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We discuss a numerical discretization of Hamilton-Jacobi equations on networks. The latter arise for example as reformulation of the Lighthill-Whitham-Richards traffic flow model. We present coupling conditions for the Hamilton-Jacobi equations and derive a suitable numerical algorithm. Numerical computations of travel times in a round-about are given.
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AIMS’ Journals
Volume X, Number 0X, XX 200X pp. X–XX
NUMERICAL DISCRETIZATION OF HAMILTON–JACOBI
EQUATIONS ON NETWORKS
Simone G¨
ottlich
University of Mannheim
School of Business Informatics and Mathematics, A5-6
68131 Mannheim, GERMANY
Michael Herty and Ute Ziegler
RWTH Aachen University
IGPM, Templergraben 55
52056 Aachen, GERMANY
(Communicated by the associate editor name)
Abstract. We discuss a numerical discretization of Hamilton–Jacobi equa-
tions on networks. The latter arise for example as reformulation of the Lighthill–
Whitham–Richards traffic flow model. We present coupling conditions for the
Hamilton–Jacobi equations and derive a suitable numerical algorithm. Numer-
ical computations of travel times in a round-about are given.
1. Introduction. Traffic flow models, especially on road networks, have been in-
tensively studied in the mathematical [13,16,20,3,6] as well as in the engineering
community [1,27,28,30,10,9,18,14,7,19] during the last years. We are in-
terested in first–order macroscopic models based on partial differential equations
for the traffic density [26,24] with the prototype being the Lighthill–Whitham–
Richards (LWR) model. When considering a traffic network the crucial point is
the coupling at a traffic junction leading to coupling conditions. Well–posedness
results for those conditions have been obtained for example in [15]. When consid-
ering car trajectories the Hamilton–Jacobi (HJ) reformulation of the LWR model
can be used. The discussion of traffic model in HJ form has been introduced in
engineering literature as for example [10,8,30]. Therein [29] the solution is also
known as Moskowitz function. Recently, there has been an intense discussion on
analytical properties of this equation in the context of traffic flow networks. For
example, in [2] the HJ formulation has been used to deduce optimal starting times
for a congested road among other results. In [22] general coupling conditions for
systems of HJ equations have been studied analytically. A particular application of
the well–posedness result therein is the coupling condition for traffic flow using a
fixed ratio for incoming and outgoing traffic flows. For question of reconstruction
of parameters of traffic models the HJ approach has been succesfully discussed and
applied in [1,28].
We are interested in a numerical scheme combining the existing coupling condi-
tions for LWR models [15,21] stated in the density and flow variables with a suitable
2000 Mathematics Subject Classification. Primary: 90B20, 35L50; Secondary: 35F21.
Key words and phrases. Networks, Traffic Flow, Hamilton-Jacobi Equations.
1
2 SIMONE G ¨
OTTLICH AND MICHAEL HERTY AND UTE ZIEGLER
numerical method for the HJ equation. The advantage of the numerical computa-
tion of the HJ model is to directly represent the trajectories of particular traffic
members and to compute the duration of specific journeys through the network.
Further, we introduce a new coupling condition for merging junctions replacing the
common right of way parameter [6,5] by a priority rule. For the numerical method
we extend the numerical algorithm of [23]. Other numerical approaches have been
discussed in [1,10,30].
2. Traffic flow network model. We give a brief review on the LWR traffic flow
model [26,31] for road networks and consider the coupling conditions for several
specific junctions.
tρ+xf(ρ) = 0,
ρ(x, 0) = ρ0(x).(1)
Here, ρ: (x, t)7→ ρ(x, t)[0, ρmax]R+denotes the density of cars, x[0, L]
R+describes the location on the road, Lis the length of the road (possibly being
) and tR+is time. As in [21,15] we assume that the flux function (also called
fundamental diagramm) f(ρ) is concave with a unique maximum at the designated
point ρ[0, ρmax]. This allows to define the function τ: [0, ρmax][0, ρmax]
mapping the density to a distinct density value with equal flux:
f(ρ) = f(τ(ρ)),with τ(ρ)6=ρ, if ρ6=ρ.
We denote by f1
l:f7→ f1
l(f)[0, ρ] and f1
r:f7→ f1
l(f)[ρ, ρmax] the
inverse of the flow function, respectively.
Remark 1. Typical flow functions are for example
f(ρ) = v(ρ)·ρ, (2)
where
v(ρ) = vmax
ρmax (ρmax ρ),
is the velocity which cars are assumed to have depending on the actual traffic
density, vmax is the maximal allowed velocity for the road and ρmax is the maximal
traffic density (bumper-to-bumper density) [3]. Another broadly used function with
constant velocity λis the triangular flow function [1,10,28] (and in the context of
telecommunication networks [12]):
f(ρ) = λ·ρif ρ[0, ρ]
λ·(2 ·ρρ) if ρ[ρ, ρmax].(3)
A road network is a directed graph G(E, V ), where Edenotes the set of edges
which represent the roads and Vthe set of vertices which represent the traffic
intersections or junctions. The length of each road i, leading from one junction to
the next, is given by LiR+. The roads are unidirectional. Lanes for different
directions can be described by separate edges. Incoming roads of each junction vare
denoted by δin
vand outgoing by δout
v. The density at the junction for an incoming
edge iwill be denoted by ˆρi(t) = ρi(Li, t) and the density for an outgoing edge j
is ¯ρj(t) = ρj(0+, t). As coupling condition the conservation of mass is imposed:
X
iδin
v
f(ˆρi(t)) = X
jδout
v
f(¯ρj(t)),t > 0.(4)
HAMILTON–JACOBI ON NETWORKS 3
For simplicity we use the following notation for the flow at junctions: ˆγi:= f( ˆρi)
for incoming edges and ¯γj:= f( ¯ρj) for outgoing edges. At dispersing junctions
(junctions with more than one outgoing road), we assume to have given (possibly
time-dependent) distribution parameters 0 αij 1 indicating the percentage of
the cars coming from road igoing to road j. Obviously, P
iδout
v
αij (t) = 1.The
condition (4) does not guarantee a unique traffic densities at the junction.
