5.4 Examples
5.4.2 Traffic flow models
5.4.2 Traffic flow models
5.4. Examples
(a) cuu/tuu (b) au/ru (c) cdu/tdu
Figure 5.9: Top row: spectra (black), branch pointsλcsatisfying the pinching condition (crosses) and absolute spectra (blue) for the microscopic Bando model. Bottom row: Dou-ble spatial eigenvalues ξc (crosses) and contour lines of log|χ(λc, ξ)| (blue) and log|χ(λc+ε, ξ)| (red), demonstrating that the pinching condition is satisfied. Note the triple points in λ ≈ −0.4 in (a) and λ≈ −0.1 in (b). In (c), choosing a higher value ofεwould move the two leftmost spatial eigenvalues to the left of the imaginary axis but obscure the fact that the second and third eigenvalue are meeting in ξc for λ=λc.
(a) cuu/tuu (he= 2, a= 1.6) (b) au/ru (he= 2, a= 0.8) (c) cdu/tdu (he= 3, a= 0.3)
Figure 5.10: Examples from instability classes (see Fig. 5.8) for the microscopic Bando model:
In the Eulerian frame, a small initial perturbation localised in x= 0 at initial time evolves as predicted by the classification
Example 5.4 (A model with two lanes)
We demonstrate that in a traffic flow context, we may encounter situations where the distinction between transient and convective instability becomes relevant, similar to the RDA example in Ex. 5.2.
Consider a situation where two different kinds of vehicles use adjacent lanes. On each lane, we impose Assumptions 2.1 and 2.2 regarding the indices, i.e. there is no overtaking or changing of lanes. However, there may still be interactions when the vehicles come close or pass each other.
Such an interaction depends of course on the context and may be modelled in different ways.
Maybe the simplest form is to assume that, due to the narrowness of the road, a high density of one class slows down the other and vice versa. This may be the case everywhere or only on some sections of the road.
Within this example, we denote the different species by subscripts 1,2 and write the vehicle index as an argument: xi(j, t) is the position at time tof vehiclej from species i, which in turn has the index set Ji.
5.4. Examples
x1(j+ 1) x1(j) x1(j−1)
x2(l−1) x2(l)
x2(l+ 1) x2(l−2)
Figure 5.11: Setup for two-lane model. In the depicted situation, the headwayh1→2 that vehicle j observes on the adjacent lane is calculated as a weighted average between the distances x2(l −1)−x2(l) and x2(l)−x2(l+ 1), ensuring continuity when x1(j) overtakes x2(l) or is itself overtaken by x2(l+ 1)
For fixed tand J1,2=Z, the function l1→2 :J1 →J2
j 7→min
l∈J2{l∈J2 :x2(l, t)≥x1(j, t)}
is well-defined and returns the indexl of the vehicle on lane 2 that is next to or directly ahead of vehiclejon lane 1; on the circular road, we may define an analogous function in a straightforward fashion. The functionl2→1:J2→J1 is defined symmetrically.
The vehicle at positionxj cuts the headwayh2(l1→2(j)) by a certain ratio α1→2(j) α1→2(j) := x2(l1→2(j))−x1(j)
x2(l1→2(j))−x2(l1→2(j) + 1) (5.67) which we may use for a weighted average betweenh2(l1→2(j)) andh2(l1→2(j)−1):
h1→2(j) :=α1→2(j)·h2(l1→2(j)) + (1−α1→2(j))·h2(l1→2(j)−1). (5.68) For this example, we prefer this approach over alternative ways along the lines discussed in Sec. 3.3.1 because it is relatively easy to implement and continuous when the vehicles move relative to each other. However, the exact definition is not crucial here.
We may now set
v˙1(j) =a1[W1(h1(j), h1→2(j), x1(j))−v1(j)] (5.69a) v˙2(j) =a2[W2(h2(j), h2→1(j), x2(j))−v2(j)] (5.69b) whereW1,2 are modified optimal velocity functions with
W1(h1, h2, x1) =V(h1)·(1−ε·f(h2)·g(x1)) ; (5.70) here,f(h) describes how the optimal speed is reduced by the headway on the adjacent lane and g(x) describes the heterogeneity of the narrowness along the road. With f ≡1, this reduces to two independent versions of the bottleneck model fromGasser and Werner (2010).
Assume now that the interaction takes only place in a certain region, i.e. g(x) has compact
support. Away from the narrow regions, the lanes behave independently and perturbations behave like discussed in Ex. 5.3.
Most interesting is the coupling of two lanes that are convectively up- and downstream unstable when considered separately at several points. In combination, their behaviour will be remnantly unstable, as in Ex. 5.2: The slightest coupling will cause a perturbation that is convected down-stream on one lane to reappear on the other lane, where it will travel updown-stream and “re-ignite” a perturbation on the downstream lane at the next coupling. This means that part of the perturba-tion is trapped and amplifies between the couplings, similar to the situaperturba-tion in a laser (Fig. 5.12).
