Uncertainty Quantiﬁcation in Hierarchical Vehicular Flow
Models
Michael Herty∗and Elisa Iacomini∗
Abstract
We consider kinetic vehicular traﬃc ﬂow models of BGK type [24]. Considering dif-
ferent spatial and temporal scales, those models allow to derive a hierarchy of traﬃc
models including a hydrodynamic description. In this paper, the kinetic BGK–model is
extended by introducing a parametric stochastic variable to describe possible uncertainty
in traﬃc. The interplay of uncertainty with the given model hierarchy is studied in detail.
Theoretical results on consistent formulations of the stochastic diﬀerential equations on
the hydrodynamic level are given. The eﬀect of the possibly negative diﬀusion in the
stochastic hydrodynamic model is studied and numerical simulations of uncertain traﬃc
situations are presented.
Keywords.
Traﬃc ﬂow, BGK models, stochastic Galerkin, Aw-Rascle-Zhang model,
kinetic equations.
1
Introduction
The mathematical description of vehicular traﬃc ﬂow is possible at diﬀerent spatial and
temporal scales ranging from models for individual cars [17] up to a description of aggregated
quantities like the traﬃc density [3, 33, 11, 16]. Recent works present models on those scales as
well as methods to traverse the existing hierarchy, see e.g. [4, 1, 23, 15, 12, 25] and references
therein. We are particularly interested in two scales, the hydrodynamic or ﬂuid–like models for
aggregated quantities and a statistical description of traﬃc as e.g. proposed in [22, 28, 6, 29].
Our contribution is mainly based on the recently introduced hierarchy [24] where in particular
a class of BGK (Bhatnagar, Gross and Krook [5]) models have been considered. The ﬂuid–
like models considered are second-order Aw-Rascle-Zhang type models [1, 48]. The hierarchy
presented in [24] has been deterministic assuming that all model parameters and initial data
are known exactly. However, often there is need to take uncertainties into account, e.g. due
∗Institut f¨ur Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55,
52062 Aachen, Germany, {herty, iacomini}@igpm.rwth −aachen.de
1
arXiv:2108.07589v1  [math.NA]  17 Aug 2021

to noisy measurements and due to variations in the behavior of vehicular traﬃc leading to
uncertainties. Then, it is necessary to extend the concepts to the stochastic case to consider
probability laws or statistical moments. The treatment of stochastic models can be either
non–intrusive, e.g., based on sampling (Monte–Carlo) [32, 42, 41] or based on collocation
[2], or intrusive [31, 45]. In the later approach, stochastic input is represented by a series of
orthogonal functions, known as generalized polynomial chaos (gPC) expansions [44, 7, 46],
substituted in the governing equations and then projected using a Galerkin projection. We
follow this intrusive approach in order to investigate how uncertainty propagates between the
kinetic and the ﬂuid ﬂow hierarchy of description. The possible links are depicted in Figure
1. Recently, results using the intrusive approach for kinetic equations have been presented
and we refer to [40, 26, 8, 49, 27, 9, 47] for corresponding results. For hyperbolic models on
the ﬂuid type description there have also been recent results [14, 10, 21, 37, 35, 20, 18, 30]
– mostly centered at the question of hyperbolicity of the underlying gPC expanded system
of partial diﬀerential equations. For the presented investigation we in particular refer to [19]
where a gPC expansion for the Aw-Rascle-Zhang has been established. Therein, it has been
shown that for a particular choice of orthogonal functions, the resulting expanded system is
hyperbolic, see [19, Theorem 2.2]. In this paper we will investigate the link between stochastic
BGK and stochastic second order traﬃc ﬂow models. In [24] the diﬀusivity coeﬃcient has been
used to classify possible unstable traﬃc regimes. We will show that the discussion translates
to the stochastic case and allows to characterize possible traﬃc zones of high risk. Here, we
also investigate the dynamic case compared with the previous publication. Our presentation
follows the diagram shown in Figure 1, in particular, the indicated blue connections.
The propagation of uncertainty through hierarchies has also been explored e.g. in the case
of the Vlasov-Poisson-Fokker-Planck system [27]. Contrary to the approach here, however,
the resulting diﬀusive system has been shown to be well–posed without further assumptions.
Due to the nonlinear hyperbolic structure the presented results therein do not extend directly
to the present case. Also, in [43] the propagation of uncertainty is discussed but the origin
and treatment of uncertainty is very diﬀerent to the presented work.
2
Hierarchical stochastic traﬃc ﬂow models
A kinetic traﬃc ﬂow model reads
∂tf(t, x, v) + v∂xf(t, x, v) = 1
εQ[f, f](t, x, v),
(1)
where f(t, x, v) : R+ × R × [0, VM] →R+ is the mass distribution function of traﬃc. The
operator Q encodes the detailed car–to–car interactions and it will be modeled in the following
as a linear operator of BGK type. The quantity ε is positive, and yields a relaxation rate
weighting the relative strength between the convective term and source term. The spatial
2

BGK
ARZ
µ(ρ)
BGK(ξ)
ARZ(ξ)
µ(ρ(ξ))
Kinetic level
Fluid level
Sec. 2
Thm. 2.2
Sec. 2.1
Sec. 2.2
Sec. 2.2
Figure 1: Outline of the model hierarchy. The left two columns indicate the kinetic and
ﬂuid description of traﬃc ﬂow as presented in [24]. The third column refers to the diﬀusion
coeﬃcient µ(ρ) to classify traﬃc instabilities. The green hierarchy is deterministic while the
blue includes a parametric uncertainty ξ. The indicated links are established in this paper.
variable is denoted by x ∈R and the velocity v is assumed to be bounded by zero and a
maximum speed VM. Finally, t ≥0 is time and we assume w.l.o.g. that the initial datum
f0(x, v) is such that the density ρ0 is bounded by one, i.e.,
Z VM
0
f0(x, v)dv =: ρ0(x) ≤1 ∀x ∈R.
(2)
BGK type collision operators prescribe a relaxation to equilibrium at rate ε. In the space
homogeneous case, the equilibrium is characterize by a function Mf(v; ρ) called Maxwellian
possibly depending on ρ = ρ0. The Maxwellian deﬁnes the mean speed of vehicles at equilib-
rium through the relation
U(ρ) = 1
ρ
Z VM
0
vMf(v; ρ)dv.
(3)
The precise modeling of Q as well as the existence of suitable Maxwellians has been discussed
intensively in the literature and we refer e.g. to [38, 24]. Integrating equation (1) in velocity
space, and provided that
R VM
0
Q[f, f]dv = 0, and one obtains the evolution equation for the
density ρ(t, x) =
R VM
0
f(t, v, x)dv as
∂tρ(t, x) + ∂x
Z VM
0
vf(t, x, v)dv