As in [3,4,15,21] the following construction is used to obtain unique traffic
densities ˆρi,¯ρjat the junction given the respective traffic flows ˆγiand ¯γj. For sake
of clarity, we skip the time variable in the following notation. The densities at the
junction are ˆρiand ¯ρj, respectively. The admissible sets for those densities depend
on the initial densities at the junction (ρi(Li) and ρi(0), respectively) and are then
for incoming roads
ˆρi{ρi(Li)}∪]τ(ρi(Li)), ρmax
i] if 0 ρi(Li)ρ
i
[ρ
i, ρmax
i] else ,(5)
and for outgoing roads
¯ρj[0, ρ
j] if 0 ρj(0) ρ
j
{ρj(0)} ∪ [0, τ (ρj(0))[ else .(6)
Given ˆγi(or ¯γj) the uniquely defined densities ˆρi(or ¯ρj) within the admissible sets
are
ˆρi=ρi(Li) if f1
il γi) = ρi(Li)
f1
ir γi) else ,(7)
and
¯ρj=ρj(0) if f1
jr γj) = ρj(0)
f1
jl γj) else ,(8)
respectively. We refer to Figure 1for an example.
ρi(Li)ρ
i:
ρi
f(ρi)
ρ
iρmax
i
ρi(L)
ρi(Li)
i:
ρi
f(ρi)
ρ
iρmax
i
ρi(L)
ρj(0)ρ
j:
ρj
f(ρj)
ρ
jρmax
j
ρj(0)
ρj(0)
j:
ρj
f(ρj)
ρ
jρmax
j
ρj(0)
Figure 1. Admissible sets for the coupling density for incoming
road iand outgoing road j(depicted by the thick black line) for
given initial densities ρi(Li) and ρj(0).
The admissible sets for the densities at the junction yield upper bounds for the
possible flux
γmax
i:= fi(ρi(Li)) if 0 ρi(Li)ρ
i
fi(ρ
i) else.,(9)
and
γmax
j:= fj(ρ
j) if 0 ρj(0) ρ
j
fj(ρ(0)) else .(10)
4 SIMONE G ¨
OTTLICH AND MICHAEL HERTY AND UTE ZIEGLER
In the following paragraphs, we explicitly state coupling conditions in order to
obtain unique traffic flows ˆγiand ¯γj. Those, by the above discussion, then lead to
unique traffic densities which in turn can be used as boundary conditions for (1).
Coupling conditions for traffic fluxes have been proposed by [3,15]. Here, we focus
on priority rules and round-abouts.
Two connected roads. In a bottleneck situation, the capacity of the traffic load may
decrease at a certain point. This can be viewed as a network model with two
connected roads as depicted in Figure 2such that maxρfj(ρ)<maxρfi(ρ) :
Figure 2. Bottleneck situation
The flows at the junction ˆγi:= γ1and ¯γj:= γ2are obtained as solution to
max γ1
such that γ2=γ1,0γ1γmax
1,and 0 γ2γmax
2
,(11)
where γmax
1and γmax
2are given by (9) and (10), respectively. The unique solution
is
ˆγi= ¯γj= min{γmax
1, γmax
2}.(12)
Figure 3. Schematic view of a dispersing junction with δin
v={1}
and δout
v={2,3}.
Dispersing junction. We assume the distribution rate αij for i= 1, j ∈ {2,3}at
the junction to be previously known. As in [9] we assume, that all the flow at the
junction is restricted as soon as one of the outgoing roads is not able to absorb the
designated flow. This corresponds to a first-in-first-out rule of cars and is a realistic
assumption, since a car waiting at the junction blockes all the traffic behind until
it continues. A mathematical model respecting this property stated in terms of the
flows ˆγiand ¯γjat the vertex is:
max γ1
such that γ2=α1,2γ1, γ3=α1,3γ1,0γkγmax
k, k ∈ {1,2,3}.(13)
This linear programming problem is solved explicitly:
¯γ2= min{α1,2γmax
1, γmax
2,α1,2
α1,3
γmax
3},¯γ3= min{α1,3γmax
1,α1,3
α1,2
γmax
2, γmax
3},
ˆγ1= ¯γ2+ ¯γ3,(14)
where γmax
1is given by (9) and γmax
2and γmax
3are given by (10). The resulting
boundary densities at the junction are again given by (7) and (8).
