Hydrodynamic traffic flow models including random accidents: A kinetic derivation

Abstract

We present a formal kinetic derivation of a second order macroscopic traffic model from a stochastic particle model. The macroscopic model is given by a system of hyperbolic partial differential equations (PDEs) with a discontinuous flux function, in which the traffic density and the headway are the averaged quantities. A numerical study illustrates the performance of the second order model compared to the particle approach. We also analyse numerically uncertain traffic accidents by considering statistical measures of the solution to the PDEs.

Full text

Hydrodynamic traffic flow models including random
accidents: A kinetic derivation
F. A. Chiarello1, S. G¨ottlich2, T. Schillinger2, and A. Tosin3
1University of L’Aquila, DISIM, 67100 L’Aquila, Italy
2University of Mannheim, School of Business Informatics and Mathematics, 68159 Mannheim, Germany
3Politecnico di Torino, Department of Mathematical Sciences “G. L. Lagrange”, 10129 Torino, Italy
Abstract
We present a formal kinetic derivation of a second order macroscopic traffic model from a
stochastic particle model. The macroscopic model is given by a system of hyperbolic partial
differential equations (PDEs) with a discontinuous flux function, in which the traffic density
and the headway are the averaged quantities. A numerical study illustrates the performance
of the second order model compared to the particle approach. We also analyse numerically
uncertain traffic accidents by considering statistical measures of the solution to the PDEs.
Keywords: particle models, Follow-the-Leader, macroscopic traffic models, random acci-
dents, uncertainty quantification
Mathematics Subject Classification: 35Q20, 35Q70, 90B20
1 Introduction
Traffic flow can be modelled at different scales for example using ordinary differential equations
(ODEs), kinetic equations or partial differential equations (PDEs). ODE-based models describe
microscopically the behaviour of individual vehicles. In this paper, we consider especially the
class of Follow-the-Leader (FTL) models [11], in which the vehicle dynamics are influenced by the
distance to the vehicle in front.
The modelling of vehicular traffic can also be based on the statistical representation of inter-
acting particle systems along the lines of the collisional kinetic theory. In this approach, vehicles
are regarded as indistinguishable particles, whose pairwise interactions produce speed variations.
The indistinguishability assumption allows one to describe them by means of the statistical dis-
tribution of their speed, pretty much like in the kinetic approach to gas dynamics introduced
by Boltzmann. The pioneer of the kinetic approach to vehicular traffic was Prigogine [22,23],
who in the 1960s proposed to adapt the classical concepts of the statistical physics of gases to
vehicles along a road, so as to obtain a mathematical representation of traffic which could serve
as a link between the microscopic vehicle-wise and the macroscopic fluid dynamic descriptions.
Since then, several improvements have been proposed, such as the use of Enskog-type rather than
Boltzmann-type kinetic equations to capture the non-locality of vehicle interactions, see e.g. [18],
up to more general Povzner-Boltzmann-type equations, which are able to explain the genesis of
non-local macroscopic traffic models, see e.g. [6]. The kinetic approach has proved useful also in
upscaling microscopically controlled vehicle dynamics to the macroscopic scale, whereby hydro-
dynamic traffic models have been deduced, which incorporate consistently the effect of driver-assist
or automated vehicles on the mean flow, see e.g. [4,8].
Macroscopic models, which do not focus on individual vehicles but on the traffic density as
an aggregate quantity, take inspiration from fluid dynamics and therefore are based on hyperbolic
conservation laws. They were first introduced by Lighthill, Whitham and Richards [19,24] in the
1

1950s. Since then, first order models have been extended in several directions. Aw, Rascle and
Zhang [1,28] introduced second order traffic models where density and speed are considered as
the averaged quantities.
In this paper, starting from FTL microscopic dynamics, we adopt an Enskog-type kinetic
approach to obtain a second order macroscopic model in which, besides the vehicle density, the
traffic flow is described by the mean headway among the vehicles. The resulting hydrodynamic
model is original in that the mean headway is treated as a second aggregate variable independent
of the traffic density, whereas in classical traffic models it is empirically assumed to be proportional
to the inverse of the latter. Moreover, thanks to the kinetic approach, the model is obtained as
aphysical limit of fundamental particle dynamics. This introduces a remarkable difference with
respect to the mainstream in the reference literature, where the link between microscopic and
macroscopic descriptions of traffic is usually established by showing that selected versions of the
former may be used as numerical discretisation of the latter, with convergence in appropriate
particle limits. Furthermore, our model allows for an additional space dependence in the flux
function for varying road capacities to model traffic accidents. For the sake of completeness, we
note that hyperbolic partial differential equations with an additional space dependence in the flux
have been studied before, e.g., in [16,17,27], and for a first order traffic model in [13], where a
corresponding microscopic model and its convergence are discussed.
As mentioned above, as an application we use our space-dependent macroscopic model to de-
scribe traffic accidents, which we understand as capacity drops in the flux function. This idea
was developed in [12,13]. Further approaches to traffic accident modelling can be found in the
literature from various disciplines: for instance, by using kinetic models [9] or by constructing
Bayesian networks [20,30] and recently also neural networks [10,29]. In our case, the physical
limit mentioned above, along with the probabilistic/statistical setting of the kinetic theory, allows
us to treat accidents as capacity drops in random, viz. uncertain, locations along the road, which
translates into a macroscopic model featuring a realistically uncertain flux. For the sake of com-
pleteness, we report that the effect of uncertain quantities on traffic dynamics has already been
taken into account in a number of other papers. Without pretending to be exhaustive, we mention
that in [14] uncertain lateral speeds, orthogonal to the main traffic stream, are introduced to model
the displacement of vehicles across the lanes of a multi-lane road; in [15] uncertain vehicle interac-
tions are considered, in a homogeneous Boltzmann-type kinetic modelling framework, to explain
the emergence of equilibrium speed distributions comparable with those shown by rough traffic
data; in [26] a theoretical investigation, still based on concepts and tools of the collisional kinetic
theory, is proposed concerning the ability of autonomous vehicles to mitigate the impact of uncer-
tain vehicle interactions on the aggregate traffic predictions; finally, in [2] analytical properties of
conservation laws with uncertain and discontinuous flux functions are discussed.
In more detail, the plan of this work is as follows. In Section 2we introduce the underlying
microscopic FTL traffic model, which we use in Section 3to derive stochastic particle dynamics
and therefrom the second order macroscopic limiting model with density and mean headway, of
which we discuss some relevant analytical properties. We also show that in an appropriate regime
of the parameters of the particle dynamics, the second order model relaxes towards a first order
model, in which the main aggregate quantity is the traffic density while the mean headway is
expressed as a function of the density derived from the interaction rules among the particles.
We complete the picture by illustrating numerically the results. In Section 4we extend the
multiscale modelling framework above with the inclusion of random accidents and we undertake a
computational analysis of the performances of the model with accidents using different numerical
simulation strategies to capture the expected values of the traffic density and mean headway at the
various scales. Finally, in Section 5we draw some conclusions and briefly sketch possible research
developments.
2

