1. The standard traffic model for the car density ρ(x, t) is
ρt + Fx = 0 where v = 1 − ρ F(ρ) = ρv = ρ − ρ
Write down the strong solution to the initial-value problem with ρ(x, 0) = A(x) and find
the exact solution for the straight line initial conditions ρ(x, 0) = αx with constant α.
2. For the standard traffic model from question 1, the characteristic wave speed
c(ρ) = F
(ρ) = 1 − 2ρ.
Show that for strong solutions the Burgers equation holds for c(x, t), i.e.,
ct + ccx = 0.
Are the week solutions of this equation identical to those from question 1? (Hint: you
need to compare the shock speeds to answer this question.)
3. Draw the graph of F(ρ) for equation from question 1 and give a graphical interpretation of the shock speed formula
F(ρR) − F(ρL)
ρR − ρL
in terms of the slope of a secant. Also give a graphical interpretation of the Lax stability
c(ρL) ≥ cs ≥ c(ρR)
using the same graph. This allows determining the stability of shocks as a function of the
pairs (ρL, ρR).
4. A useful highway driving safety guideline is to keep a distance to the car in front
of you that is proportional to your speed, so 1/ρ ∝ v. A simple model of this rule
(including a speed limit) is
v(ρ) = (
1 0 ≤ ρ ≤ 1
1/ρ 1 < ρ
Here ρ ∈ [0,∞] is allowed. Compute and sketch F(ρ) and c(ρ) for this model. For strong
solutions, what does this model predict for the initial-value problem in which ρ(x, 0) is
specified? (Hint: you need to distinguish between characteristics on which ρ is bigger or
smaller than unity.) Compute the shock speed for this model and use the Lax criterion
to find the stability of shocks. (Hint: it is easiest to use the graphical construction from
the previous questions.)
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