= 0.
(4)
If the system approaches equilibrium, f →Mf, then
∂tρ(t, x) + ∂x (ρ(t, x)U(ρ(t, x))) = 0.
(5)
The previous equation and the initial data ρ(0, x) = ρ0(x) provides a level of description on
an aggregated, ﬂuid–like level. If, however, the system is not at equilibrium, the equation (4)
is still coupled to the kinetic equation (1). In the case ε →0, the interactions of cars are
3

so frequent to instantaneously relax f to the local equilibrium distribution Mf. Instead, we
expect that if ε > 0 is small but positive, we are in a regime where the kinetic equation is
given by an extended continuum hydrodynamic equations as for example the Aw-Rascle and
Zhang model [1, 48]. Studying stability properties of traﬃc patterns in terms of an asymptotic
analysis in terms of the parameter ε has been conducted e.g. [24, 39] using a Chapman-Enskog
expansion. The Aw–Rascle–Zhang equations are a system of hyperbolic equations for traﬃc
density ρ and (average) velocity v = v(t, x) = q(t,x)
ρ(t,x) for ρ > 0. Here, q(t, x) is the ﬂux and
the function h will be introduced below in equation (10). For some given equilibrium velocity
V eq : R+ →R+ decreasing in its argument, the equations read for ρ = ρ(t, x) and q = q(t, x)
with x ∈R, t ∈R+ :
∂tρ + ∂x(q −ρh(ρ)) = 0,
(6)
∂tq + ∂x
q2
ρ −qh(ρ)

= 1
ϵ (ρVeq(ρ) + ρh(ρ) −q) .
(7)
In the limit ϵ →0 we formally obtain a (ﬁrst–order) consistent approximation of solutions to
(6) to (4) by deﬁning for ρ ≥0
V eq(ρ) = U(ρ).
(8)
In the following we are interested in the link between (6) and (1), resp. (11) in the
stochastic case. A key observation in the deterministic analysis [24] has been the link between
a discretization of the kinetic equation (1) using a ﬁnite number of particles and the Aw–
Rascle–Zhang traﬃc ﬂow model.
This connection as been established using the variable
w ∈W = [wmin, ∞)
w = v + h(ρ).
(9)
Here, wmin = h(0) and h(ρ) : R+ →R+ is an increasing, diﬀerentiable function of the density
called hesitation or pressure function [16]. We assume that for γ ∈{1, 2}
h(ρ) = ργ.
(10)
The quantity of w can be understood as a driver’s preference that is Lagrangian quantity [1,
Section 4].
Based on a particle description the link between the kinetic equation for g :
R+ × R × W →R+
∂tg(t, x, w) + ∂x
h
(w −h(ρ(t, x)))g(t, x, w)
i
= 1
ε

Mg(w; ρ(t, x)) −g(t, x, w)

,
(11)
4

and the Aw–Rascle–Zhang equations (6) for the density ρ and ﬂux q
ρ(t, x) =
Z
W
g(t, x, w)dw,
q(t, x) =
Z
W
w g(t, x, w)dw
(12)
has been established using asymptotic analysis in ε. The Maxwellian Mg can be related to
Mf, which is assumed to fulﬁll for any ρ ∈R
Z
W
Mg(w; ρ) dw = ρ,
(M1)
Z
W
w Mg(w; ρ) dw = ρVeq(ρ) + ρh(ρ).
(M2)
The function Veq : R →R+
0 is the previously introduced equilibrium velocity. As discussed we
are interested in the description of vehicular traﬃc on the kinetic (11) and ﬂuid–dynamic (6)
level in the presence of parametric uncertainty ξ. This uncertainty may have many origins but
for now we simply assume that it can be described by a (possibly multi-dimensional) random
variable ω. Let the random variable ω be deﬁned on the probability space (Ωω, F(Ω), P).
Further, we denote by ξ = ξ(ω) : Ωω →Ω⊂Rd a (possibly d-dimensional) real-valued random
variable. Assume further that ξ is absolutely continuous with respect to the Lebesgue measure
on Rd and denote by fΞ(ξ) : Ω→R+ the probability density function of ξ. For simplicity
we assume that the uncertainty enters only in the initial data g0(x, w, ω) that is now random
ﬁeld deﬁned on R × W × Ωω that is denoted by g0(x, w, ξ) = g0(x, w, ξ(w)) : R × W × Ω→R.
Further, we assume that g0(x, w, ·) ∈L2(Rd, fΞ) a.e. in (x, w). Then, we are interested in
the evolution of the random ﬁeld g(t, x, w, ξ) : R+ × R × W × Ωgoverned by a BGK–kinetic
equation (11) with uncertain initial data g0. For the following derivations it is suﬃcient to
assume that ﬁrst and second moment g w.r.t. to w exist as well as up to second moment in
ξ. Further, the derivation is based on the assumption that the random ﬁeld g fulﬁlls (13)
pointwise a.e. in (t, x, w) as well as fΞ a.s. in ξ :
∂tg(t, x, w, ξ) + ∂x
h
(w −h(ρ(t, x, ξ)))g(t, x, w, ξ)
i
= 1
ε

Mg(w; ρ(t, x, ξ)) −g(t, x, w, ξ)