2 Microscopic dynamics with headway
We consider the following Follow-the-Leader (FTL) microscopic model introduced in [13]
˙xi(t) = c(xi(t)) ˜
VL
xi+1(t)−xi(t), i = 1,2, . . . (1)
where L > 0 is a reference vehicle length, xi(t)∈Rthe position of the i-th vehicle at time tand
˜
Va given speed function. Moreover, c=c(x) : R→[0,1] is a prescribed function modulating the
actual speed of the vehicles depending on the road capacity in the point x∈R.
We define the distance between two consecutive vehicles iand i+ 1, i.e. the headway of the
i-th vehicle, as
si(t) := xi+1(t)−xi(t),
whence, using (1),
˙si(t) = c(xi+1(t)) ˜
VL
si+1(t)−c(xi(t)) ˜
VL
si(t).
If the vehicles i,i+ 1 participating in the interaction described by this equation are meant to
be representative of any pair of interacting vehicles, we can drop the indices i,i+ 1 and define
the generic positions x:= xi(t), x∗:= xi+1(t) and the pre-interaction headways s:= si(t),
s∗:= si+1(t). Furthermore, if we assume that the variation of the headway in consequence of
an interaction takes place in a small time interval of size γ > 0 we can approximate ˙si(t)≈
si(t+γ)−si(t)
γ, where we identify s′:= si(t+γ) as the post-interaction headway. Finally, we convert
the previous ODE into the following algebraic binary interaction rule
s′=s+γc(x∗)V(s∗)−c(x)V(s),(2)
where we have denoted V(s) := ˜
V(L
s).
In addition to the FTL interaction dynamics described by (2), we consider also a spontaneous
relaxation of the headway of each vehicle towards an optimal/recommended headway H, which
we assume to be given as a function of the global density ρof traffic: H=H(ρ). Hence we couple
to (2) a second update rule of the headway of the form
s′′ =s+a(H(ρ)−s),(3)
where a > 0 is a relaxation parameter.
On the functions V,Hwe make the following assumptions:
Assumption 2.1. We assume that the speed Vis a non-negative function of the headway s≥0
with the following characteristics:
(i) it is differentiable and monotonically increasing:
V′(s)>0,∀s∈R+;
(ii) there exists a constant C > 0such that
0≤V(s)≤Cs, ∀s∈R+.
Furthermore, we assume that the optimal/recommended headway His a non-negative, differenti-
able and monotonically decreasing function of the traffic density ρ≥0.
Assumption 2.1(i) is quite natural from the modelling point of view: the larger the distance
between two consecutive vehicles the faster they travel. Analogously, the assumption on the
monotonicity of Hhas a modelling value, because it implies that the more congested the traffic
the closer the vehicles are forced to stay.
Conversely, Assumption 2.1(ii) is needed in order to guarantee the physical consistency of the
binary interaction rule (2), in particular the fact that s′≥0 for all s, s∗≥0. Using Assump-
tion 2.1(ii), we easily check that this condition is satisfied if γ≤C. Finally, as far as the update
rule (3) is concerned, we notice that the analogous condition of physical consistency s′′ ≥0 for all
s≥0 is guaranteed if a≤1, thanks to the non-negativity of H.
3

3 Enskog-type kinetic description and hydrodynamics
We consider the superposition of the FTL interaction dynamics (2) and the Optimal Headway (OH)
relaxation dynamics (3) and we show that, in the hydrodynamic limit, two types of macroscopic
models can be obtained. If OH dynamics happen at a much slower rate than FTL dynamics
then we get an inhomogeneous second-order macroscopic model featuring the traffic density and
the mean headway as hydrodynamic parameters. Conversely, OH and FTL dynamics happen at
comparable rates then we get a first order Lighthill-Whitham-Richards type-model featuring only
the traffic density as hydrodynamic parameter.
To obtain these results, we rely on an Enskog-type collisional kinetic description of the system
of interacting particles subject to the rules (2), (3).
3.1 Slow relaxation regime
We consider a large ensemble of indistinguishable vehicles, each of which is identified by the
dimensionless position Xt∈Rand dimensionless headway St∈R+at time t > 0. Motivated by
the rules (2), (3), we consider the following discrete-in-time stochastic particle model:
(Xt+∆t=Xt+c(Xt)V(St)∆t,
St+∆t=St+γΘc(X∗
t)V(S∗
t)−c(Xt)V(St)+ Ξa(H(ρ)−St),(4)
where ∆t > 0 is a (small) time step. Moreover, Θ,Ξ∈ {0,1}are Bernoulli random variables
describing whether during the time step ∆ta randomly chosen vehicle with microscopic state
(Xt, St): (i) updates (Θ = 1) or not (Θ = 0) its headway owing to an FTL interaction with the
leading vehicle in X∗
t; (ii) updates (Ξ = 1) or not (Ξ = 0) its headway because of an OH relaxation
towards the optimal headway H(ρ). In more detail, we let
Θ∼Bernoulli(∆t),Ξ∼Bernoulli(ε∆t),(5)
thereby assuming that the probability for either updates to happen is proportional to ∆t. The
parameter ε > 0 is used to differentiate the rate of OH relaxation from that of FTL interactions.
In particular, here we assume that OH relaxation is much slower than FTL interactions, i.e.
ε≪1.
Furthermore, we need ∆t≤1 for consistency.
To reach an aggregate statistical description of our particle system we introduce the kinetic
distribution function f=f(x, s, t) : R×R+×(0,+∞)→R+of the microscopic state (x, s) of
a generic representative vehicle at time t. In essence, f(x, s, t)dx ds gives the probability that
a vehicle has a position comprised between xand x+dx and a headway comprised between s
and s+ds at time t. Then, by standard arguments, see e.g., [21], averaging (4) and taking the
continuous-time limit ∆t→0+we obtain formally that fsatisfies the following equation:
∂tZ+∞
0
φ(s)f(x, s, t)ds +∂xc(x)Z+∞
0
φ(s)V(s)f(x, s, t)ds
=1
2Z+∞
0Z+∞
0
(φ(s′)−φ(s))f(x, s, t)f(x+η, s∗, t)ds ds∗
+εZ+∞
0
(φ(s′′)−φ(s))f(x, s, t)ds, (6)
for every choice of φ:R+→R, which here plays the role of a test function, where s′and s′′ on
the right-hand side are given by (2), (3), respectively.
Notice that (6) is the weak form of a collisional kinetic equation in which we have assumed
that vehicles interact when they are at a distance η > 0 from each other (cf. the first term on the
4

right-hand side). In other words, with respect to the notation used in the interaction rule (2), we
have assumed x∗=x+η. Therefore, (6) is an Enskog-type kinetic equation, which, as discussed
in [18], is more appropriate than a Boltzmann-type equation to model vehicular traffic. The main
reason is that a non-locality in space of the interactions is necessary in order to reproduce density
waves possibly travelling backwards in spite of the non-negativity of the microscopic car speeds.
In order to make (6) more amenable to further analytical investigations, we assume that the
non-locality ηis sufficiently small so that we can approximate
f(x+η, s∗, t)≈f(x, s∗, t) + η∂xf(x, s∗, t).(7)
An analogous approximation applies also to the term
c(x∗) = c(x+η)≈c(x) + c′(x)η, (8)
contained in s′, cf. (2), whence, assuming φsmooth,
φ(s′) = φs+γc(x∗)V(s∗)−c(x)V(s)
=φ˜s′+γηc′(x)V(s∗)
≈φ(˜s′) + φ′(˜s′)γηc′(x)V(s∗),
where we have set
˜s′:= s+γc(x)V(s∗)−V(s).(9)
Plugging these approximations into (6) and enforcing the equality we get the following approxim-
ated kinetic equation:
∂tZ+∞
0
φ(s)f(x, s, t)ds +∂xc(x)Z+∞
0
φ(s)V(s)f(x, s, t)ds
=1
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))f(x, s, t)f(x, s∗, t)ds ds∗
+γη
2c′(x)Z+∞
0Z+∞
0
φ′(˜s′)V(s∗)f(x, s, t)f(x, s∗, t)ds ds∗
+η
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))f(x, s, t)∂xf(x, s∗, t)ds ds∗
+γη2
2c′(x)Z+∞
0Z+∞
0
φ′(˜s′)f(x, s, t)∂xf(x, s∗, t)ds ds∗
+εZ+∞
0
(φ(s′′)−φ(s))f(x, s, t)ds. (10)
3.1.1 Hydrodynamic limit
To pass from the kinetic description (10) to the hydrodynamic regime, we use the parameter εas
a sort of Knudsen number. Specifically, we scale time and space as
t→t
ε, x →x
ε,(11)
5