HAMILTON–JACOBI ON NETWORKS 5
(a) Merging junction (b) Priority road
Figure 4. Merging junction with δin
v={1,2}and δout
v={3}.
The priority road is i= 1.
Merging junction. In contrast to the previous discussion problem (13) may have
multiple solutions in the case of a merging junction depicted in Figure 4(a). There-
fore, additional conditions have to be imposed as constraints to (13). In [3,5,20,9]
propose a right of way parameter q]0,1[ prescribing the proportion of flow coming
from road 1 and 2. Here, we formulate a priority rule, where the traffic of the main
road always is prioritized over the traffic of a side road. As soon as road 3 reached a
state of dense traffic, cars from road 1 are preferred. As a mathematical formulation
we propose to replace (13) by (15) where the prioritization of the flow coming from
road 1 is obtained by using a weighting parameter ω > 1 in the objective function:
max ω·γ1+γ2
such that γ3=γ1+γ2,0γkγmax
k,k∈ {1,2,3}.(15)
Lemma 2.1. There exists a unique solution of (15) given by
¯γ3= min{γmax
1+γmax
2, γmax
3},(16a)
ˆγ1= min{γmax
1, γmax
3},(16b)
ˆγ2= ¯γ3ˆγ1,(16c)
where γmax
1and γmax
2are given by equation (9) and γmax
3is given by equation (10).
The proof of Lemma 2.1 can be found in Appendix A.
(a) (b) (c)
Figure 5. Model of a round–about (a) as combination of dispers-
ing and merging junctions. The round-about is composed of four
junctions (b) which in the present framework are modelled by (c).
Round–about. We consider a round–about depicted in Figure 5(a). For simplicity
we assume it is composed of four junctions with two incoming and two outgoing
6 SIMONE G ¨
OTTLICH AND MICHAEL HERTY AND UTE ZIEGLER
roads each. Additionally, the central ring of the round–about has priority over the
connecting roads, e.g., road 1 is prioritized over road 2 in Figure 5(c). Furthermore,
we assume road 2, no car is going to road 3, but all go to road 4. Hence, the
distribution parameters for the dispersing junction are α2,3= 0 and α2,4= 1.
Finally, the traffic distribution from road 1 to road 3 and 4 is also prescribed by
α1,3and α1,4, respectively. Therefore, the coupling condition for one part of the
round-about is
max ω·γ1+γ2
such that
γ3=α1,3γ1+α2,3γ2, γ4=α1,4γ1+α2,4γ2,0ˆγkγmax
k,k= 1,...4.
(17)
Lemma 2.2. Let α2,36= 0 and α2,4= 1.If α1,3α1,46= 0,then, problem (17) has a
unique solution. the solution is given by
ˆγ1= min{γmax
1,1
α13
γmax
3,1
α14
γmax
4},(18a)
ˆγ2= min{γmax
2, γmax
4α1,4ˆγ1},(18b)
¯γ3=α13 ˆγ1,(18c)
¯γ4=α14 ˆγ1+ ˆγ2.(18d)
The proof of Lemma 2.2 can be found in Appendix B. Note, that in [3,4,6] the
considered distribution parameters αhave to be strictly larger than zero and strictly
smaller than 1. The proof of Lemma 2.2 especially considers the case, where α2,3= 0
and α2,4= 1.
3. Numerical scheme for the Hamilton-Jacobi reformulation. As in [28,
29,8,10], the traffic network model can be reformulated as HJ equation. In this
section, we briefly recall the relation between LWR and HJ, and derive a numerical
scheme [23]. A HJ equation with Hamiltonian fis given by
Mt(x, t) + f(Mx(x, t)) = 0.(19)
If we consider roads on which vehicles cannot overtake, it is possible to number
them according to the order, they pass a certain point of the road. In [4,27,29,30]
a continuous function is considered, where the space-time trajectory of each car is
given by its the curves of cumulative counts. In detail, if we start counting with
the foremost car at time t=t0we get
N(x, t0) = ZL
x
ρ(x, t0)dx,
and for a general point in time t, the car number at (x, t) is given by
N(x, t) = ZL
x
ρ(x, t)dx+N(L, t) = N(0, t)Zx
0
ρ(x, t)dx.(20)
where the value of the left boundary is given by
N(0, t) = Zt
t0
f(ρ(x, t))dt.
Consequently, the curve given by
{(x, t) : N(x, t) = n}
HAMILTON–JACOBI ON NETWORKS 7
describes the tra jectory of the nth car. Obviously, M(x, t) = N(x, t). From (20)
we obtain Mx(x, t) = ρ(x, t),(x, t)[0, L]×[t0,+).¿From (19) the continuity
equation used in the LWR-model (1) holds. Differentiation of (19) with respect to
xyields:
0 = Mtx +f(Mx)x=Mxt +f(Mx)x.
Consequently, if we find an Mthat satisfies (19), ρ:= Mxalso satisfies (1). On
traffic problems we have ρ0, hence Mis monotonically increasing in x.
