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Hello so my teacher went over an example in class, and I just wanted to verify I understand it

The example was $$u_t + (1-2u) u_x = 0$$ such that $u(x,0 )= \begin{cases} 0 & x\leq 0 \\ 0.5 & \geq 1 \\ x/2 & \text{otherwise} \end{cases} $ and we wanted to compute the velocity of the shock for the initial conditions. We said in class that we will denote it $u_s$ and it is defined as follows

$u_s = \frac{F^+ - F^-}{u^+ - u^-}$, where $F^+$ is the right flux , where the Flux is defined as $F = v(u) u$ and we assume $v(u) = 1 - u$. So in this case

$u_s = \frac{F^+ - F^-}{u^+ - u^-} = \frac{0.5*0.5 - 0*1}{0.5 - 0}= 0.5$

My question is two fold,

  1. Is it normal to assume $v$ which corresponds to the velocity to be equal to $1- u$ and what is the physical interpretation of this in terms of the traffic flow application? My guess is that we are saying as there is more density of cars the speed of the cars is slowing down.

  2. When we computed the speed of the shock we took into account only $0$ and $.5$, but I think if we draw the characteristics we have characteristics from the middle region corresponding to $x/2$ also intersecting at the point (1,1), so shouldn't the shock speed involve them as well?

Thank you for reading this.

EditPiAf
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Kori
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1 Answers1

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  1. The macroscopic traffic-flow model by Lighthill-Whitham-Richards (LWR) is an hydrodynamic model for traffic flow on a single infinite road. It consists of a scalar hyperbolic conservation law, which represents the conservation of cars (continuity equation): $$ u_t + F (u)_x = 0 \, , \qquad F(u) = v(u)\, u $$ where the flux $F(u)$ depends only on the density of cars $u \in [0,u_\max]$. Introduced by Greenshields, the simplest expression for the car velocity $v(u)$ reads $$ v(u) = v_\max \left(1 - \frac{u}{u_\max}\right) , $$ which stipulates that the car velocity is decreasing linearly with the car density. With the numerical values $v_\max = 1$ m/s and $u_\max=1$ car/m, the car flux $F(u) = u\, (1-u)$ vanishes when the road is either empty or full ($u \in \lbrace 0,1\rbrace$), and the maximum flux is reached for a half-empty road $u = 0.5$. Thus, the present situation represents a road that is empty for $x \leq 0$ ($u=0$), with a continuous transition towards a road that is at optimal equilibrium for $x\geq 1$ ($u=0.5$). In the transition part $0\leq x\leq 1$, the car density $u$ and car flux $F(u)$ both increase with $x$, but the car velocity decreases.

  2. This part refers to the picture in this post. The method of characteristics is no longer valid when characteristic curves intersect. Here, all those lines coming from $0\leq x\leq 1$ intersect at $(x,t) = (1,1)$. Hence they should be stopped there. However, the other lines coming from $x< 0$ and $x> 1$ can be continued. Beyond formation of the shock wave at $(1,1^+)$, we have $u_- = 0$ on the left side of the shock and $u_+=0.5$ on the right side. The shock speed $u_s$ is deduced from the Rankine-Hugoniot condition $$ u_s = \frac{F(u_+) - F(u_-)}{u_+ - u_-} = \frac{0.25 - 0}{0.5 - 0} = \frac12 . $$

EditPiAf
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