,
(13)
g(0, x, w, ξ) = g0(x, w, ξ),
(14)
ρ(t, x, ξ) =
Z
W
g(t, x, w, ξ)dw.
(15)
Next, we turn to the description of the intrusive approach in order to establish the hierarchy
indicated in Figure 1. A random ﬁeld g(t, x, w, ·) ∈L2(Ω, fΞ) can be expressed by a spectral
5

expansion [7]
g(t, x, w, ξ) =
∞
X
k=0
˜gi(t, x, w)φi(ξ),
(16)
where φi ∈L2(Ω, fΞ) are basis functions, typically chose orthonormal with respect to the
weighted scalar product, and {˜gi(t, x, w)}∞
i=0 is a set of coeﬃcients:
˜gi(t, x, w) =
Z
Ω
g(t, x, w, ξ)φi(ξ)fΞ(ξ)dξ.
(17)
The previous expansion is truncated at K to obtain an approximation with K + 1 moments.
The projection of g(t, x, w, ·) to the span of the K + 1 base functions is denoted by
GK(g(t, x, w, ·))(ξ) :=
K
X
i=0
˜gi(t, x, w)φi(ξ) a.s.ξ ∈Ω.
(18)
The expansion (16) is called generalized polynomial chaos expansion (gPC). In particular, for
kinetic equations, also more involved than the given BGK equation, this has been explored
recently in a series of paper, see e.g. [40, 26, 8, 49, 27, 9, 47]. Therein, also conditions on
{g0,i}∞
i=0 have been developed to allow for existence of a (weak) stochastic solution g.
Next, we establish the connection between the random BGK model (11) and the stochastic
Aw–Rascle–Zhang system. Assume g is a pointwise a.e. and integrable solution to the system
(13). Then, the density ρ and ﬂux q allow for gPC expansion for all i ∈N :
ρ(t, x, ξ) =
Z
W
g(t, x, w, ξ)dw =
∞
X
i=0
˜ρiφi(ξ), ˜ρi = ˜ρi(t, x) =
Z
W
˜gi(t, x, w)dw,
(19)
q(t, x, ξ) =
Z
W
wg(t, x, w, ξ)dw =
∞
X
i=0
˜qiφi(ξ), ˜qi = ˜qi(t, x) =
Z
W
w˜gi(t, x, w)dw.
(20)
As in [35, 36, 20] we introduce the Galerkin production for any ﬁnite K > 0 and any u, z ∈
L2(Ω, fΞ), ˜u = (˜ui)K
i=0, ˜z := (˜zi)K
i=0 and for all i, j, ℓ= 0, . . . , K :
GK[u, z](t, x; ξ) :=
K
X
k=0
(˜u ∗˜z)k(t, x)φk(ξ),
(˜u ∗˜z)k(t, x) :=
K
X
i,j=0
˜ui(t, x)˜zj(t, x)Mℓ,
(Mℓ)i,j :=
Z
Ω
φi(ξ)φj(ξ)φℓ(ξ)fΞ(ξ)dξ.
Note that Mℓis a symmetric matrix of dimension (K+1)×(K+1) for any ﬁxed ℓ∈{0, . . . , K}.
6

The Galerkin product GK is not the only possible projection of the product of random variables
u, z on the subspace span{ φ0, . . . , φK}. However, this choice (and additional assumptions on
the base functions) have shown to be suﬃcient to guarantee hyperbolicity of the p−system
[20] as well as the Aw–Rascle–Zhang system [19]. Furthermore, we have ˜u ∗˜z = P(˜u)˜z for
P ∈RK+1×K+1 and ˜u ∈RK+1 deﬁned by
P(˜u) :=
K
X
ℓ=0
˜uℓMℓ.
(21)
The Galerkin product is symmetric, but not associative [13, 34, 41]. Finally, we assume that
the chosen functions {φi}i fulﬁll the following properties [18, A1-A3]
(A1) The matrices Mℓand Mk commute for all ℓ, k = 0, . . . , K.
(A2) The matrices P(bu) and P(˜z) commute for all ˜u, ˜z ∈RK+1.
(A3) There is an eigenvalue decomposition P(˜u) = V D(˜u)V T with constant eigenvectors V .
It has been shown that for example the one–dimensional Wiener–Haar basis and piecewise
linear multiwavelets fulﬁll the previous assumptions, but, Legendre and Hermite polynomials
do not fulﬁll those requirements.
Similar to [40, 26] and for any ﬁxed K we derive a system of equations for the evolution
of ˜gi(t, x, w) : R+ × R × W →R for i = 0, . . . , K by projection the operators of equation (13)
to the space span{φi : i = 0, . . . , K }. We further assume that the set of base functions is
orthonormal and fulﬁlls the assumptions (A1)–(A3).
∂t ˜gi(t, x, w) + ∂x

wId −P (h(˜ρ (t, x)))

˜g(t, x, w)

i = 1
ε

f
Mi (w; ˜ρ(t, x)) −˜gi(t, x, w)

,
(22)
˜gi(0, x, w) =
Z
Ω
g0(t, x, w, ξ)φi(ξ)fΞ(ξ)dξ
(23)
In the derivation of the previous system (22) we have used the following results: Under
assumptions (A1)-(A3) h, as given by equation (10), fulﬁlls [19]:
K
X
j=0
Z
Ω
h
 K
X
ℓ=0
˜ρℓφℓ(ξ)
!
˜gjφj(ξ)φi(ξ)fΞ(ξ)dξ = (P(h(˜ρ))˜g)i , ∀i = 0, . . . , K.
(24)
Further, we deﬁne for i = 0, . . . , K
7

f
Mi (w; ˜ρ(t, x)) :=
Z
Ω
Mg

w;
K
X
ℓ=0
˜ρℓ(t, x)φℓ(ξ)

φi(ξ)fΞdwdξ,
(25)
c
Mg(w, ˜ρ(t, x), ξ) :=Mg
 
w;
K
X
i=0
˜ρi(t, x)φi(ξ)
!
.
(26)
2.1
Derivation of Stochastic Aw–Rascle–Zhang Model
In [24] a connection between two levels of description, i.e., (11) and (6) has been established
under the assumption that the Maxwellian fulﬁlls (M1) and (M2). The next lemma shows
that those assumptions extend to directly to the stochastic case.
Lemma 2.1. Let K > 0. Consider a base functions φi and i = 0, . . . , K fulﬁlling (A1)–(A3).
Furthermore, assume that the functions Mg, V eq fulﬁll the assumptions (M1)-(M2). Let g be
expanded in a gPC series with K + 1 modes as given by equation (16). Then, f
Mg deﬁned by
(26) fulﬁll for any i = 0, . . . , K, t ≥0, and x ∈R :
Z
W
f
Mi (w; ˜ρ(t, x)) dw = ˜ρi(t, x),
(UM1)
Z
W
w f
Mi (w; ˜ρ(t, x)) dw =