Extension to the network case. For the network model of HJ equation we provide
the additional index eindicating the current road. We have e∈ {1, ..., |E|}. Then,
the complete model reads
eE
tMe+f(xMe) = 0
xMe(x, 0) = ρ0(x) initial condition
xMe(0, t)(=: x¯
Me(t)) = ¯ρe(t) left boundary condition
xMe(Le, t)(=: xˆ
Me(t)) = ˆρe(t) right boundary condition
.(21)
The values of ¯ρeand ˆρeare obtained by the coupling conditions. They therefore
depend on the adjacent arcs of the node. Within the coupling conditions the car
density ρe(x, t) for x= 0 and x=Lhas to be evaluated. Therefore, in the numerical
algorithm we first reconstruct at every time step tnthe density ρ(·, tn).Using the
coupling conditions we then compute the ¯ρe(tn+1) and ˆρ(tn+1 ) to finally obtain the
boundary conditions for Me.
3.1. Numerical scheme for a single road. Consider a single road first. We
introduce a spatial grid i∈ {0,...,nx},xi= ∆x·i, nx=L
x, and a temporal
grid t∈ {0,...,nt}, where the gridsize ∆tis set according to the CFL-condition for
piecewise differentiable functions:
tx
maxρ|f(ρ)|.(22)
The superindex denotes the time step and the subindex the spatial point. We set
ρt
i:= ρ(∆x·i, t ·t) and Mt
j=M((j1
2)∆x, t ·t) where j={0, nx+ 1}. Note,
that the grid points for the discretization of M(x, t) are shifted by x
2c compared
to the grid for ρ(x, t). For the discretization of HJ we use the central first order
scheme [23]. The time evolution of Mat the inner grid points 0 < j < nx+ 1 is
computed as follows:
Mt+1
j=Mt
jt
2"fMt
j+1 Mt
j
x+fMt
jMt
j1
x#
+t
2∆xat
j(Mt
j+1 2Mt
j+Mt
j1),(23)
and due to the CFL condition
an
jmax
x[(j1)∆x,(j+1)∆x]|f(Mx)|.
The coupling is done in terms of densities (at least at the boundary of the do-
main). Therefore we need to reconstruct the derivative of Mat the junction. This
is done by finite differences and we define the density ρt
iby
ρt
i:= Mt
j+1 Mt
j
x,(24)
8 SIMONE G ¨
OTTLICH AND MICHAEL HERTY AND UTE ZIEGLER
where (j1
2)∆x=ix. We have to add an additional index efor all eEto M
and ρ, respectively, when considering the network formulation. However, for the
sake of readability, we will drop this index whenever the context is clear.
3.2. Discretization of the boundary condition. Due to dispersion effects of
the discretization scheme [25], the densities at the junction might not always be
captured correctly. Therefore we need to introduce suitable ghost-cells added on
both ends of each road. The wave fronts travel along the roads until they reach the
next junction, providing the coupling condition with information about the actual
density on the road. We let the waves run through these artificial cells, but take
the value at the road boundary to compute the coupling condition (Figure 6). This
finally leads to the correct density information at the junction, see below for more
details.
This method only works, when the number of ghost-cells is large enough to
absorb the dissipation of the front dissipation. In the sequel we will show, that two
ghost-cells on each side of the road are sufficient. A correlation between spatial and
temporal grid and a partocular choice of parameter an
jis required to ensure this
result.
Lemma 3.1. If the parameter an
jof the HJ-Scheme (23) is
an
j:= max
ρ|f(ρ)| ∀j, n, (25)
and the grid size tis set to the maximal possible value satisfying the CFL-condition
(22), then the HJ scheme (23) is equivalent to the Lax-Friedrich scheme:
ρt+1
i=1
2(ρt
i+1 +ρt
i1)t
2∆x(f(ρt
i+1)f(ρt
i1)).(26)
Proof. Scheme (23) and equation (24) allow for the following calculation:
ρt+1
iρt
i
t
(24)
=Mt+1
j+1 Mt
j+1
t·xMt+1
jMt
j
t·x
(23)
=1
2∆x"fMt
j+2 Mt
j+1
xfMt
jMt
j1
x#
+a
2∆xMt
j+2 Mt
j+1
x2·Mt
j+1 Mt
j
x+Mt
jMt
j1
x
(24)
=1
2∆xhf(ρt
i+1)f(ρt
i1)i+a
2∆xhρt
i+1 2ρt
i+ρt
i1i
(25)&(22)
ρt+1
i=1
2ρt
i+1 +ρt
i1t
2∆x·hf(ρt
i+1)f(ρt
i1)i,
which is precisely the Lax-Friedrich scheme (26).
Lemma 3.2. Assume that fis given by equation (3) and assume the density at
time tis of Heaviside type. i.e., ˆx[0, L]such that
ρ(x, t) = l, x ˆx
r, x > ˆx.
Then, for the Lax-Friedrich scheme and
t:= x
maxρ[0max]|f(ρ)|=x
λ,(27)
the dispersion over time of the wave front will not exceed two grid cells.
HAMILTON–JACOBI ON NETWORKS 9
Proof. The Lax-Friedrich scheme yields
ρt+1
i=1
2(ρt
i+1 +ρt
i1)t
2∆x(f(ρt
i+1)f(ρt
i1))
(27)
=1
2(ρt
i+1 +ρt
i1)1
2λ(f(ρt
i+1)f(ρt
i1)).(28)
The spatial grid is given such that the discontinuity of the initial condition is lo-
cated between grid point iand gridpoint i+ 1. Hence, the density values at the
discontinuity at timestep tare given by:
ρt= (l,..., l
i1, l
i, r
i+1, r
i+2,...,r).
