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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?

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  • $\begingroup$ Just thinking out loud, here. Looks like $\rho_r$ goes imaginary when $D>1/100,$ based on your value for $\rho_l.$ Can you edit your question to include what you had in mind for a shock term? Maybe we can work around the lack of information problem you mentioned. $\endgroup$ – Adrian Keister Dec 31 '18 at 16:24
  • $\begingroup$ For $x\not=0,$ your pde reduces down to $\rho_t+\rho_x-2\rho\rho_x=0.$ The IC, of course, makes the trivial solution not work. Again, still just thinking out loud, here. $\endgroup$ – Adrian Keister Dec 31 '18 at 16:29
  • $\begingroup$ This is a hyperbolic conservation law, which doesn't necessarily have a unique solution. The setup you have is a generalised Riemann problem, which may include shocks and rarefaction fans. $\endgroup$ – Eddy Dec 31 '18 at 16:35
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    $\begingroup$ However, the shock condition and jump condition give you 2 equations, and there are 2 unknowns, so you should only have finitely many solutions. $\endgroup$ – Eddy Dec 31 '18 at 16:37
  • $\begingroup$ The constant solution $\rho=2/5$ satisfies the pde and IC when $x\not=0$. $\endgroup$ – Adrian Keister Dec 31 '18 at 16:42
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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 $\rho_r = \frac12 + \sqrt{\frac{1}{100} + D}$.

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