I am reading this paper, describing an ODE-based traffic model. Initially it starts with a PDE model, which is then simplified to obtain a more tractable ODE one.
I'm trying to understand the initial PDE model. I am new to traffic modeling (and PDEs really), so forgive my ignorance.
The paper, and many resources I found, mention that Ligthill-Whitham models assume the car flow is solely a function of the car density $\rho$, which is supposed to be:
- A continuously differentiable, strictly concave function
- With value zero when the density is zero (as there are no cars) and when the density is maximal (as the cars are bumper to bumper) -- This maximal density on the $j$-th road of the network is denoted: $\rho_{\max, j}$.
Given such a flow function $f$, the paper states that:
the macroscopic Lighthill–Whitham model for traffic flow on a road j is given by the non-linear conservation law:
$$ \dfrac{\partial \rho_j(x, t)}{\partial t} + \dfrac{\partial f(\rho_j(x, t))}{\partial x} = 0 $$ For all $x$ in $[a_j, b_j]$ (interval representing the $j$-th road) and $t$ in the time frame considered $[0, T]$, and: $$ \rho_j(x, 0) = \boxed{\rho_{j, 0}}(x) \quad\quad \forall x \in [a_j, b_j] $$
I do not understand what the boxed term $\rho_{j, 0}$ is. The notation is not introduced anywhere in the paper, so I believe it may be some standard notation I'm not aware of. I believe it may be some boundary or initial condition, but I don't quite get its role.
