Riemann problem for traffic flow

Question

I would like to solve for $u_t+(1-2u)u_x=0$ with initial condition $$u(x,0)=g(x)=\begin{cases}1, x<0\\ 0, x\ge0\end{cases}.$$ I used method of characteristic to get $\frac{dx}{dt}=1-2u$, $\frac{du}{dt}=0$ with initial conditions $u(x_0,0)=g(x_0)$, $x(0)=x_0$. Directly solving, we get $$u(x,t)=\begin{cases}0, x\ge t\\ 1, x<-t\end{cases}.$$ But what happens when $-t\le x< t$? Did I miss anything when solving the PDE?

Answer 1

The Lighthill-Whitham-Richards (LWR) traffic flow model is governed by a quasilinear PDE of the first order $u_t + a(u) u_x =0$, which can be rewritten as a scalar conservation law $u_t + f(u)_x =0$ with a suitable flux function $f(u) = u(1-u)$. The present initial-value problem describes a road that is empty after a red light ($x>0$) and completely saturated before the red light ($x<0$). Here cars progress towards increasing $x$. The red light turns green... The resolution of the Riemann problem described in this post leads to a rarefaction wave solution, i.e. the cars gradually start to move and pass the green light: $$u(x,t) = \begin{cases} 1, & x \leq -t,\\ \tfrac12 (1-x/t), & -t\leq x\leq t,\\ 0 , & x\geq t. \end{cases}$$