whence ∂t→ε∂t,∂x→ε∂x,c′(x)→εc′(x), and consequently we rewrite (10) as
∂tZ+∞
0
φ(s)fε(x, s, t)ds +∂xc(x)Z+∞
0
φ(s)V(s)fε(x, s, t)ds
=1
2εZ+∞
0Z+∞
0
(φ(˜s′)) −φ(s))fε(x, s, t)fε(x, s∗, t)ds ds∗
+γη
2c′(x)Z+∞
0Z+∞
0
φ′(˜s′)V(s∗)fε(x, s, t)fε(x, s∗, t)ds ds∗
+η
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))fε(x, s, t)∂xfε(x, s∗, t)ds ds∗
+εγη2
2c′(x)Z+∞
0Z+∞
0
φ′(˜s′)V(s∗)fε(x, s, t)∂xfε(x, s∗, t)ds ds∗
+Z+∞
0
(φ(s′′)−φ(s))fε(x, s, t)ds,
(12)
where fε(x, s, t) := f(x
ε, s, t
ε) denotes the distribution function parameterised by ε. Next, we
introduce the following definition.
Definition 3.1. We call collisional invariant of the kinetic equation (12) any quantity φ:R+→R
such that Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))fε(x, s, t)fε(x, s∗, t)ds ds∗= 0.
It is not difficult to check that φ(s) = 1 and φ(s) = sare collisional invariants in the sense of
Definition 3.1. Plugging these two quantities into (12) we obtain therefore, for every ε > 0,















































∂tZ+∞
0
fε(x, s, t)ds +∂xc(x)Z+∞
0
V(s)fε(x, s, t)ds= 0,
∂tZ+∞
0
sfε(x, s, t)ds +∂xc(x)Z+∞
0
sV (s)fε(x, s, t)ds
=γη
2c′(x)Z+∞
0Z+∞
0
V(s∗)fε(x, s, t)fε(x, s∗, t)ds ds∗
+γη
2c(x)Z+∞
0Z+∞
0
(V(s∗)−V(s))fε(x, s, t)∂xfε(x, s∗, t)ds ds∗
+εγη2
2c′(x)Z+∞
0Z+∞
0
V(s∗)fε(x, s, t)∂xfε(x, s∗, t)ds ds∗
+aZ+∞
0
(H(ρ)−s)fε(x, s, t)ds.
(13)
On the other hand, in the hydrodynamic limit ε→0+equation (12) implies that the limit
distribution f0satisfies formally
Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))f0(x, s, t)f0(x, s∗, t)ds ds∗= 0,(14)
for every observable quantity φ. We call a solution f0to this equation a local Maxwellian, i.e. an
equilibrium distribution of the headway sfor fixed x,t.
Assume we are given a local Maxwellian. Then, passing to the hydrodynamic limit also in (13)
6

we discover:





































∂tZ+∞
0
f0(x, s, t)ds +∂xc(x)Z+∞
0
V(s)f0(x, s, t)ds= 0,
∂tZ+∞
0
sf0(x, s, t)ds +∂xc(x)Z+∞
0
sV (s)f0(x, s, t)ds
=γη
2c′(x)Z+∞
0Z+∞
0
V(s∗)f0(x, s, t)f0(x, s∗, t)ds ds∗
+γη
2c(x)Z+∞
0Z+∞
0
(V(s∗)−V(s))f0(x, s, t)∂xf0(x, s∗, t)ds ds∗
+aZ+∞
0
(H(ρ)−s)f0(x, s, t)ds.
(15)
Clearly, any local Maxwellian is defined up to the collisional invariants. This means, in partic-
ular, that f0is parameterised by:
•the traffic density
ρ(x, t) := Z+∞
0
f0(x, s, t)ds;
•the mean headway
h(x, t) := 1
ρ(x, t)Z+∞
0
sf0(x, s, t)ds.
Indeed equation (14), which for φ(s) = 1, s is trivially satisfied because of Definition 3.1, cannot
determine univocally the zeroth and first order s-moments of f0. Consequently, if an explicit
expression of f0is available, system (15) may provide the evolution equations for ρ,h, thus the
macroscopic counterpart of the particle traffic model (4).
It is not difficult to check, by direct substitution in (14), that the monokinetic distribution
f0(x, s, t) = ρ(x, t)δ(s−h(x, t)),(16)
δ(·) being the Dirac distribution, is a local Maxwellian. Uniqueness of (16) is hard to obtain
for a completely general speed function V. Nevertheless, it can be recovered, under additional
assumptions on V, in the quasi-invariant regime of the particle dynamics (4), i.e. a regime
reminiscent of the grazing collision regime of the classical kinetic theory in which every particle
interaction produces a little change of microscopic state but interactions are quite frequent. See [5]
for further details.
Using (16) in (15) we get, after some computations, the second order macroscopic model





∂tρ+∂xc(x)V(h)ρ= 0,
∂t(ρh) + ∂xc(x)V(h)ρh=γ
2ρ2η∂xc(x)V(h)+aρ(H(ρ)−h),
(17)
in the hydrodynamic parameters ρ,h.
3.1.2 Analytical insights into the macroscopic system
Despite the general derivation of a macroscopic description incorporating both FTL and relaxation
dynamics, here we focus on system (17) without relaxation term:





∂tρ+∂xc(x)V(h)ρ= 0,
∂t(ρh) + ∂xc(x)V(h)ρh=γ
2ρ2η∂xc(x)V(h),
(18)
7

with (x, t)∈R×[0,+∞). System (18) constitutes the hydrodynamic counterpart of the original
microscopic FTL model (1).
We notice that we can write (18) in conservative form defining the pressure function p(ρ) = γ
2ηρ,
i.e. 


∂tρ+∂xc(x)V(h)ρ= 0,
∂tρ(h+p(ρ))+∂xc(x)V(h)ρ(h+p(ρ))= 0.
(19)
Now we rewrite system (19) as









∂tρ+∂xcV (h)ρ= 0,
∂tρ(h+p(ρ))+∂xcV (h)ρ(h+p(ρ))= 0,
∂tc= 0.
(20)
This is a 3 ×3 system for V:= (ρ, ρ(h+p(ρ)), c)Tand we complement it with the initial datum
¯u(x) = (¯ρ, ¯
h, ¯c)(x) s.t. ¯ρ(x)>0,¯
h(x)>0,¯c(x)>0.(21)
The quasilinear vector form of the system (20) is
∂tU+A(U)∂xU=0,
with U:= (ρ, h, c)Tand
A(U) := 

cV (h)V′(h)cρ V (h)ρ
0cV(h)−γ
2ηρV ′(h)−γ
2ηρV (h)
0 0 0 
.
The eigenvalues λ1, λ2, λ3,and eigenvectors r1, r2, r3,of this matrix are
λ3=cV (h) with r3= (1,0,0),
λ2=cV(h)−γ
2ηρV ′(h)with r2=1,−γ
2η, 0,
and
λ1= 0 with r1=V(h)ρ, −γ
2ηV (h)ρ, c ργ
2ηV ′(h)−V(h).
Since the eigenvalues are real and A(U) is diagonalisable, system (17) is hyperbolic. Furthermore,
under the Assumptions V′(h)>0, V (h)>0 and V(h)>γ
2ηρV ′(h), it results λ1< λ2< λ3,
then the system is strictly hyperbolic. No characteristic speed is greater than the flow speed.
Hence (18) complies with the Aw-Rascle consistency condition. The first characteristic field and
the third characteristic field are linearly degenerate: ∇λ1·r1= 0,∇λ3·r3= 0, thus the associated
waves are contact discontinuities. Conversely, the second characteristic field is genuinely nonlinear:
∇λ2·r2= 0, hence the associated waves are either shocks or rarefactions, if the velocity function is
such that V′(h)−ρ
2V′′(h)= 0. It is worth noticing that, choosing V(h) := h
1+hwe get ∇λ2·r2<0.
In this setting, in order to prove the global existence of entropy solutions, we can apply [3, Theorem
7.1] under a quite restrictive assumption on the total variation of the initial datum.
Theorem 3.2. Let us consider the set Ωsuch that for every (ρ, h, c)∈Ωthe following conditions
are verified:
•V(h), c and V′(h)are strictly greater than zero;
•V(h)−γ
2ηρV ′(h)= 0;
•V′(h)−ρ
2V′′(h)= 0.
8