The scheme preserves the density values inside the constant regions, because (28)
yields
if ρt
j1=ρt
j+1 ρt+1
j=ρt
j1
for an arbitrary space grid point j. Hence, it is sufficient to consider the density
evolution next to the discontinuity. For this purpose we distinguish several cases:
Case 1: l[0, ρ]r[0, ρ] :
Applying (28) to ρt, we get
ρt+1 = (l,..., l
i1, l
i, l
i+1, r
i+2,...,r),
Hence, we get a sharp forward traveling front without any dispersion.
Case 2: l[ρ, ρmax]r[ρ, ρmax ] :
Applying (28) to ρt, we get
ρt+1 = (l, . . . , l
i1, r
i, r
i+1, r
i+2,...,r),
Hence, we get a sharp backwards traveling front without any dispersion.
Case 3: l[0, ρ]r[ρ, ρmax] : This case is slightly more involved. We show the
claim in two steps:
i) Computing the next time step via Lax-Friedrich leads to
ρt+1 = (l, . . . , l
i1, m
i, m
i+1, r
i+2,...,r),
with m=l+rρ[l, r].
ii) Given the densities
ρ¯
t= (l, . . . , l
i1, m
i, m
i+1, r
i+2,...,r),
with an abitrary m[l, r].
a) If m[0, ρ], the density for the next time step evolves to
ρ¯
t+1 = (l,...,l
i,ˆm
i+1,ˆm
i+2, r
i+3,...,r),
with ˆm=m+rρ.Due to the assumption made for Case 3, we
have
ˆm=m+rρml
Furthermore, we have
ˆm=m
|{z}
a) ρ
ρ+rr.
Consequently, we get ˆm[l, r].
10 SIMONE G ¨
OTTLICH AND MICHAEL HERTY AND UTE ZIEGLER
b) If m[ρ, ρmax], the density values for the following time step are
ρ¯
t+1 = (l, . . . , l
i2,˚m
i1,˚m
i, r
i+1,...,r),
with ˚m=l+mρ.We have
˚m=l+m
|{z}
b)ρ
ρl
and
˚m=l
|{z}
(Case 3)ρ
+mρmr
˚m[l, r].
Hence, ρ¯
t+1 again fulfills the assumptions imposed to ρ¯
t, with the shape
shifted by one space step either to the left or to the right. Therefore, by
applying the Lax-Friedrich scheme iteratively over time, the dispersion
will never become greater than two cells.
Case 4: l[ρ, ρmax]r[0, ρ] :
i) Computing timestep t+ 1 via Lax-Friedrich yields:
ρt+1 = (l, . . . , l
i1, ρ
i
, ρ
i+1
, r
i+2,...,r).
ii) Applying again (28) the densities for timestep t+ 2 are given by:
ρt+2 = (l, . . . , ρ
i1
, ρ
i
, r
i+1, r
i+2,...,r).
Hence, the resulting wave front is moving backwards carrying along two middle
density values ρ.
3.3. Details of the numerical algorithm in the network case. The complete
numerical scheme for solving HJ equations on road networks is described in Algo-
rithm 1. Some steps are particularly illustrated in Figure 6.
A crucial point is the computation are the coupling conditions as depicted in
Figure 6(c). As denoted in line 16 of Algorithm 1, equations (9) to (18) are used.
The detailed procedure is the following: Consider a junction vwith at most two
incoming roads (δin
v) and at most two outgoing roads (δout
v). The leftmost
gridpoints of the incoming roads and the rightmost gridpoints of the outgoing roads
in terms of the density ρhave been computed for timestep t+ 1, see Figure 6(b).
Hence, the values for ρt+1
e,nxeδin
vand ρt+1
e,0eδout
vare given corresponding to
ρe(L) and ρe(0) in the continuous case. Now, we use equations (9) and (10) to
obtain the maximal possible flow γmax
efor all roads econnected to the junction.
Depending on the junction type we compute the coupling flows ˆγeeδin
vand
¯γeeδout
vusing equations (12), (14), (16) or (18), respectively. The density
boundary values ˆρeeδin
vand ¯ρeeδout
vare uniquely given by (7) and (8).
An illustration of this procedure is given in Figure 7. Some further remarks on the
algorithm 1in order:
line 2: Note that the Godunov scheme [17] does not need any ghostcells to compute
the coupling condition. The presented scheme introduces numerical diffusion
such that the ghost cells need to be sufficiently large. Its size has been dis-
cussed in the previous Lemma. The length of the ghost cells is chosen equal
HAMILTON–JACOBI ON NETWORKS 11
t
t+ 1
Mnx+1 ˜
Mr1˜
Mrngˆ
M¯
M˜
Mlng˜
Ml1M1
incoming road right ghostcells left ghostcells outgoing road
junction
(a) Computation of the next time step for the inner cells in terms of M, see algorithm 1, line 12
t
t+ 1
Mnx+1 ˜
Mr1˜
Mrngˆ
M¯
M˜
Mlng˜
Ml1M1
ρnxρ1
incoming road right ghostcells left ghostcells outgoing road
junction
(b) Computation of density value at last grid point before the ghost-cells, see 1, line 14
t
t+ 1
Mnx+1 ˜
Mr1˜
Mrngˆ
M¯
M˜
Mlng˜
Ml1M1
ρnx
ˆρ¯ρ
ρ1
incoming road right ghostcells left ghostcells outgoing road
junction
(c) Computation of the coupling density values ˆρiand ¯ρj1, line 16
t
t+ 1
Mnx+1 ˜
Mr1˜
Mrngˆ
M¯
M˜
Mlng˜
Ml1M1
ρnx
ˆρ¯ρ
ρ1
incoming road right ghostcells left ghostcells outgoing road
junction
(d) Computation of the coupling values in terms of M, namely ˆ
Miand ¯
Mj1, line 19
Figure 6. The numerical algorithm exemplarily for one incoming
and one outgoing road including the two ghost cells. For the sake
of readability, we have skipped the road index.