P(Veq(˜ρ(t, x)))˜ρ(t, x) + P(h(˜ρ(t, x)))˜ρ(t, x)

i.
(UM2)
Proof. Due to (M1)-(M2) and (19) we obtain for a.e. (t, x, ξ) ∈R+ × R × Ω.
Z
W
Mg(w; ρ(t, x, ξ)) dw = ρ(t, x, ξ),
(27)
Z
W
w Mg(w; ρ(t, x, ξ)) dw = ρ(t, x, ξ)Veq(ρ(t, x, ξ)) + ρ(t, x, ξ)h(ρ(t, x, ξ)).
(28)
Integration with respect to dw yields
Z
W
f
Mi(w, ˜ρ(t, x)) dw =
Z
Ω
Z
W
Mg

w;
K
X
j=0
˜ρj(t, x)φj(ξ)

φi(ξ)dwfΞ(ξ)dξ =
Z
Ω
K
X
j=0
˜ρj(t, x)φj(ξ)φi(ξ)fΞ(ξ)dξ = ˜ρi(t, x).
The similar computation yielding (UM2) is omitted.
In the following result we derive a gPC formulation of the ﬂuid model obtained by the
stochastic BGK model (13). Further, we compare this model with the stochastic Aw–Rascle–
8

Zhang model derived in [19]. The theorem shows that under assumption (29) the derived
gPC model is equivalent to the stochastic model of [19]. Therein, it has also been shown that
the partial diﬀerential equation is hyperbolic.
Theorem 2.2. Let K > 0, ϵ > 0. Assume the base functions {φ0, . . . , φK} fulﬁll (A1)–(A3)
and assume that the functions Mg, V eq fulﬁll the assumptions (M1)-(M2) and let h(·) fulﬁll
(10). Let ˜gi be a strong solution to (22) and (25) for i = 0, . . . , K. Further, assume that for
i = 0, . . . , K and (t, x) ∈R+ × R
Z
W
w2 ˜gi(t, x, w)dw = (P(˜q(t, x))P−1(˜ρ(t, x))˜q(t, x))i,
(29)
where (˜ρ, ˜q)i are the ﬁrst and second moment of ˜gi as in (19)–(20) and P is deﬁned by (21).
Then, the functions (˜ρ, ˜q) formally fulﬁll pointwise in (t, x) ∈R+ × R and for all i =
0, . . . , K the second–order traﬃc ﬂow model
∂t˜ρi(t, x) + ∂x [˜qi(t, x) −(P(˜ρ(t, x))˜ρ(t, x))i] = 0
(30a)
∂t˜qi(t, x) + ∂x

(P(˜q(t, x))P−1(˜ρ(t, x))˜q(t, x))i −(P(˜ρ(t, x))˜q(t, x))i

=
(30b)
1
ϵ

P(Veq(˜ρ(t, x)))˜ρ(t, x) + P(h(˜ρ(t, x)))˜ρ(t, x)

i −˜qi(t, x)

(30c)
˜ρi(0, x) =
Z
W
˜g0,i(t, x, w)dw,
(30d)
˜qi(0, x) =
Z
W
w ˜g0,i(t, x, w)dw.
(30e)
The system (30) is hyperbolic for ˜ρi > 0.
Let the random ﬁelds (ρ, q) = (ρ, q)(t, x, ξ) : R+ × R × Ω→R2 be a pointwise a.e. solution
with second moments w.r.t. to ξ of the stochastic Aw–Rascle–Zhang system with random initial
data:
∂tρ + ∂x(q −ρh(ρ)) = 0,
(31a)
∂tq + ∂x
q2
ρ −qh(ρ)

= 1
ϵ (ρVeq(ρ) + ρh(ρ) −q) ,
(31b)
ρ(0, x, ξ) = ρ0(x, ξ), q(0, x, ξ) = q0(x, ξ).
(31c)
Under the previous assumptions on the base functions { φ0, . . . , φK} and provided that for all
i = 0, . . . , K
9

Z
Ω
ρ0(x, ξ)φi(ξ)fΞdξ =
Z
W
˜g0,i(t, x, w)dw,
Z
Ω
q0(x, ξ)φi(ξ)fΞdξ =
Z
W
w˜g0,i(t, x, w)dw, (32)
we have
GK (ρ(t, x, ·)) (ξ) =
K
X
i=0
˜ρi(t, x)φi(ξ) and GK (q(t, x, ·)) (ξ) =
K
X
i=0
˜qi(t, x)φi(ξ),
(33)
where (˜ρ, ˜q) fulﬁll equation (30).
Some remarks are in order.
The assumption (29) is a closure relation and has been
presented in the deterministic case [24]. The result on hyperbolicity of the system (30) has
been presented in [19]. Therein, also the system for the coeﬃcients ˜ρ, ˜q of a gPC expansion
of the stochastic case of (6) has been derived, i.e., the system (30). Condition (32) states the
consistency of initial data of both systems.
Proof. The proof is similar to [24] and given here for completeness.
For a pointwise a.e.
solution ˜g and corresponding densities ˜ρ and ﬂuxes ˜q according to (19)–(20) we obtain for
each i ∈{0, . . . , K} by (22) and after integration on W
∂t
Z
W
˜gi(t, x, w)dw + ∂x
Z
W
w˜gi(t, x, w) −

P(h(˜ρ(t, x)))˜g(t, x, w)

idw
(34)
= 1
ε
Z
W
˜
Mi(w; ˜ρ(t, x)) −˜gi(t, x, w)dw

.
(35)
Since ˜g →P(˜ρ)˜g is linear and by equation (UM1) of Lemma 2.1
∂t˜ρi(t, x) + ∂x

˜qi(t, x) −

P(h(˜ρ(t, x)))˜ρ(t, x)

i

= 0.
(36)
Furthermore, we integrate (22) w.r.t. to w dw on W to obtain
∂t
Z
W
w˜gi(t, x, w)dw + ∂x
Z
W
w2˜gi(t, x, w) −w

P(h(˜ρ(t, x)))˜g(t, x, w)

idw
(37)
= 1
ε
Z
W
wf
Mi(w; ˜ρ(t, x)) −w˜gi(t, x, w)dw

.
(38)
This yields
10

∂t˜qi(t, x) + ∂x
Z
W
w2˜gi(t, x, w)dw −

P(h(˜ρ(t, x)))˜q(t, x)

i
(39)
= 1
ε
Z
W
wf
Mi(w; ˜ρ(t, x))dw −˜qi(t, x)