For every compact K⊂Ωthere exists a constant δ > 0with the following property. For every
initial condition ¯uwith
TV(¯u)≤δ, lim
x→−∞ ¯u(x)∈K, (22)
the Cauchy problem (20)-(21)has a weak entropy solution u(t, x)=(ρ, h, c)(t, x),defined for all
t≥0.If a convex entropy ˜ηis given, then the Cauchy problem (20)-(21)admits an ˜η−admissible
solution.
In order to give more analytical insights, let us consider the Riemann problem for our sys-
tem (20), with initial data
¯u(0, x) := ((ρ−, h−, c−), x < 0,
(ρ+, h+, c+), x > 0,
assuming ρ±, h±, c±>0.By r1, r2, r3the rarefaction curves through (ρ−, h−, c−) are obtained
solving the following Cauchy problems. From r1we get





˙ρ=V(h)ρ,
˙
h=−γ
2ηV (h)ρ,
˙c=cργ
2ηV ′(h)−V(h).
This yields to





˙ρ=V(h)ρ,
˙
h=−γ
2η˙ρ,
˙c=cργ
2ηV ′(h)−V(h),
then, we end up with the implicit curve
R1:σ1→ρ−eRσ1
0V(h(σ1))dσ1, h−−γ
2η(ρ(σ1)−ρ−), c−eRσ1
0(−V(h(σ1))+ γ
2ηρV ′(h(σ1)))dσ1.
From r2we write





dh
dρ =−γ
2η,
h(ρ−) = h−,
˙c= 0.
This yields to the curve
R2:σ2→(σ2+ρ−, h−−γ
2ησ2, c−),
that can be rewritten as
R2=n(ρ, h, c) : h−h−=−γ
2ηρ−ρ−, c =c−o.
From r3we have





˙ρ= 1,
˙
h= 0,
˙c= 0.
This yields the curve
R3:σ3→(σ3+ρ−, h−, c−).
The shock curves S1,S2and S3through (ρ−, h−, c−) are derived from the Rankine-Hugoniot
conditions
λ(ρ−−ρ) = c−V(h−)ρ−−cV (h)ρ,
9

λρ−(h−+p(ρ−)) −ρ(h+p(ρ))=c−ρ−V(h−)h−+p(ρ−)−cρV (h) (h+p(ρ)) ,
λ(c−−c) = 0.
We can observe by a straightforward computation that S1coincides with the rarefaction curve
R1,S2coincides with the rarefaction curve R2and S3coincides with R3,being the characteristic
fields associated to r1and r3linearly degenerate.
Remark 3.3.It is interesting to notice that system (20) reduces to the following 2×2 system when
cis constant: 


∂tρ+c∂xV(h)ρ= 0,
∂tρ(h+p(ρ))+c∂xV(h)ρ(h+p(ρ))= 0.
This system of conservation laws belongs to the Temple class, see [7,25], and deserves more
analytical attention, for this reason a deeper study will be done in a future work.
3.2 Fast relaxation regime
We now analyse the case in which the FTL and the relaxation dynamics take place at the same
rate. This means that, in place of (5), we consider
Θ,Ξ∼Bernoulli(∆t),
which produces the following weak form of the Enskog-type kinetic equation:
∂tZ+∞
0
φ(s)f(x, s, t)ds +∂xc(x)Z+∞
0
φ(s)V(s)f(x, s, t)ds
=1
2Z+∞
0Z+∞
0
(φ(s′)−φ(s))f(x, s, t)f(x+η, s∗, t)ds ds∗
+Z+∞
0
(φ(s′′)−φ(s))f(x, s, t)ds. (23)
in place of (6). The difference with respect to the latter is that the two terms at the right-hand
side are now of the same order of magnitude. Repeating the approximations (7), (8) under the
assumption of small ηyields
∂tZ+∞
0
φ(s)f(x, s, t)ds +∂xc(x)Z+∞
0
φ(s)V(s)f(x, s, t)ds
=1
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))f(x, s, t)f(x, s∗, t)ds ds∗
+γη
2c′(x)Z+∞
0Z+∞
0
φ′(˜s′)V(s∗)f(x, s, t)f(x, s∗, t)ds ds∗
+η
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))f(x, s, t)∂xf(x, s∗, t)ds ds∗
+γη2
2c′(x)Z+∞
0Z+∞
0
φ′(˜s′)f(x, s, t)∂xf(x, s∗, t)ds ds∗
+Z+∞
0
(φ(s′′)−φ(s))f(x, s, t)ds, (24)
in place of (10), ˜s′being again given by (9).
10

3.2.1 Hydrodynamic limit
Under the scaling (11), the kinetic equation (24) in the scaled distribution function fεbecomes:
∂tZ+∞
0
φ(s)fε(x, s, t)ds +∂xc(x)Z+∞
0
φ(s)V(s)fε(x, s, t)ds
=1
2εZ+∞
0Z+∞
0
(φ(˜s′)−φ(s))fε(x, s, t)fε(x, s∗, t)ds ds∗
+γη
2c′(x)Z+∞
0Z+∞
0
φ′(˜s′)V(s∗)fε(x, s, t)fε(x, s∗, t)ds ds∗
+η
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))fε(x, s, t)∂xfε(x, s∗, t)ds ds∗
+εγη2
2c′(x)Z+∞
0Z+∞
0
φ′(˜s′)fε(x, s, t)∂xfε(x, s∗, t)ds ds∗
+1
εZ+∞
0
(φ(s′′)−φ(s))fε(x, s, t)ds. (25)
At this point, we introduce the following new definition of collisional invariants, which for (25)
replaces Definition 3.1:
Definition 3.4. We call collisional invariant of the kinetic equation (25) any quantity φ:R+→R
such that
1
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))fε(x, s, t)fε(x, s∗, t)ds ds∗+Z+∞
0
(φ(s′′)−φ(s))fε(x, s, t)ds = 0.
It is immediate to check that φ(s) = 1 is again a collisional invariant whereas φ(s) = sis not.
Therefore, plugging φ(s) = 1 into (25) we are left, for every ε > 0, with the equation
∂tZ+∞
0
fε(x, s, t)ds +∂xc(x)Z+∞
0
V(s)fε(x, s, t)ds= 0,
which, passing to the hydrodynamic limit ε→0+, yields formally
∂tZ+∞
0
f0(x, s, t)ds +∂xc(x)Z+∞
0
V(s)f0(x, s, t)ds= 0,(26)
f0being a local Maxwellian which satisfies
1
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))f0(x, s, t)f0(x, s∗, t)ds ds∗+Z+∞
0
(φ(s′′)−φ(s))f0(x, s, t)ds = 0,
for every observable quantity φ:R+→R. For φ(s) = sthis relationship gives
ρ(H(ρ)−h)=0,
ρ,hbeing the density and the mean of f0. Next, by direct substitution we discover that the
monokinetic distribution
f0(x, s, t) = ρ(x, t)δs−H(ρ(x, t)),
parameterised by ρ(i.e., the macroscopic parameter associated with the only collisional invariant)
is a local Maxwellian. Again, uniqueness of such a local Maxwellian is hard to obtain for a generic
speed function Vbut can be guaranteed at least in the quasi-invariant regime under additional
assumptions on V, cf. [5].
Inserting the f0above into (26) we finally get the first order macroscopic model
∂tρ+∂xc(x)ρV (H(ρ))= 0,(27)
in the hydrodynamic parameter ρ. Notice that with H(ρ)=1/ρ this becomes the equation
considered in [13].
11