to the size of the interior cells. In order to have the same speed of propagation
those cells do not enter the computation of the length of the road.
line 3: Choose the size of the timegrid such that the CFL-condition holds.
line 8:Mis initialized from right to left on each road.
4. Computational results for round-abouts. We apply the merging and dis-
persing junction model to a small network consisting of eight roads, see Figure 8.
The network describes a small traffic round–about examined in [3]. We use the
same setting as in [3], where the flow is given by f(ρ) = ρ(1 ρ) and initial as well
12 SIMONE G ¨
OTTLICH AND MICHAEL HERTY AND UTE ZIEGLER
incoming roads eδin
v:
ρt+1
e,nx
(9)γmax
e
(12), (14), (16) or (18)ˆγt+1
e
(7)ˆρt+1
e
outgoing roads eδout
v:
ρt+1
e,0
(10)γmax
e¯γt+1
e
(8)¯ρt+1
e
Figure 7. Equations involved in the numerical computation of
the coupling conditions depending on the type of junction. It is as-
sumed that from the values of Mt
e,j the values of the corresponding
densities ρt
e,i are already recovered using (24).
as boundary data a given as follows:
boundary density of incoming roads: ρ1(x, 0) = 0.25, ρ3(x, 0) = 0.4
initial density of incoming roads: ρ1(0, t) = 0.25, ρ3(0, t) = 0.4
initial density of outgoing roads: ρ2(0, t) = ρ4(0, t) = 0.5
initial density of inner circle: ρi(0, t) = 0.5,i= 5,6,7,8
In [3] this test case is compared for different right-of-way parameters q]0,1[,
determining the proportion of cars coming from each road at merging junctions.
The priority rule used in this paper corresponds to the case q= 0.
Figure 8shows the traffic density exemplarily for four roads at four different
points in time. Since the boundary condition is constant, the density evolution
reaches an equilibrium and does not change for t > 5. The traffic at the inner circle
has priority, therefore, the round–about does not get blocked. This is qualitatively
the same behaviour as in [3], when a small parameter qis used. Since our model
uses strict priorities, the equilibrium state is reached faster compared with [3]. We
have used the proposed method for simulations and reconstructed the density values
as in (24). The thin lines show the result obtained by using the Godunov scheme,
which we use as a benchmark. For road 4 at time t= 2 it can clearly be seen,
that by Godunov a sharper resolution of the shock wave is obtained. However, the
actual density levels are equivalent for both schemes.
Next, we consider a larger round–about composed of four junctions with two
incoming and two outgoing roads as in Figure 9(a).
According to the enumeration, roads 1 to 4 are leading towards the inner circle
which is composed of roads 5 to 8. Roads 9-12 point out of the round–about.
Usually, drivers cannot drive as fast inside the inner circle as on the other roads. For
this reason, we describe these roads by different triangular fundamental diagrams.
As before, the traffic density belongs to the interval between 0 (no traffic) and 1
(maximal dense traffic). Since we assume that the usual speed of the cars is faster at
the outer roads than inside the circle, the corresponding flow function has a steeper
slope outside the inner circle. We prescribe the left boundary data for the incoming
roads 1 to 4. We assume, that road 1 and 3 are slightly more busy than roads 2
and 4. For simplicity we use the same boundary data for each road pair. Figure 10
gives a detailed overview of the boundary data at an average working day from 5am
HAMILTON–JACOBI ON NETWORKS 13
Algorithm 1: HJ-scheme for networks
/* Input: Road network with length and flow function for each
road, initial and boundary conditions in terms of ρ, time
horizon T, grid size x, number of ghostcells ng*/
/* Output: Simulation of the traffic in terms of density */
1begin
/* Compute number of gridpoints */
2number of spacesteps: nxe =Le
x+ 1 ng,eE;
3timegridsize: ∆t=x
maxeE{maxρ|f
e(ρ)|} ;
4number of timesteps: nt=T
t+ 1;
/* Transfer initial values from ρto M. */
5forall the eEdo
6ˆ
M0
e= 0; /* right boundary value */
7M0
e,rng=ˆ
M0
exˆρ0
e;/* rightmost ghost-cell */
8M0
e,j =Me,j+1 0
e,i gridpoints j(including ghost-cells);
9¯
M0
e=M0
e,0x¯ρ0
e;/* left boundary value */
10 for t= 0,...,nt1do
/* Compute next time iteration for each road e*/
11 forall the eEdo
12 Compute Mt+1
e,j by (23)gridpoints j(including ghost-cells)
/* see Figure 6(a) */
/* Transfer Mto ρ*/
13 forall the eEdo
14 ρt+1
e,i =Mt+1
e,j+1 Mt+1
e,j
x,gridpoints j/* see Figure 6(b) */
/* Compute coupling at junctions */
15 forall the vVdo
16 Compute coupling for timestep taccording to junction type using
density values next to ghost-cells. /* see Figure 6(c) and 7
*/
/* Get boundary value in terms of M*/
17 forall the eEdo
18 left: ¯
Mt+1
e=˜
Mlng¯ρt+1
ex;
19 right: ˆ
Mt+1
e=˜
Mrng+ ˆρt+1
ex/* see Figure 6(d) */
to 1pm. This is a test setting attempting to tackle the qualitative traffic behaviour
taking the morning rush hour into account.