(a) Lane 1 (b) Lane 2
(c) Lane 1 (d) Lane 2
Figure 5.12: Results of a two-lane simulation on a circular road with L = 100 and couplings at x= L2 ±10L. Subfigures (a,b) and (c,d) show velocity v and log|v−ve|, respectively.
A situation of special interest is that of two lanes with opposing directions. If the individual lanes are convectively upstream unstable in this scenario, the system will be remnantly unstable provided there are couplings between the lanes, which can hardly be avoided in practice.
5.4. Examples
5.4.2.2 Macroscopic models
Example 5.5 (Instability classification for the models linked to the Bando model) For the low-order macroscopic equivalents to the Bando model we can find the boundaries between convectively and absolutely unstable parameter regions analytically:
Withfh =aV0(he) =:aβ,fv =a, andf∆v = 0 for the Bando model, (5.24) becomes (λ−cz) (λ−cz+a)−aβ z+ z2
2
!
. (5.71)
Then setting ∂χ∂z = 0 yields
λ−cz=−a
2 +d(1 +z), whered:=−aβ
2c. (5.72)
We plug (5.72) into (5.71); the result
d2(1 +z)2− a2
4 −aβ z+ z2 2
!
= 0 (5.73)
is then solved for z:
z=−1± v u u
t1−d2−a42
d2−aβ2 . (5.74)
Usingλ = (d+c)z−a2 +d, we easily see that a pair of purely imaginary branch points crosses the imaginary axis for a = 2c (au-cdu). However, one of these eigenvalues returns if the line d2 = aβ2 is crossed. This corresponds to a= cβ2 (au-cuu). The resulting classification is displayed in Fig. 5.13(a).
For the Lagrangian frame we have to set c= 0. Starting again from (5.71), we have ∂χ∂z = 0 for z=−1 andλ2+aλ+aβ2 = 0, leading to branch pointsλ± =−a2±qa42 − aβ2 . Note that when the radicant is multiplied by 4a, we recover the string instability criterion, meaning that the radicant is negative for all relevant parameter values. So all of these are convectively unstable in a frame that is moving with the vehicles.
For the higher-order models the classification can be done by numerical continuation of the branch points satisfying the pinching condition. Since they have the same characteristic polynomial (cf.
Ex. 4.6), the classification for ND and FLD yields the same results (Fig. 5.13); however, the classification for the IHD is only marginally different (Fig. 5.17). Simulations at the representa-tive parameter values lead to very similar results for ND (Fig. 5.15), FLD (Fig. 5.16), and IHD (Fig. 5.18).
Overall we may conclude that, while the absolutely unstable parameter region starts out notably smaller in the first-order macroscopic models (Fig. 5.13 (a), Fig. 5.17 (a)) than in the microscopic model (Fig. 5.8), already at third order (Fig. 5.13 (c), Fig. 5.17 (c)) the classifications are visually indistinguishable from each other.
(a)O(ε1) (b)O(ε2) (c)O(ε3)
Figure 5.13: Stability classification in parameter space for ND/FLD Bando model equivalents up to third orders in ε. The classification in (a) was obtained analytically, (b) and (c) were classified by numerical continuation of branch points
(a) cuu (b) au (c) cdu
Figure 5.14: Examples from instability classes for O(ε1)-ND/IHD Bando model equivalent; same parameter values as in Fig 5.10 (cf. Appendix A, Fig. 1 for density plots)
5.4. Examples
(a) cuu (b) au (c) cdu
Figure 5.15: Examples from instability classes for O(ε2)-ND Bando model equivalent, (cf. Ap-pendix A, Fig. 2 for density plots)
(a) cuu (b) au (c) cdu
Figure 5.16: Examples from instability classes for O(ε2) FLD Bando model equivalent; same pa-rameter values as in Fig 5.10 (cf. Appendix A, Fig. 3 for density plots)
(a) 1st order (cf. Fig. 5.13(a)) (b) 2nd order (c) 3rd order
Figure 5.17: Stability classification in parameter space for IHD Bando model equivalents for dif-ferent orders inε
(a) cuu (b) au (c) cdu
Figure 5.18: Examples from instability classes for O(ε2)-IHD Bando model equivalent; same pa-rameter values as in Fig 5.10 (cf. Appendix A, Fig. 4)
III Shift-invariant solutions
Chapter 6
Finding solutions
In this chapter, we are studying a special kind of travelling wave solutions in which the vehicle trajectories are shifted copies of each other with a constant translation vector between them.
Such solutions are typical for periodic solutions on the circular road. In Sec. 6.1 we are going to recall some of the existing theory on periodic solutions on the circular road and try to generalise it to the infinite lane withJ =Z. This will also help us to find heteroclinic solutions, a topic we are going to discuss in Sec. 6.2.
Much of the material presented in this chapter is currently under review in similar form in von Allw¨orden and Gasser(submitted 2018).