3.3 Numerical simulations
3.3.1 Discretisations
Consider an equispaced spatial grid (˜xi)i=1,..., ˜
Nwith step size ∆x > 0 and a temporal grid
(tj)j=1,...,M and step size ∆t > 0. The particle model (4) can directly be solved using a sufficiently
small step size of ∆t≤min{1,1
ε}. The microscopic model (1) is solved using the explicit Euler
scheme, i.e.
xj+1
i=xj
i+ ∆tc(xj
i)˜
V L
xj
i+1 −xj
i
,!,
with stepsize ∆t≤1
∥c∥∞∥˜
V∥∞
and xj
idenotes the position of the i-th vehicle at time tj. The
velocity function is set to ˜
V(x)=1−x.
We apply the Lax-Friedrichs scheme to investigate the numerical behaviour of the proposed
macroscopic models (17) and (27). Even though the scheme has diffusive properties it rebuilds
the main properties of the model, especially for small stepsizes. For the second order macroscopic
model (17) we define zj
i=ρj
ihj
iand use a splitting approach for the second equation. First, the
advection step is performed, then the diffusion part is taken into account.

















ρj+1
i=ρj
i−1+ρj
i+1
2−∆t
2∆xc(xi+1)V(hj
i+1)ρj
i+1 −c(xi−1)V(hj
i−1)ρj
i−1,
˜zj+1
i=ρj
i−1hj
i−1+ρj
i+1hj
i+1
2−∆t
2∆xc(xi+1)V(hj
i+1)ρj
i+1hj
i+1 −c(xi−1)V(hj
i−1)ρj
i−1hj
i−1,
zj+1
i= ˜zj
i+ ∆tγ
2(ρj
i)2ηc(xi+1)V(hj
i+1)−c(xi)V(hj
i)
∆x+aρj
i(H(ρj
i)−hj
i),
hj+1
i=zj+1
i
ρj+1
i
.
The discretization of the first order model (27) is similarly given by
ρj+1
i=ρj
i−1+ρj
i+1
2−∆t
2∆xc(xi+1)V(hj
i+1)ρj
i+1 −c(xi−1)V(hj
i−1)ρj
i−1.
For stability of the numerical scheme the CFL-condition is ensured for both the first and second
order model
∆t
∆x∥c∥∞∥V∥∞≤1.
3.3.2 Particle-macro and micro-macro comparison
We consider a road given by the interval [−4,4] with periodic boundary conditions and set the
initial conditions
ρ0(x) := (0.15 x < 0
0.1x≥0,h0(x) := (0.8x < 0
0.95 x≥0.
Let δ > 0, then we set
croad(x) = 








1x∈[−4,−2−δ)∪[2 + δ, 4]
0.6x∈[−2 + δ, 2−δ]
−0.2
δx+ 0.8−0.4
δx∈[−2−δ, −2 + δ]
0.2
δx+ 0.8−0.4
δx∈[2 −δ, 2 + δ].
In the following we set δ=1
10 . The road capacity is reduced on the area of [−2,2] which for example
may be caused by an accident. For the microscopic simulation croad needs to be sufficiently smooth.
12

-4 -3 -2 -1 0 1 2 3 4
x
0
0.2
0.4
0.6
0.8
1
1.2
density
Density particle and second order macroscopic
particle model
second order model
-4 -3 -2 -1 0 1 2 3 4
x
0
0.2
0.4
0.6
0.8
1
1.2
density
Density microscopic and both macroscopic
microscopic model
first order model
second order model
-4 -3 -2 -1 0 1 2 3 4
x
0.8
0.85
0.9
0.95
headway
Headway particle and second order macroscopic
particle model
second order model
-4 -3 -2 -1 0 1 2 3 4
x
0.8
0.85
0.9
0.95
headway
Headway microscopic and both macroscopic
microscopic model
first order model
second order model
Figure 1: Comparison of density and headway between the particle model and the second order
macroscopic model (left) and the microscopic model together with first and second order macro-
scopic model (right) without relaxation (a= 0).
Therefore, we add a linear interpolation in an δ-range around the discontinuities. In the numerical
tests we consider the following functions
V(h) := h
h+ 1 and H(ρ) := 1
1 + ρ.(28)
We consider a time horizon of T= 10. We set ∆x= 10−2,ε= 10−3, ∆t=εand 106particles
in the particle simulation and ∆x= 2 ·10−4,∆t= 2 ·10−4,γ= 0.5, η= 10−2in the macroscopic
simulations to reduce the diffusion of the Lax-Friedrichs-Scheme. In the microscopic model we
consider N= 104vehicles and a vehicle length of L=1
N. The initial vehicle positions are arranged
such that the local densities correspond to ρ0. To visualize the effects of the relaxation parameter
aof the second order macroscopic model (17) we show simulations for a= 0 (Figure 1) and a= 1
(Figure 2). Only in the particle and the second order macroscopic model, the headway is an
state variable in the system. To be able also to compare headways from the microscopic and first
order macroscopic model, in these models we compute the headways artificially according to H(ρ)
defined in (28).
In the left half of Figure 1we observe a good match of the particle model and the second order
macroscopic model for both, the density and the headway. On the right half for the comparison
of the microscopic model and the first and second order macroscopic model, the microscopic and
the first order macroscopic model show the same behaviour for on both scales. In the density,
the second order model is still close, but shows significant deviations in for example the increase
around x=−0.5 which has been captured by the particle simulation but not the microscopic or
first order macroscopic model. The difference is driven by the different evolution of the headway
in the second order model. Comparing Figure 1and 2we observe that increasing the relaxation
parameter apushes the particle and second order macroscopic model closer to microscopic model
and first order macroscopic model. For both the second order macroscopic model and the particle
model, the peak in the density around x=−0.5 is reduced. A similar behaviour is observable
in the headway illustration. Compared to a= 0, in Figure 2the headways of the second order
macroscopic model and especially the particle model very strongly tend to the ones from the
microscopic model.
13

Figure 2: Comparison of density and headway between the particle model and the second order
macroscopic model (left) and the microscopic model together with first and second order macro-
scopic model (right) with relaxation parameter a= 1.
4 Kinetic and macroscopic descriptions including random
accidents
In this section, we go back to the particle model (4)without relaxation (Ξ ≡0) and we include
random accidents understood as a reduction of the road capacity in uncertain locations. In more
detail, we consider:
(Xt+∆t=Xt+c(Xt;Y)V(St)∆t,
St+∆t=St+γΘc(X∗
t;Y)V(S∗
t)−c(Xt;Y)V(St),(29)
where Y∈R+is a bounded random variable parameterising the road capacity function c, such
that [−Y, Y ] is the uncertain interval within which an accident taking place at x= 0 affects the
traffic flow by reducing the road capacity, see Figure 3. Thus, 2Yis the uncertain size of the
accident.
Remark 4.1.Contrary to (4), here we disregard the relaxation term in the particle model (29)
because for this application we do not intend to compare first and second order macroscopic
dynamics. Therefore, we stick to the original motivating FTL model (1).
4.1 Enskog-type kinetic description and its hydrodynamic limit
The (weak) Enskog-type description of the particle dynamics (29) is formally analogous to (6) up
to dropping the second integral at the right-hand side:
∂tZ+∞
0
φ(s)f(x, s, t;Y)ds +∂xZ+∞
0
φ(s)c(x;Y)V(s)f(x, s, t;Y)ds
=1
2Z+∞
0Z+∞
0
(φ(s′)−φ(s))f(x, s, t;Y)f(x+η, s∗, t;Y)ds ds∗.
(30)
14