Figure 11 shows the traffic density along the inner circle for exemplary points in
time. Since the traffic at the inner roads always has the priority at junctions and
outgoing roads are not blocked in our setting, no jams appear inside the round–
about. However, we observe, that at peak time the traffic density all along the inner
circle is at value ρ= 0.5, which means that the traffic moves to maximal possible
flow. Also at peak time, traffic jams occur at roads leading to the inner circle.
Particularly from 7am to shortly after 11am, the traffic entering the roundabout is
14 SIMONE G ¨
OTTLICH AND MICHAEL HERTY AND UTE ZIEGLER
Figure 8. Small round–about. Comparison of the Godunov
scheme for LWR and the presented numerical methods. Results
are given for the fundamental diagram f(ρ) = ρ(1 ρ).
quite dense. However, since the incoming traffic reduces drastically around 11am
(see boundary condition depicted at Figure 10(a)) the jam is resolved again a while
after the incoming traffic reduces.
It is easy to derive the trajectories of cars from the HJ, since we only have to
compute the contour lines of function Me(x, t). In Figure 13, the trajectories of
3 cars moving along the roads 1-5-6-11 are depicted. When entering the traffic
network before 6:59am the cars move freely and leave the system already about 1
minute. In contrast to that, another driver, who enters the system only 4 minutes
later, already encounters dense traffic on the road and needs more than 4 minutes
to move to the end of road 11. Figure 14 shows the duration of the route 1-5-6-11
HAMILTON–JACOBI ON NETWORKS 15
(a) Round–about
ρ
f(ρ)
0.5 1
1
λ=2
Road 1-4 and 9-12
ρ
f(ρ)
0.5 1
0.5
λ=1
road 5-8
(b) Fundamental diagram for
outer and inner roads of the
roundabout
Figure 9. Model of a round–about with different fundamental
diagrams on different roads.
5am 7am 9am 11am 1pm
0.2
0.4
0.6
0.8
1left boundary of road 1 and 3
time
traffic density
(a) Boundary density of road 1 and 3
5am 7am 9am 11am 1pm
0
0.2
0.4
0.6
0.8
1left boundary of road 2 and 4
time
traffic density
(b) Boundary density of road 2 and 4
Figure 10. Incoming traffic over time.
depending on the starting time of the journey. While it takes only 1 minute to
traverse the route during light traffic times, cars need up to 4.7 minutes between
7 and 9 am. Hence, it takes more than 4 times longer to traverse the given route
during the rush-hour.
16 SIMONE G ¨
OTTLICH AND MICHAEL HERTY AND UTE ZIEGLER
road 5 −> road 6 −> road 7 −> road 8 −>
0.25
0.5
0.75
1Density evolution in inner traffic circle
x
density
6:00:50 AM (begin early traffic)
7:00:50 AM (begin rush hour)
7:01 AM − 10:10 AM (peak time)
10:10:03 AM (begin of traffic reduction)
11:40 AM − T (midday traffic)
Figure 11. Evolution of the traffic density in the inner circle.
Figure 12. Traffic evolution at the junction.
HAMILTON–JACOBI ON NETWORKS 17
Figure 13. Single car tracking for three cars on the above route
starting at different times.
5 AM 7 AM 9 AM 11 AM 1 PM
10
15
20
25
30
35
40
45
departure time
duration of crossing (minutes)
Figure 14. Travel time (in minutes) for the route depicted in
Figure 13, depending on the starting time of the journey.
18 SIMONE G ¨
OTTLICH AND MICHAEL HERTY AND UTE ZIEGLER
Acknowledgements. This work was financially supported by the DAAD research grants no.
50756459 and the DFG project HE5386/7-1. Furthermore, the author U. Ziegler likes to
thank the RWTH Aachen University for the scholarship.
REFERENCES
[1] A. M. Bayen and C. G. Claudel, Convex Formulations of Data Assimilation Problems for a
Class of Hamilton-Jacobi Equations, SIAM J. Control Optim., 49 (2011), 383–402.
[2] A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM J. Math.
Anal., 43 (2011), 2384–2417.
[3] G. Bretti, R. Natalini and B. Piccoli, Numerical approximations of a traffic flow model on
networks, Netw. Heterog. Media, 1(2006), 57-84.
[4] G. Bretti and B. Piccoli, A Tracking Algorithm for Car Paths on Road Networks, SIAM J.
Appl. Dyn. Syst., 7(2008), 510–531.
