Can the Method of Characteristics Solve Complex Traffic Flow Models?

[Carla1985]

I am creating a traffic flow model, modelling the flow of traffic as fluid with density and velocity. I am using the method of characteristics to solve the model but I have run into a problem.

The model I have is:
$\frac{\partial\rho}{\partial t}+c(\rho)\frac{\partial\rho}{\partial x}=0$, $\rho(x,0)=f(x)$, $v=v(\rho)$, $c(\rho)=v+\frac{\partial v}{\partial \rho}\rho$

$\rho$ is density, $\rho_c$ is some density between 0 and max density which I have set at 1. $v$ is velocity.

so using:
\(\displaystyle
v(\rho)= \left\{
\begin{array}{1 1}
\ 70 && 0\leq\rho<\rho_c \\
&& \\
\frac{70}{\rho \ln(\frac{1}{\rho_c})}\ln(\frac{1}{\rho}) && \rho\leq\rho\leq 1
\end{array}\right.
\)

\(\displaystyle
f(x)= \left\{
\begin{array}{1 1}
\ 0 && x>\epsilon \\
1-\frac{(x+\epsilon)}{2\epsilon} && -\epsilon\leq x\leq\epsilon \\
\ 1 && x<-\epsilon
\end{array}\right.
\)

I have already worked out that:

Case 1:
\(\displaystyle x_0>\epsilon\ \Rightarrow x>70t\ \Rightarrow x_0=x-70t\ \Rightarrow \rho(x,t)=0
\)

Case 2:
\(\displaystyle x_0<-\epsilon\ \Rightarrow x<\left(\frac{70}{\ln(\rho_c})\right)t\ \Rightarrow x_0=x-\left(\frac{70}{\ln(\rho_c})\right)t\ \Rightarrow \rho(x,t)=1
\)

Case 3 (this is where it gets complicated):
\(\displaystyle -\epsilon\leq x\leq\epsilon
\)
but $\rho_c$ falls within this range so I split it into 2 sections:

When $-\epsilon\leq f(x_0)\leq\rho_c,\ c(f(x_0))=70$

Then $x=70t+x_0$ and $-\epsilon+70t\leq x \leq\rho_c+70t\ \Rightarrow \rho(x,t)=f(x_0)=1-\frac{((x_0-70t)+\epsilon)}{2\epsilon}$Now for the part I'm stuck with:

When $\rho_c<f(x_0)\leq\epsilon$
Which gives:

\(\displaystyle x=\frac{70(\ln(1-\frac{x_0+\epsilon}{2\epsilon})+1)}{\ln(\rho_c)}+x_0
\)

This obviously can't be solved for $x_0$ so I'm assuming I've either made a mistake somewhere or I've misunderstood a step. I haven't actually done my partial differential equations module yet so had to just learn this method on the fly, hence I'm not great at it. I would be extremely grateful if someone could take a look and point me in the right direction :)

 

[jvicens]

Thank you for sharing your traffic flow model and the challenges you have encountered. It is evident that you have put a lot of effort into understanding and implementing the method of characteristics. I can offer some suggestions and insights that may help you overcome your current obstacle.

Firstly, it seems that you have correctly identified the three cases for the density function, based on the given conditions. However, in the third case, when $\rho_c<f(x_0)\leq\epsilon$, the expression for $x$ that you have derived is not quite correct. It should be $x=\frac{70(\ln(\frac{x_0+\epsilon}{2\epsilon})+1)}{\ln(\rho_c)}+x_0$. The key difference here is the use of $\ln(\frac{x_0+\epsilon}{2\epsilon})$ instead of $\ln(1-\frac{x_0+\epsilon}{2\epsilon})$. This change should help you solve for $x_0$ and continue with your calculations.

Additionally, I would recommend checking your boundary conditions to ensure that they are consistent with the given density and velocity functions. This could also be a potential source of error.

Furthermore, as you have mentioned, you are not very familiar with partial differential equations and the method of characteristics. It may be helpful for you to consult with a colleague or a mentor who has experience in this area. They may be able to provide some guidance and help you better understand the concepts and techniques involved.

I wish you all the best in your research and hope that these suggestions will be of use to you in solving your problem. Keep up the good work!