.
(40)
Using now (29) and equation (28) of Lemma 2.1 we obtain the momentum equation of the
second–order traﬃc ﬂow model (30).
Under the assumptions (A1)–(A3) we obtain (30) is hyperbolic as proven in [19]. Therein,
also the assertion (33) has been established.
Remark 2.3.
Introducing stochasticity also allows for more general Maxwellians. In partic-
ular, the Maxwellian Mg could also depend on ξ directly. Hence, we may assume that
Mg(w, ξ; ρ) := M(w; ρ(t, x, ξ), ξ).
(41)
The previous derivation can be also conducted for Maxwellians of the previous type. In order
to conserve mass it is necessary to assume that M fulﬁlls (UM1). Then, we obtain a gPC
expansion in coeﬃcients ¯ρ = (¯ρi)K
i=0 and ¯q = (¯qi)K
i=0 as
∂t¯ρi(t, x) + ∂x [¯qi(t, x) −(P(¯ρ(t, x))¯ρ(t, x))i] = 0
(42a)
∂t¯qi(t, x) + ∂x
Z
W
w2¯gi(t, x, w) dw −(P(¯ρ(t, x))¯q(t, x))i

= 1
ϵ
Z
W
wMi dw −¯qi(t, x)

(42b)
¯ρi(0, x) = ˜ρi(0, x) ¯q(0, x) = ˜qi(0, x), Mi =
Z
Ω
M(w; ρ(t, x, ξ), ξ)φi(ξ)fΞ(ξ)dξ.
(42c)
Clearly, applying assumption (29) leads for the transport part of the system the same ﬂux
as for the Aw–Rascle–Zhang system. A direct identiﬁcation of the source term with ﬂuid
dynamic quantities is no longer possible but depends on the precise dependence of M on ρ
and ξ.
2.2
Stability analysis
We extend the stability analysis [24, Section 3.2] to the stochastic case. Recall, the stochastic
PDE (13) for the random ﬁeld g = g(t, x, w, ξ) is given by
11

∂tg(t, x, w, ξ) + ∂x
h
(w −h(ρ(t, x, ξ)))g(t, x, w, ξ)
i
= 1
ε

Mg(w; ρ) −g(t, x, w, ξ)

,
(43)
where we assume that Mg fulﬁlls (M1) and (M2). The stability analysis in [24] is based on a
Chapman Enskog expansion of g in terms of ϵ. Here, we similarly assume that
g(t, x, w, ξ) = Mg(w; ρ(t, x, ξ)) + εg1(t, x, w, ξ),
(44)
where a.e. (x, w) and a.s. in ξ
Z
W
g1(t, x, w, ξ)dw = 0.
(45)
Up to terms of order ϵ2 the perturbation g1 fulﬁlls
g1(t, x, w, ξ) = −∂tMg(w; ρ(t, x, ξ)) −∂x (w −h(ρ(t, x, ξ))) Mg(w; ρ(t, x, ξ)).
(46)
The formal computations presented in [24, Section 3.2] extend to the above equations (43) and
(46) since they only rely on integration with respect to w and the properties (M1) and (M2).
Those are independent of ξ. Hence, after integrating (43) with respect to w, substituting g1
by (46) as well as subsequent diﬀerentiation leads to
∂tρ + ∂x (ρVeq(ρ)) = ϵ∂x

−D(ρ)∂xρ + ∂x
Z
W
w2Mg(w, ρ)dw

,
(47)
D(ρ) = (∂ρQeq(ρ) + ∂ρ(h(ρ)ρ)) (∂ρQeq(ρ) + h(ρ)) + ∂ρh(ρ) (Qeq(ρ) + h(ρ)ρ)
(48)
Qeq(ρ) = Veq(ρ)ρ,
(49)
where ρ = ρ(t, x, ξ). In [24] it is assumed that the Maxwellian Mf, see Section 2, and Mg are
related. In this case
Mf(v, ρ) := Mg(v + h(ρ), ρ) ∀v ∈V, ρ ≥0,
(50)
where Mf is a Maxwellian such that
R
V Mf(v, ρ)dv = ρ and
R
V vMf(v, ρ)dv = Qeq(ρ). Using
those properties equation (47) simpliﬁes and we obtain
12

∂tρ + ∂x (ρVeq(ρ)) = ϵ∂x (µ(ρ)∂xρ) ,
(51)
µ(ρ) =
 −∂ρQeq(ρ)2 −∂ρh(ρ)∂ρQeq(ρ)ρ + Qeq(ρ)∂ρh(ρ)

+
Z
V
v2∂ρMf(v, ρ)dv.
(52)
Note that in the presented case µ is in fact a random ﬁeld through its dependence on ρ =
ρ(t, x, ξ). Therefore, compared with the deterministic case, we may now infer information on
e.g. expectation, conﬁdence bands or the probability of µ to be non-positive. In particular,
the later is relevant for qualitative assessment of traﬃc ﬂow since it shows where possible
instabilities may occur. Hence, for ﬁxed t ≥0 and x ∈R consider
Pt,x(µ ≤0) :=
Z
Ω
H(−µ(ρ(t, x, ξ))fΞ(ξ)dξ.
(53)
It has been argued in [24] that (53) indicates regions of traﬃc situations of high risk. Further,
points (t, x) where Pt,x(·) > 0 holds might lead to the rise of stop–and–go waves. A numerical
investigation of (53) will be presented in the forthcoming section.
Note that the computation of (53) requires to reconstruct the stochastic ﬁeld ρ(t, x, ξ).
This can be obtained by reconstruction of g given by (16) where ˜g are given by equation (22).
For particular choices of Veq(·) and h(·) the gPC expansion of the ﬁrst terms in µ can be
obtained directly using the moments ˜ρ. In fact, assume h(ρ) = ρ and Veq = ρmax −ρ. Then,
Qeq = ρ(ρmax −ρ) is the ﬂux of the Lighthill-Whitham–Richards model and equation (51)
simpliﬁes
µ(ρ) = R(ρ) +
Z
V
v2∂ρMf(v, ρ)dv,
(54)
R(ρ) := −(ρmax −2ρ)2 −(ρmaxρ −ρ2) + ρ(ρmax −ρ).
(55)
Hence, we obtain
GK(R(ρ(t, x, ·)))(ξ) =
K
X
i=0
˜Ri(t, x)φi(ξ)
(56)
where ˜R is expressed in terms of ˜ρ and 1 = (1, . . . , 1)T ∈RK+1
˜R(t, x) = −P(ρmax1 −2˜ρ)(ρmax1 −2˜ρ) −ρmax˜ρ + P(˜ρ)˜ρ + P(˜ρ)(ρmax1 −˜ρ)
(57)
13