Y
−Y
1
x
c(x;Y)
0
Figure 3: Prototypical road capacity function parameterised by the random extent of an accident.
Outside the uncertain stretch [−Y, Y ] we have c= 1, whereas within the uncertain stretch [−Y, Y ]
we have c < 1.
However, now also the interaction rule (2) is parameterised by the random variable Y, which plays
indeed the role of a random parameter in the whole equation. Therefore, in (30) it results
s′=s+γc(x∗;Y)V(s∗)−c(x;Y)V(s).
On the whole, the solution fto (30) depends on Yas a parameter, thus we write f=f(·,·,·;Y).
If the non-locality ηof the interactions is sufficiently small, we may approximate (30) by
(cf. (10)):
∂tZ+∞
0
φ(s)f(x, s, t;Y)ds +∂xc(x;Y)Z+∞
0
φ(s)V(s)f(x, s, t;Y)ds
=1
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))f(x, s, t;Y)f(x, s∗, t;Y)ds ds∗
+γη
2∂xc(x;Y)Z+∞
0Z+∞
0
φ′(˜s′)V(s∗)f(x, s, t;Y)f(x, s∗, t;Y)ds ds∗
+η
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))f(x, s, t;Y)∂xf(x, s∗, t;Y)ds ds∗
+γη2
2∂xc(x;Y)Z+∞
0Z+∞
0
φ′(˜s′)f(x, s, t)∂xf(x, s∗, t;Y)ds ds∗,
where
˜s′:= s+γc(x;Y)V(s∗)−V(s).
Under the scaling (11) this yields
∂tZ+∞
0
φ(s)fε(x, s, t;Y)ds +∂xc(x;Y)Z+∞
0
φ(s)V(s)fε(x, s, t;Y)ds
=1
2εZ+∞
0Z+∞
0
(φ(˜s′)−φ(s))fε(x, s, t;Y)fε(x, s∗, t;Y)ds ds∗
+γη
2∂xc(x;Y)Z+∞
0Z+∞
0
φ′(˜s′)V(s∗)fε(x, s, t;Y)fε(x, s∗, t;Y)ds ds∗
+η
2Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))fε(x, s, t;Y)∂xfε(x, s∗, t;Y)ds ds∗
15

+εγη2
2∂xc(x;Y)Z+∞
0Z+∞
0
φ′(˜s′)fε(x, s, t)∂xfε(x, s∗, t;Y)ds ds∗.
Based on Definition 3.1, that we may reapply in this case, we see that φ(s)=1, s are collisional
invariants for this kinetic equation. Using them we get, for every ε > 0, the system of equations



































∂tZ+∞
0
fε(x, s, t;Y)ds +∂xc(x;Y)Z+∞
0
V(s)fε(x, s, t;Y)ds= 0,
∂tZ+∞
0
sfε(x, s, t;Y)ds +∂xc(x;Y)Z+∞
0
sV (s)fε(x, s, t;Y)ds
=γη
2∂xc(x;Y)Z+∞
0Z+∞
0
V(s∗)fε(x, s, t;Y)fε(x, s∗, t;Y)ds ds∗
+γη
2c(x;Y)Z+∞
0Z+∞
0
(V(s∗)−V(s))fε(x, s, t;Y)∂xfε(x, s∗, t;Y)ds ds∗
+εγη2
2∂xc(x;Y)Z+∞
0Z+∞
0
V(s∗)fε(x, s, t;Y)∂xfε(x, s∗, t;Y)ds ds∗,
which, in the hydrodynamic limit ε→0+, converges formally to























∂tZ+∞
0
f0(x, s, t;Y)ds +∂xc(x;Y)Z+∞
0
V(s)f0(x, s, t;Y)ds= 0,
∂tZ+∞
0
sf0(x, s, t;Y)ds +∂xc(x;Y)Z+∞
0
sV (s)f0(x, s, t;Y)ds
=γη
2∂xc(x;Y)Z+∞
0Z+∞
0
V(s∗)f0(x, s, t;Y)f0(x, s∗, t;Y)ds ds∗
+γη
2c(x;Y)Z+∞
0Z+∞
0
(V(s∗)−V(s))f0(x, s, t;Y)∂xf0(x, s∗, t;Y)ds ds∗,
(31)
f0being the local Maxwellian, which satisfies
Z+∞
0Z+∞
0
(φ(˜s′)−φ(s))f0(x, s, t;Y)f0(x, s∗, t;Y)ds ds∗= 0,
for every observable φ.
Similarly to Section 3.1, the monokinetic distribution
f0(x, s, t;Y) = ρ(x, t;Y)δ(s−h(x, t;Y)),
turns out to be a local Maxwellian, whose uniqueness can be established e.g., in the quasi-invariant
regime along the lines of the theory developed in [4]. Notice that the uncertainty brought by the
random variable Ynaturally translates on the hydrodynamic parameters ρ,hassociated with the
two collisional invariants above. With such an f0, (31) specialises as





∂tρ+∂xc(x;Y)V(h)ρ= 0,
∂t(ρh) + ∂xc(x;Y)V(h)ρh=γ
2ρ2η∂xc(x;Y)V(h),
(32)
which, as a matter of fact, coincides with the second order macroscopic model without relaxa-
tion (18) but, in this case, with an uncertain solution (ρ, h) due to the uncertain extent of the
accident parameterising the road capacity function c.
In the classical spirit of uncertainty quantification, the family of uncertain solutions {(ρ, h)}Y
to (32) can be post-processed to average the uncertainty out to a deterministic macroscopic de-
scription. This can be done by computing the expectations of ρ,hwith respect to the law of Y.
16

In more detail, assume that the latter is expressed by a probability distribution g=g(y), then
the following mean density and headway can be defined:
EY[ρ(x, t;Y)] := ZR
ρ(x, t;y)g(y)dy, EY[h(x, t;Y)] := ZR
h(x, t;y)g(y)dy. (33)
Notice that both EY[ρ(x, t;Y)], EY[h(x, t;Y)] are functions of x,tbut they are not, in general, a
solution to either (32) or any other specific macroscopic model.
4.2 Microscopic and macroscopic numerical simulations
We consider the microscopic model from (1) in which the capacity function cadditionally depends
on the accident size random variable Yintroduced in Section 4
˙xi(t;Y) = c(xi(t;Y); Y)˜
VL
xi+1(t;Y)−xi(t;Y), i = 1,2, ... (34)
where
c(x;Y)=1−0.4·
1
[−Y, Y ](x).(35)
The accident size Yis set to Y= 2Z+ 1, where Zis a Beta distributed random variable with
parameters α, β > 0 taking values on [0,1], i.e., the probability density function of Zfor x∈[0,1]
is given by
φY(x, α, β) = Γ(α+β)
Γ(α) + Γ(β)xα−1(1 −x)β−1.
Note that choosing α=β= 1 results in the special case of a uniform distribution on the interval
[1,3] for Y. This construction corresponds to an accident centered at x= 0 with size 2Y. In this
chapter we are interesting in the evolution of the local densities
ρ(N)
i(t;Y) = L
x(N)
i+1 (t;Y)−x(N)
i(t;Y), i = 1, . . . , N −1,(36)
dependent on the realization of the random variable Y. Especially, we consider the piecewise
constant function
ρ(N)(x, t) = ρ(N)
i(t), x ∈[xi(t), xi+1(t)) (37)
which will be used to compare to the evolution of the macroscopic densities. Parameters are chosen
as in Section 3.3.1.
In a second step, we consider the second order macroscopic model from (32) and set the
accident capacity function as in (35). We are interested in the quantities EY[ρ(x, T ;Y)] and
EY[ρ(N)(x, T ;Y)], where ρ(x, T ;Y), ρ(N)(x, T ;Y) are random variables of the densities depending
on the realization of Y.
A Monte Carlo simulation of 2 ·103samples is used to approximate the expectation of the
densities at each point of the spatial grid. We choose the same parameters for the microscopic
and macroscopic model as in Section 3.3, except for the temporal and spatial step sizes of the
Lax-Friedrichs-scheme we set ∆x= ∆t= 10−3to reduce the computational effort.
To describe the perturbed systems, we consider not only the mean realization of the densities,
but confidence intervals in the Figures 4and 5in the special case of the uniform distribution
(α=β= 1). The upper dashed lines show the level of the five percent highest densities in the
Monte Carlo run, whereas the lower dashed lines represent the five percent lowest densities in the
simulation. The green curve shows again the mean realization, whereas the red curve shows the
median representing the 50 percent highest densities.
17

