-Rascle model from (Bharathi et. al., 2022) and add diffusion using Burger's Equation to the model to account for unscheduled delays, such as traffic jams along a transportation route. This diffusion factor is added to account for the delay propagation of unscheduled delays earlier in the trip, and also to account for drivers attempting to catch up to the scheduled stops each trip is required to perform. We then perform numerical simulations, adding in random noise using a Wiener process to simulate the randomness of traffic in our selected transit route, to show that our model is more accurate in comparison to a diffusion-less model derived and presented in Bharathi et. al., 2022.