However, in the numerical simulations we use an Maxwellian Mf obtained by a discrete
velocity model, see below for the details. Therein, ρ enters within a rational polynomial and
a simple expression as above also for the expansion of ∂ρMf(v, ρ)dv seems to be not possible.
3
Computation results
Numerically, we are interested in indicating and forecast, regions of high risk of congestion or
instabilities. For this reason we focus on the simulation of (53). First, we perform a steady
state analysis and investigate parameters inﬂuencing regions of high probability. Secondly,
we show the evolution of this probability in time.
As Maxwellian we choose a discrete velocity distribution with N velocities as in [24]:
Mf(v; ρ) =
N
X
j=1
f∞
j (ρ)δvj(v).
(58)
The weights are normalized to ensure equation (27), i.e., PN
j=1 f∞
j (ρ) = ρ for any ρ > 0. The
set of velocities is {vj}N
j=1. Then, for ﬁxed ρ > 0 the weights are recursively deﬁned by
f∞
j
=





0
ρ ≥1
2
−2(1−ρ) Pj−1
k=1 f∞
k +(1−2ρ)ρ+
q
[(1−2ρ)ρ−2(1−ρ) Pj−1
k=1 f∞
k ]2+4ρ(1−ρ)ρf∞
k
2(1−ρ)
else
,
(59)
j = 1, . . . , N −1,
f∞
N =ρ −
N−1
X
j=1
f∞
j .
(60)
We use Veq = 1−2ρ and h(ρ) as indicated in the tests below. The Maxwellian Mg is obtained
through relation (50). Since the previous Maxwellian is a rational polynomial of ρ, an explicit
expression of µ(·) in terms of the moments of ρ might not be feasible. Therefore, we evaluate
(53) numerically using quadrature with Nξ number of points.
The gPC Aw–Rascle–Zhang system is discretized as in [19], i.e., employing a local Lax
Friedrichs scheme to solve (30).
The numerical parameters are as follows. We consider the space interval x ∈[a, b] = [0, 2]
and deﬁne the uniform spatial grid of size ∆x = 2 · 10−2. Moreover, let Tf = 1 be the ﬁnal
time of the simulations and ∆t the time step, which is chosen in such a way that the CFL
condition is fulﬁlled. By Nt we denote the number of the time steps needed to reach Tf. The
random variable ω is assumed be uniform distributed on (0, 1), i.e., fΞ = 1 and Ω= (0, 1).
As basis functions we consider the Haar basis, which are known to fulﬁll (A1)–(A3). The
numerical quadrature of (53) is conducted with a uniform discretization of (0, 1) in ξ with
14

Nξ = 104 quadrature nodes.
Whenever necessary the random ﬁelds density and ﬂux are
approximated up to a speciﬁed order K by ρ(t, x, ξ) = PK
i=0 ˜ρi(t, x)φi(ξ) and similarly for
q(t, x, ξ).
3.1
Steady state analysis
The Maxwellian Mf depends on two parameters, the number of discrete velocities nv governing
the level of description of traﬃc as well as the local density 0 < ρ < 1. In the steady state
case the density ρ has been a constant in the deterministic case [24], however, it is now a
random parameter. Since we are interested in the stability of traﬃc patterns we setup the
steady state problem as follows: We assume a constant traﬃc density ρ0 that is perturb by
(possibly small) perturbation
ρ(ξ) = ρ0 + σ(ξ −1
2),
(61)
where σ > 0 controls the standard deviation and the factor 1
2 is included to have zero mean
for ξ uniformly distributed. We are interested in the probability (53) for the previous choice
of ρ and ρ0 ∈[0.1; 0.9] with σ = 0.1. The resulting P(µ ≤0) is shown in Figure 2 (red curve).
In the free ﬂow regime, i.e., ρ0 < 1
2, the probability of instabilities is zero, it is increasing
until its maximum transition regime, and decreasing in the congested are ρ >
1
2.
It is
interesting to observe that in the congested region the probability of µ < 0 is close to zero
with the interpretation that the traﬃc propagates at low speed and and no free space to
accelerate. As expected, the highest probability for instabilities is in the transitional regime
ρ0 ≈(0.5, 0.6).
Moreover, in Figure 2, we compare the predictions for nv = 3 and nv = 10. In case of a
small and large number of discrete velocities. In case of only three velocities the transition
region stretches up to the maximal density due to the limit choices of velocities the drivers
can attain. For nv = 10, we observe the highest probability for ρ between 0.5 and 0.6 as
detailed above.
Further, the dependence of P(µ < 0) for ﬁxed values of ρ0 but diﬀerent standard deviations
σ is shown. We observe a diﬀerent behavior if we start from ρ0 = 0.4 and ρ0 = 0.6, see Figure
3. In the latter, the probability is decreasing with the possible explanation that the density is
spreading far from the transition area. In the former, we are in the opposite situation, since
we approach the transition region for increasing value for the standard deviation.
Furthermore, we compare also the eﬀect of diﬀerent hesitation functions In Figure 4,
h(ρ) = ρ(blue line) and h(ρ) = ρ3 are considered. The observed behavior is very similar.
15

0.2
0.4
0.6
0.8
Density
0
0.2
0.4
0.6
0.8
1
P(7<0)
nv=3
nv=10
Figure 2: Probability (53) for a Maxwellian Mf with diﬀerent number of discrete velocities:
nV = 3, nV = 10. On the x-axis ρ0 is shown, see (61).
0
0.05
0.1
0.15
0.2
Standard deviation
0
0.1
0.2
0.3
0.4
0.5
0.6
P(7<0)
nv=3
nv=10
0
0.05
0.1
0.15
0.2
Standard deviation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
P(7<0)
nv=3
nv=10
Figure 3: Probability (53) for diﬀerent velocities samples: nv = 3 (blue line), nV = 10(red
line) for diﬀerent values of ρ0 = 0.4(left) ρ0 = 0.6(right) when the standard deviation σ ranges
from zero to 0.2.
3.2
Time–dependent problems
We investigate numerically if the dynamics ampliﬁes the probability of instabilities starting
from a Riemann problem. In order to evaluate (53) for a temporal and spatially dependent
ρ we need to reconstruct the density and therefore we ﬁrst show the convergence in K. In all
following computations we set nv = 5, Nξ = 102 and deﬁne the Riemann problem:
16