Density second order macroscopic model
-4 -3 -2 -1 0 1 2 3 4
x
0
0.1
0.2
0.3
0.4
0.5
0.6
density
Mean
Median
5 percent quantile
95 percent quantile
Figure 4: Mean, median and 90 percent con-
fidence interval of the density evolution of the
second order macroscopic model (32) at T= 10
for uniformly distributed Y.
Density microscopic model
-4 -3 -2 -1 0 1 2 3 4
x
0.05
0.1
0.15
0.2
0.25
0.3
density
Mean
Median
5 percent quantile
95 percent quantile
Figure 5: Mean, median and 90 percent con-
fidence interval of the density evolution of the
microscopic model (34) at T= 10 for uniformly
distributed Y.
Headway second order macroscopic model
-4 -3 -2 -1 0 1 2 3 4
x
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
headway
Mean
Median
5 percent quantile
95 percent quantile
Figure 6: Mean, median and 90 percent con-
fidence interval of the headway evolution of the
second order macroscopic model (32) at T= 10
for uniformly distributed Y.
Headway microscopic model
-4 -3 -2 -1 0 1 2 3 4
x
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
headway
Mean
Median
5 percent quantile
95 percent quantile
Figure 7: Mean, median and 90 percent con-
fidence interval of the headway evolution of the
microscopic model (34) at T= 10 for uniformly
distributed Y.
We observe that there are areas in which densities and headways vary in a small range and
are almost deterministic. But there are also road sections in which the densities show a large
variance, as can be seen in Figure 4for the macroscopic model in the range of x∈[−1,0]. Due
to the density increases that have already been observed in Figure 1high densities are attained
frequently. This is not the case in the microscopic simulation in Figure 5. It is also striking that
the headways in Figure 6of the second order macroscopic model behave very stable with regard
to the changed accident sizes.
To shortly illustrate the behaviour for a different distribution of Y= 2Z+1, we set α= 5, β = 2
for Zcorresponding to a right-skewed Beta distribution of the accident size on the interval [1,3].
The results for the densities of the second order macroscopic model (32) and the microscopic
model (34) are presented in Figures 8-11. In comparison to the uniform distribution, we mainly
observe two effects: on the one hand, the increase of the densities is slightly shifted to the left due
to the right-skewed distribution of the accident size that makes larger accidents more likely. On
the other hand, the 90 percent confidence intervals of the density and headway are thinner due to
the probability density function decaying to zero as we reach the boundary values of Y. But the
overall shape of the quantities is very similar to the results for the uniform distribution.
18

Density second order macroscopic model
-4 -3 -2 -1 0 1 2 3 4
x
0
0.1
0.2
0.3
0.4
0.5
0.6
density
Mean
Median
5 percent quantile
95 percent quantile
Figure 8: Mean, median and 90 percent con-
fidence interval of the density evolution of the
second order macroscopic model (32) at T= 10
for shifted and scaled beta distributed Y.
Density microscopic model
-3 -2 -1 0 1 2 3
x
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
density
Mean
Median
5 percent quantile
95 percent quantile
Figure 9: Mean, median and 90 percent con-
fidence interval of the density evolution of the
microscopic model (34) at T= 10 for shifted
and scaled beta distributed Y.
Headway second order macroscopic model
-4 -3 -2 -1 0 1 2 3 4
x
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
headway
Mean
Median
5 percent quantile
95 percent quantile
Figure 10: Mean, median and 90 percent con-
fidence interval of the headway evolution of the
second order macroscopic model (32) at T= 10
for shifted and scaled beta distributed Y.
Headway microscopic model
-4 -3 -2 -1 0 1 2 3 4
x
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
headway
Mean
Median
5 percent quantile
95 percent quantile
Figure 11: Mean, median and 90 percent con-
fidence interval of the headway evolution of the
microscopic model (34) at T= 10 for shifted
and scaled beta distributed Y.
Apart from the Monte Carlo simulations, we approximate the expectations of the densities in
both models using a polynomial chaos expansion. Note that we are not particularly interested in
providing a deep analysis on the polynomial chaos expansion itself but exploit this approach as
an alternative way of approximating the expected value of the densities.
From now on we stick to the case of Ybeing uniformly distributed in [1,3]. The Legendre
polynomials form an orthonormal basis for a uniformly distributed random variable on [−1,1].
For our purpose a simple transformation by an additive shift transforms Yto the interval [1,3].
Then orthonormality means
Z3
1
1
2ϕi(y)ϕj(y)dy =δi,j ,
where ϕkdenotes the k-th Legendre polynomial. We consider the polynomial chaos expansion for
the second order macroscopic model in conservative form (19), where
z(x, t) = ρ(x, t)h(x, t) + γ
2ηρ(x, t).(38)
19

Then, the system is described in the solution subspace span {ϕk}K
k=0 , K ∈Nby
ρK(x, t;Y)
zK(x, t;Y)=
K
X
k=0 ˆρk(x, t)ϕk(Y)
ˆzk(x, t)ϕk(Y),
where at the initial time ˆρk(x, 0) and ˆzk(x, 0) are the modes of the expansion and can be determined
as follows:
ˆρk(x, 0) = Z3
1
1
2ρ(x, 0, y)ϕk(y)dy, ˆzk(x, 0) = Z3
1
1
2z(x, 0, y)ϕk(y)dy. (39)
The propagation of the modes up to order Kcan be described by the system
0 = ∂t







ˆρ0(x, t)
ˆz0(x, t)
.
.
.
ˆρK(x, t)
ˆzK(x, t)







+∂x











R3
1c(x)VPK
k=0 ˆzk(x,t)ϕk(y)
PK
k=0 ˆρk(x,t)ϕk(y)−γ
2ηPK
k=0 ˆρk(x, t)ϕk(y)(PK
k=0 ˆρk(x, t)ϕk(y))ϕ0(y)1
2dy
R3
1c(x)VPK
k=0 ˆzk(x,t)ϕk(y)
PK
k=0 ˆρk(x,t)ϕk(y)−γ
2ηPK
k=0 ˆρk(x, t)ϕk(y)(PK
k=0 ˆzk(x, t)ϕk(y))ϕ0(y)1
2dy
.
.
.
R3
1c(x)VPK
k=0 ˆzk(x,t)ϕk(y)
PK
k=0 ˆρk(x,t)ϕk(y)−γ
2ηPK
k=0 ˆρk(x, t)ϕk(y)(PK
k=0 ˆρk(x, t)ϕk(y))ϕK(y)1
2dy
R3
1c(x)VPK
k=0 ˆzk(x,t)ϕk(y)
PK
k=0 ˆρk(x,t)ϕk(y)−γ
2ηPK
k=0 ˆρk(x, t)ϕk(y)(PK
k=0 ˆzk(x, t)ϕk(y))ϕK(y)1
2dy











