Traffic flow with Dirac-$\delta$ source (on ramp) I have been trying to solve the traffic flow equation with a singular source ($D>0$ large):
$$
\rho_t + f(\rho)_x = D\delta(x)
$$
with the flux $f(\rho)=\rho(1-\rho)$ and the initial data $\rho(x,0)=0.4$.
I understand that the jump condition for small $D$ is $f(\rho_r)-f(\rho_l)=D$, where $\rho_l=0.4$. This gives me a real value for $\rho_r$. But when $D$ is large (eg:0.012), it does not give me a real value for $\rho_r$, so there should be something non trivial happening, like a shock (which makes sense physically too). However, I am not able to work this out explicitly. I tried to incorporate a shock term to the jump condition, but that gave me two unknowns in one equation. 
Does anyone have any ideas on how I could proceed?
 A: One approach consists in approximating the Dirac delta by a sequence of rectangular functions (see this post). The traffic flow problem with a rectangular on-ramp is addressed here.
Another approach consists in tackling the problem directly. We write the corresponding Rankine-Hugoniot condition. Consider that $\rho$ has a single discontinuity with jump $\rho_r-\rho_l$ at $x=\gamma(t)$ between $x_1$ and $x_2$. Since $\rho$ is nonsmooth, we go back to the integral definition of the conservation law
$$
\frac{\text d}{\text d t} \int_{x_1}^{x_2} \rho\,\text d x =
\left\lbrace\begin{aligned}
& f(\rho)|_{x=x_1} - f(\rho)|_{x=x_2} & &\text{if}\quad 0 \notin [x_1,x_2] \\
& f(\rho)|_{x=x_1} - f(\rho)|_{x=x_2} + D & &\text{if}\quad 0 \in [x_1,x_2]
\end{aligned}\right.
$$
where $f(\rho) = \rho(1-\rho)$.
Then, the identity
$$
\frac{\text d}{\text d t} \int_{x_1}^{x_2} \rho\,\text d x =  \int_{x_1}^{\gamma(t)} \rho_t\,\text d x + \int_{\gamma(t)}^{x_2} \rho_t\,\text d x + \gamma'(t)\left(\rho_l-\rho_r\right) 
$$
and the PDE yield the modified Rankine-Hugoniot condition
$$
\left\lbrace\begin{aligned}
\gamma'(t) &= \frac{f(\rho_r)-f(\rho_l)}{\rho_r-\rho_l} & &\text{if}\quad \gamma\neq 0 \\
D &= {f(\rho_r)-f(\rho_l)} & &\text{if}\quad \gamma = 0
\end{aligned}\right.
$$
as $x_2 \to x_1$.
The solution to the initial-value problem is made of two elementary wave solutions:


*

*At $x=0$, there is a static shock. If $D<1/100$, then the jump has left state $\rho_l = 2/5$ and right state $\rho_r = \frac12 - \sqrt{\frac{1}{100} - D}$ deduced from the Rankine-Hugoniot condition. Thus, only downstream car densities are impacted by the on-ramp. If $D>1/100$, then the jump has right state $\rho_r = 2/5$ and left state $\rho_l = \frac12 + \sqrt{\frac{1}{100} + D}$. In this case, upstream car densities are impacted by the on-ramp.

*There is an admissible rarefaction wave if $D<1/100$. The jump has left state $\rho_l = \frac12 - \sqrt{\frac{1}{100} - D}$ and right state $\rho_r = \frac25$. If $D>1/100$, then there is an admissible shock wave, with left state $\rho_l = 2/5$ and right state $\rho_r = \frac12 + \sqrt{\frac{1}{100} + D}$.