0
0.2
0.4
0.6
0.8
1
Density
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
P(7<0)
h(;)= ;
h(;)= ;3
Figure 4: Probability (53) for diﬀerent hesitation functions: h(ρ) = ρ(blue line) and h(ρ) =
ρ3(red line).
ρ(x, 0, ξ) =



ρl ≡ξ ∼U(0.55, 0.85)
x < 1
0.2
x ≥1
,
v(x, 0, ξ) =



0.2
x < 1
0.7
x ≥1
.
(62)
In Figure 5 we show PTf,x(µ < 0) for an increasing number of base functions K. Moreover,
the probability of instabilities is increasing in time and travels backward. As explanation of
this behavior we note that the the given data leads to a rarefaction wave in (t, x) for any ﬁxed
ρl. Hence, drivers observe a free ﬂow area ahead and accelerate. The 95%−conﬁdence band
of the density ρ at the ﬁnal time Tf is shown in Figure 6(right).
As second test case we consider a (random) shock wave as given by (63).
Here, the
probability of instabilities increases and spreads both forward and backward.
A possible
explanation might be that the vehicles have to decelerate in order to avoid collisions leading
also to backwards spreading waves.
ρ(x, 0, ξ) =



ρl ≡ξ ∼U(0.15, 0.45)
x < 1
0.75
x ≥1
,
v(x, 0, ξ) =



0.7
x < 1
0.3
x ≥1
.
(63)
In Figure 7(right) the 95%−conﬁdence band of the density at the ﬁnal time Tf is shown. It is
interesting to note that even starting from P(µ < 0) ≡0 does not ensure to avoid instabilities
as time evolves, see Figure 7(left). Indeed, at time t = 0 the probability is zero. However,
as the time evolves, the density enters the transition phase and the probability of instability
grows.
17

0
0.5
1
1.5
2
Space x
0
0.01
0.02
0.03
0.04
0.05
P(7<0)
K=4
t=0
t=T/2
t=T
0
0.5
1
1.5
2
Space x
0
0.02
0.04
0.06
0.08
0.1
P(7<0)
K=8
t=0
t=T/2
t=T
0
0.5
1
1.5
2
Space x
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
P(7<0)
K=32
t=0
t=T/2
t=T
0
0.5
1
1.5
2
Space x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
P(7<0)
K=64
t=0
t=T/2
t=T
Figure 5: Probability (53) at time t = 0 (black line), t = T:f
2
(green line) t = T (blue line)
for K = 4 (top-left), K = 8 (top-right), K = 32 (bottom-left), K = 64 (bottom-right).
Acknowledgments
The authors thank the Deutsche Forschungsgemeinschaft (DFG, German Research Founda-
tion) for the ﬁnancial support through 20021702/GRK2326, 333849990/IRTG-2379, HE5386/19-
1,22-1,23-1 and under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612.
References
[1] A. Aw and M. Rascle, Resurrection of “second order” models of traﬃc ﬂow, SIAM J.
Appl. Math., 60 (2000), pp. 916–938 (electronic).
[2] I. Babuˇska, F. Nobile, and R. Tempone, A stochastic collocation method for ellip-
tic partial diﬀerential equations with random input data, SIAM Journal on Numerical
18

0
0.5
1
1.5
2
Space x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
P(7<0)
t=0
t=T/2
t=T
0
0.5
1
1.5
2
Space x
0
0.2
0.4
0.6
0.8
1
;(x,T)
P(7<0)
Figure 6: Probability of negative diﬀusion coeﬃcient in a rarefaction case at diﬀerent time:
t = 0, t = Tf
2 ,t = Tf, K = 64, and comparison with the conﬁdent region of the density at
t = Tf.
0
0.5
1
1.5
2
Space x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
P(7<0)
t=0
t=T/2
t=T
0
0.5
1
1.5
2
Space x
0
0.2
0.4
0.6
0.8
1
;(x,T)
P(7<0)
Figure 7: Density proﬁle and probability of negative diﬀusion coeﬃcient in a shock case at
t = Tf, K = 64.
Analysis, 45 (2007), pp. 1005–1034.
[3] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama, Dynamical
model of traﬃc congestion and numerical simulation, Phys. Rev. E, 51 (1995), pp. 1035–
1042.
[4] N. Bellomo and C. Dogbe, On the modeling of traﬃc and crowds: a survey of models,
speculations, and perspectives, SIAM Rev., 53 (2011), pp. 409–463.
19

[5] P. L. Bhatnagar, E. P. Gross, and M. Krook, A Model for Collision Processes in
Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems,
Phys. Rev., 94 (1954), pp. 511–525.
[6] R. Borsche and A. Klar, A nonlinear discrete velocity relaxation model for traﬃc
ﬂow, SIAM J. Appl. Math., 78 (2018), pp. 2891–2917.
[7] R. H. Cameron and W. T. Martin, The orthogonal development of non-linear func-
tionals in series of Fourier-Hermite functionals, Annals of Mathematics, 48 (1947),
pp. 385–392.
[8] J. Carrillo, L. Pareschi, and M. Zanella, Particle based gPC methods for mean-
ﬁeld models of swarming with uncertainty, Communications in Computational Physics,
25 (2019), pp. 508–531.
[9] J. Carrillo and M. Zanella, Monte Carlo gPC methods for diﬀusive kinetic ﬂocking
models with uncertainties, Vietnam Journal of Mathematics, 47 (2019), pp. 931–954.
[10] Q.-Y. Chen, D. Gottlieb, and J. S. Hesthaven, Uncertainty analysis for the steady-
state ﬂows in a dual throat nozzle, Journal of Computational Physics, 204 (2005), pp. 378–
398.
[11] R. M. Colombo, Hyperbolic phase transitions in traﬃc ﬂow, SIAM J. Appl. Math., 63
(2002), pp. 708–721.
[12] E. Cristiani and S. Sahu, On the micro-to-macro limit for ﬁrst-order traﬃc ﬂow
models on networks, Netw. Heterog. Media, 11 (2016), pp. 395–413.
[13] B. J. Debusschere, H. N. Najm, P. P. P´ebay, O. M. Knio, R. G. Ghanem, and
O. P. L. Maˆıtre, Numerical challenges in the use of polynomial chaos representations
for stochastic processes, SIAM Journal on Scientiﬁc Computing, 26 (2004), pp. 698–719.
[14] B. Despr´es, G. Po¨ette, and D. Lucor, Uncertainty quantiﬁcation for systems of
conservation laws, Journal of Computational Physics, 228 (2009), pp. 2443–2467.
[15] M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar con-
servation laws from follow-the-leader type models via many particle limit, Arch. Ration.
Mech. Anal., 217 (2015), pp. 831–871.
[16] S. Fan, M. Herty, and B. Seibold, Comparative model accuracy of a data-ﬁtted
generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), pp. 239–268.
[17] D. Gazis, R. Herman, and R. Rothery, Nonlinear follow-the-leader models of traﬃc
ﬂow, Oper. Res., 9 (1961), pp. 545–567.
20

