Let $u = u(x,t)$ be the density of cars at $x$ (ie. cars per unit length at $x$). Let also, $f(u) = u^2$ be the flux of cars at any point $x$, with some initial conditions (described below). In other words, we want to solve:
$$u_t + 2u u_x = 0 \text{ for } -\infty < x < \infty, \text{ and } 0 <t<\infty$$
under the initial conditions:
$$u(x,0) = \left\{ \begin{array}{ll} 1 & x\leq 0 \\ 1-x & 0 < x < 1 \\ 0 & 1\leq x \\ \end{array} \right. $$
This is a problem from Stanley J. Farlow's book, and to solve it he uses the method of characteristics which makes sense to me.
Using invariance along these lines he then proceeds to draw density diagrams over $x$, for different times, very much like the ones that I attached here (see picture below). Notice how $u=0$ for all $x > 1$. This seems rather strange to me.
What I find unclear is why density doesn't propagate past the point $x=1$? I mean, I understand that for $1 \leq x_0 < \infty$ the characteristics are given by $x = x_0$ (vertical lines), but how is this consistent with our intuition about density moving from left to right? (In the same direction that cars are moving)
In other words, since we have a non-zero density up to $x \leq 1$, doesn't that mean that density should propagate forward (and past the point $x=1$) at a future time? What am I missing?
What is the physical interpretation of these diagrams?
I am very surprised that any text would present this as a model of traffic flow. The flux of vehicles is $f(u)=uv(u)$ where $v(u)$ is the vehicle speed at $v=u$. Clearly $v(u)$ should be a decreasing function of $u$ and traffic engineers find it by fitting to observations. For educational purposes it is very common to take $v(u)=1-u$ which leads to a more credible outcome. The given equation is a ridiculous model of traffic behavior, and that is why you have difficulty understanding your results.
The equation as given, however, does frequently appear in mathematics books (more usually with $f(u)=\frac{1}{2}u^2$) because the algebra is extremely simple. It can be rewritten $$u_t+2uu_x=0$$ as you do. This is a statement that $u$ is constant along lines $dx/dt=2u$ and these are the characteristic lines. Characteristic lines are lines along which information travels, and are usually quite distinct from lines along which vehicles travel. Here the vehicle paths are $dx/dt=u$. This is a very fundamental point about wave propagation, that the speed of a wave is often not the speed of the medium that carries it.
The behavior for $t<\frac{1}{2}$ is just as you draw it; it is correct so long as you do not try to interpret it as traffic flow. After that a shock will form. If the states on either side of a shock are $u_L,u_R$, then the shock will move with a speed given by the form of the conservation law as $$ S=\frac{f(u_R)-f(u_L)}{u_R-u_L},$$ which for $f(u)=u^2$ is just $S=u_L+u_R$. The characteristics on either side terminate by colliding with the shock. This is thermodymically important because information is lost and entropy is created.
In a better traffic model, the characteristic lines also terminate in the shock, but the vehicle paths pass through it with a discontinuous change in velocity. The only admissible shock solutions are those through which the vehicles slow down. They represent a narrow region within which drivers slow down frantically, and its a bit like a gasdynamic shock within which molecular collisions are very frequent.
This little problem could have been very instructive. For a better treatment, get hold of R. J. LeVeque, Numerical Methods for Conservation Laws.