.
(40)
Similarly, we can set up the system for the expansion in the case of the microscopic model (34) by
x(N)
i(t, Y ) =
K
X
k=0
ˆx(N)
i,k (t)Φk(Y),
where the modes for k= 0, . . . , K at the initial time t= 0 are given by
ˆx(N)
i,k (0) = Z3
1
1
2x(N)
i(0)Φk(y)dy. (41)
Then, the evolution of the microscopic system is for r= 0 ...,K given by
Z3
1
1
2
K
X
k=0
˙
ˆx(N)
i,k (t)Φk(y)Φr(y)dy
=Z3
1
1
2c K
X
k=0
ˆx(N)
i,k (t)Φk(y)!˜
V

L
PK
k=0 ˆx(N)
i+1,k(t)−ˆx(N)
i,k (t)Φk(y)
Φr(y)dy,
(42)
which can by orthogonality of the basis functions for r= 0 ...,K be rewritten to
˙
ˆx(N)
i,r (t) = Z3
1
1
2c K
X
k=0
ˆx(N)
i,k (t)Φk(y)!˜
V

L
PK
k=0 ˆx(N)
i+1,k(t)−ˆx(N)
i,k (t)Φk(y)
Φr(y)dy. (43)
The vehicle positions can be transformed to the microscopic local density function by the
equations (36) and (37).
20

As we aim to approximate the expectation of the densities, we focus on the modes for K= 0
which exactly describe the expectation of the stochastic system. The integrals in (39)-(43) can be
approximated using Gauss-Legendre quadrature. The weights for the corresponding quadrature
rule and a root zkare given by (staying on the roots inside [−1,1])
wk=1
(1 −z2
k)·(L′
n(zk))2,
where L′
nis the derivative of the n-th Legendre polynomial. The expectation of the macroscopic
density at a position xand time tgiven by
EY[ρ(x, T ;Y)] = ˆρ0(x, T ).
Similarly, an approximation of the expected headway is given by
EY[h(x, T ;Y)] = ˆ
h0(x, T ),
where ˆ
h0is recovered from ˆz0by (38). One can proceed exactly in the same way for the microscopic
model and approximate
EY[ρ(N)(x, T ;Y)] = ˆρ(N)
0(x, T ),
where ˆρ(N)
0(x, T ) is the piecewise constant density function resulting from the vehicle position
modes ˆx(N)
i,0(T) using (36) and (37). The expectation of the headway in the microscopic model is
obtained when plugging the density into the optimal headway function from (28).
In the Figures 12 and 13 we show the approximations for the expectations using n= 1,5,9 roots
of the Legendre polynomials and compare it with the expectation of the Monte Carlo simulations
with 2·103realizations. For n= 1 the approximation coincides with the one choosing the accident
size Yto E[Y] = 2 and is not very accurate on both levels. Overall for n= 5 the approximation
shows a good performance but has still some inaccuracies in some areas, as for example in the
macroscopic density around x=−0.5. For n= 9 the approximation using the polynomial chaos
expansion shows no discernible differences for both the macroscopic and microscopic expected
density.
A convergence analysis of the expectation approximated using n= 1,...,9 nodes in the poly-
nomial chaos approach towards the result from the Monte Carlo simulation in a framework with
logarithmic values on the axis is presented in the Figures 16 and 17. On the microscopic scale
we recognize a linear relationship with approximate convergence rates of slightly larger than 2,
whereas the correlation on the macroscopic scale is not perfectly linear but still strictly decreasing
with an approximate convergence rate of 2. These results underline the approximation behaviour
in the Figures 4-7.
5 Conclusions
In this paper, we have considered classical Follow-the-Leader traffic dynamics with space-dependent
speed, which, upon reformulation as interaction dynamics of a system of stochastic particles, we
have described at the mesoscopic scale by means of an Enskog-type collisional kinetic equation.
Since Follow-the-Leader dynamics are based on the reciprocal distance of the vehicles, in our
kinetic representation we have used the headway, along with the position, as microscopic state
of the vehicles in place of the more usual speed. This has allowed us to obtain formally, in the
hydrodynamic limit, macroscopic conservation laws based on the density and the mean headway
of traffic, which constitute original models with respect to those consolidated in the reference
literature. In particular, our investigations have shown that, in the limit, one may formally get
either a first or a second order macroscopic model with space-dependent flux depending on a
certain relaxation parameter of the headway. Analytical investigations have proved that our new
second order model complies with the Aw-Rascle consistency condition and admits weak entropy
solutions at least for initial data with small total variation.
21

-4 -3 -2 -1 0 1 2 3 4
x
0
0.1
0.2
0.3
0.4
0.5
0.6
(x,T)
Approximation expected density - second order macroscopic
Mean Monte Carlo
N=1
N=5
N=9
Figure 12: Approximation of the expected dens-
ities in the second order macroscopic model (32)
using polynomial chaos expansion.
-4 -3 -2 -1 0 1 2 3 4
x
0
0.05
0.1
0.15
0.2
0.25
0.3
(x,T)
Approximation expected density - microscopic model
Mean Monte Carlo
N=1
N=5
N=9
Figure 13: Approximation of the expected dens-
ities in the microscopic model (34) using poly-
nomial chaos expansion.
-4 -3 -2 -1 0 1 2 3 4
x
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
headway
Approximation expected headway - second order macroscopic
Mean Monte Carlo
N=1
N=5
N=9
Figure 14: Approximation of the expected head-
ways in the second order macroscopic model (32)
using polynomial chaos expansion.
-4 -3 -2 -1 0 1 2 3 4
x
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
headway
Approximation expected headway - microscopic model
Mean Monte Carlo
N=1
N=5
N=9
Figure 15: Approximation of the expected head-
ways in the microscopic model (34) using poly-
nomial chaos expansion.
We have used these traffic models to describe the impact of accidents on the traffic flow. Our
derivation from principles of statistical mechanics has made it possible to include, in particular,
accidents taking place in random, viz. uncertain, positions along the road. In the hydrodynamic
limit, this has yielded a new version of the former second order macroscopic model with uncertain
flux. Numerical investigations on uncertain accident sizes have illustrated, on one hand, that
expected traffic densities can be computed efficiently using a polynomial chaos expansion and, on
the other hand, that some road sections may be much more affected by accidents, hence may face
a much larger variety of traffic scenarios, than others.
Future work may consider generalised space-dependent traffic accident models on road networks
with ad-hoc numerical simulation techniques.
Acknowledgments
F.A.C. is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Ap-
plicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). F.A.C. was par-
tially supported by the Ministry of University and Research (MUR), Italy, under the grant PRIN
2020 - Project N. 20204NT8W4, “Nonlinear evolution PDEs, fluid dynamics and transport equa-
tions: theoretical foundations and applications”. F.A.C. would like to thank Debora Amadori
22

0 0.5 1 1.5 2 2.5 3 3.5
log2(n)
-11
-10
-9
-8
-7
-6
-5
-4
-3
log2(L2-error)
Numerical convergence for L2 deviation - macroscopic
density
headway
Figure 16: Evolution of the L2-error in dens-
ity and headway using the generalized polyno-
mial chaos expansion for the second order mac-
roscopic model (32) and n= 1,...,9 nodes.
0 0.5 1 1.5 2 2.5 3 3.5
log2(n)
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
log2(L2-error)
Numerical convergence for L2 deviation - microscopic
density
headway
Figure 17: Evolution of the L2-error in density
and headway using the generalized polynomial
chaos expansion for the microscopic model (34)
and n= 1,...,9 nodes.
for useful discussions about the analytical properties. S. G. was supported by the German Re-
search Foundation (DFG) under grant GO 1920/10-1, 11-1 and 12-1. A.T. is member of Gruppo
Nazionale per la Fisica Matematica (GNFM) of INdAM, Italy.
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