[18] S. Gerster and M. Herty, Entropies and symmetrization of hyperbolic stochastic
Galerkin formulations, Communications in Computational Physics, 27 (2020), pp. 639–
671.
[19] S. Gerster, M. Herty, and E. Iacomini, Stability analysis of a hyperbolic stochastic
galerkin formulation for the aw-rascle-zhang model with relaxation, Mathematical Bio-
sciences and Engineering, 18(4) (2021).
[20] S. Gerster, M. Herty, and A. Sikstel, Hyperbolic stochastic Galerkin formulation
for the p-system, Journal of Computational Physics, 395 (2019), pp. 186–204.
[21] D. Gottlieb and D. Xiu, Galerkin method for wave equations with uncertain coeﬃ-
cients, Communications in Computational Physics, 3 (2008), pp. 505–518.
[22] M. Herty and R. Illner, Analytical and numerical investigations of reﬁned macro-
scopic traﬃc ﬂow models, Kinet. Relat. Models, 3 (2010), pp. 311–333.
[23] M. Herty and L. Pareschi, Fokker-Planck asymptotics for traﬃc ﬂow models, Kinet.
Relat. Models, 3 (2010), pp. 165–179.
[24] M. Herty, G. Puppo, S. Roncoroni, and G. Visconti, The BGK approximation
of kinetic models for traﬃc, Kinet. Relat. Models, 13 (2020), pp. 279–307.
[25] H. Holden and N. H. Risebro, The continuum limit of Follow-the-Leader models—a
short proof, Discrete Contin. Dyn. Syst., 38 (2018), pp. 715–722.
[26] J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with un-
certainty, Journal of Computational Physics, 315 (2016), pp. 150–168.
[27] S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-
Fokker-Planck system with uncertainty and multiple scales, SIAM Journal on Mathe-
matical Analysis, 50 (2018), pp. 1790–1816.
[28] A. Klar and R. Wegener, A kinetic model for vehicular traﬃc derived from a stochas-
tic microscopic model, Transport. Theor. Stat., 25 (1996), pp. 785–798.
[29]
, Enskog-like kinetic models for vehicular traﬃc, J. Stat. Phys., 87 (1997), p. 91.
[30] J. Kusch, G. Alldredge, and M. Frank, Maximum-principle-satisfying second-
order intrusive polynomial moment scheme, SMAI Journal of Computational Mathemat-
ics, 5 (2017), pp. 23–51.
[31] O. P. Le Maˆıtre and O. M. Knio, Spectral Methods for Uncertainty Quantiﬁcation,
Springer-Verlag GmbH, 2010.
21

[32] P. L’Ecuyer and C. Lemieux, Recent advances in randomized quasi-monte carlo meth-
ods, in International Series in Operations Research & Management Science, Springer US,
2002, pp. 419–474.
[33] M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traﬃc
ﬂow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), pp. 317–345.
[34] O. P. L. Maˆıtre and O. M. Knio, Spectral Methods for uncertainty quantiﬁcation,
Springer Netherlands, 1 ed., 2010.
[35] P. Pettersson, G. Iaccarino, and J. Nordstr¨om, A stochastic Galerkin method for
the Euler equations with Roe variable transformation, Journal of Computational Physics,
257 (2014), pp. 481–500.
[36]
, Polynomial chaos methods for hyperbolic partial diﬀerential equations, Springer
International Publishing, Switzerland, 2015.
[37] R. Pulch and D. Xiu, Generalised polynomial chaos for a class of linear conservation
laws, Journal of Scientiﬁc Computing, 51 (2012), pp. 293–312.
[38] G. Puppo, M. Semplice, A. Tosin, and G. Visconti, Kinetic models for traﬃc
ﬂow resulting in a reduced space of microscopic velocities, Kinet. Relat. Mod., 10 (2017),
pp. 823–854.
[39] B. Seibold, M. R. Flynn, A. R. Kasimov, and R. R. Rosales, Constructing set-
valued fundamental diagrams from jamiton solutions in second order traﬃc models, Netw.
Heterog. Media, 8 (2013), pp. 745–772.
[40] R. Shu, J. Hu, and S. Jin, A stochastic Galerkin method for the Boltzmann equation
with multi-dimensional random inputs using sparse wavelet bases, Numerical Mathemat-
ics: Theory, Methods and Applications, 10 (2017), pp. 465–488.
[41] T. J. Sullivan, Introduction to uncertainty quantiﬁcation, Texts in Applied Mathemat-
ics, Springer, Switzerland, 1 ed., 2015.
[42] K. Taimre, Botev, Handbook of Monte Carlo Methods, John Wiley and Sons, 2011.
[43] A. Tosin and M. Zanella, Boltzmann-type models with uncertain binary interactions,
Commun. Math. Sci., 16 (2018), pp. 963–985.
[44] N. Wiener, The homogeneous chaos, American Journal of Mathematics, 60 (1938),
pp. 897–936.
[45] D. Xiu, Numerical methods for stochastic computations: a spectral method approach,
Princeton University Press, Princeton, N.J, 2010.
22

[46] D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic
diﬀerential equations, SIAM Journal on Scientiﬁc Computing, 24 (2002), pp. 619–644.
[47] M. Zanella, Structure preserving stochastic Galerkin methods for Fokker–Planck equa-
tions with background interactions, Mathematics and Computers in Simulation, 168
(2020), pp. 28–47.
[48] H. M. Zhang, A non-equilibrium traﬃc model devoid of gas-like behavior, Transport.
Res. B-Meth., 36 (2002), pp. 275–290.
[49] Y. Zhu and S. Jin, The Vlasov-Poisson-Fokker-Planck system with uncertainty and a
one-dimensional asymptotic preserving method, Multiscale Modeling & Simulation, 15
(2017), pp. 1502–1529.
23
