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?