[5] Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete
Contin. Dyn. Syst. Ser. B, 5(2005), 599–630.
[6] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math.
Anal., 36 (2005), 1862–1886.
[7] R. Corthout, G. Fl¨otter¨od, F. Viti and C.M.J. Tamp`ere, Non-unique flows in macroscopic
first-order intersection models, Transportation Res. Part B, 46 (2012), 343–359.
[8] C. F. Daganzo, A variational formulation of kinematic waves: basic theory and complex
boundary conditions, Transportation Res. Part B, 39 (2005), 187–196.
[9] C. F. Daganzo, The Cel l Transmission Model, Part II: Network Traffic, Transportation Res.
Part B, 29 (1995), 79–93.
[10] C. F. Daganzo, On the variational theory of traffic flow: well-posedness, duality and appli-
cations, Networks and Heterogeneous Media, 1(2006), 601-619.
[11] G.B. Dantzig, “Linear programming and extensions,” Princeton University Press, Princeton,
1963.
[12] C. D’Apice, R. Manzo and B. Piccoli, A fluid dynamic model for telecommunication networks
with sources and destinations, SIAM J. Appl. Math., 68 (2008), 981-1003.
[13] C. D’Apice, R. Manzo and L. Rarit`a, Splitting of Traffic Flows to Control Congestion in
Special Events, Int. J. Math. Math. Sci., (2011), Art. ID 563171, 18 pages.
[14] G. Fl¨otter¨od and J. Rohde, Operational macroscopic modeling of complex urban intersections,
Transportation Res. Part B: Methodological, 45 (2011), 903–922 .
[15] M. Garavello and B. Piccoli, “Traffic flow on Networks,” American Institute of Mathematical
Sciences, Springfield, 2006.
[16] M. Garavello and B. Piccoli, Source-Destination flow on a road network, Commun. Math.
Sci., 3(2005), 261–283.
[17] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of
the equations of hydrodynamics, Mat. Sb. (N.S.), 47 (1959), 271–306.
[18] B. Haut and G. Bastin, A second order model of road junctions in fluid models of traffic
networks, Netw. Heterog. Media, 2(2007), 227–253.
[19] M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM
Journal on Scientific Computing, 25 (2003), 1066–1087.
[20] M. Herty and M. Rascle, Coupling conditions for a class of “second-order” models for traffic
flow, SIAM J. Math. Anal., 38 (2006), 595–616.
[21] H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidi-
rectional roads, SIAM J. Math. Anal., 26 (1995), 999–1017.
[22] C. Imbert, R. Monneau and H. Zidnani, A Hamilton-Jacobi approach to junction problems
and application to traffic flow, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166.
[23] A. Kurganov and E. Tadmor, New High-Resolution Semi-discrete Central schemes for
Hamilton-Jacobi Equations, J. Comput. Phys., 160 (2000), 720–742.
[24] J.-P. Lebacque and M. Khoshyaran, First order macroscopic traffic flow models for networks
in the context of dynamic assignment, Transportation Planning Applied Optimization, 64
(2004), 119–140.
[25] R.J. LeVeque, “Numerical methods for conservation laws,” Birkh¨auser Verlag, Basel, 1992.
[26] M. J. Lighthill and G. B. Whitham, On kinematic waves II. A theory of traffic flow on long
crowded roads, Proc. Royal Society London. Ser. A., 229 (1955), 317–345.
HAMILTON–JACOBI ON NETWORKS 19
[27] Y. Makigami, G. F. Newell and R. Rothery, Three-dimensional representation of traffic flow,
Transportation Science, 5(1971), 302–313.
[28] P. Mazar´e, A. Dehwah, C. Claudel and A. Bayen, Analytical and grid-free solutions to the
lighthill-whitham-richards traffic flow model, Transportation Res. Part B: Methodological, 45
(2011), 1727-1748.
[29] K. Moskowitz, Discussion of ‘freeway level of service as influenced by volume and capacity
characteristics, Highway Research Record, 99 (1965), 43–44.
[30] G. F. Newell, A simplified theory of kinematic waves in highway traffic: (i) general theory;
(ii) queuing at freeway bottlenecks; (iii) multi-destination flow, Transportation Res. Part B,
27 (1993), 281–313.
[31] P. I. Richards, Shock Waves on the Highway, Operations Res., 4(1956), 42–51.
Appendix A. Proof of Lemma 2.1.
Proof. Problem (15) is a linear optimization problem. Techniqually, it can be solved
by SIMPLEX algorithm [11]. it is also possible to directly get the optimal solution
by the considering different cases. We distinguish between three cases depending
on the size of γmax
3. The feasible region of each case is depicted in Figure 15 and
the optimal solution is indicated by the black dot.
γ1
γ2
γmax
2
γmax
1
γ1+γ2=γmax
3
ωγ1+γ2
max
i)
γ1
γ2
γmax
2
γmax
1
γ1+γ2=γmax
3
ωγ1+γ2
max
ii)
γ1
γ2
γmax
2
γmax
1
γ1+γ2=γmax
3
ωγ1+γ2
max
iii)
Figure 15. Feasible region for flows at junction.
i) γmax
3γmax
1: In this case the optimal solution of (15) is given by
ˆγ1=γmax
3,ˆγ2= 0,¯γ3=γmax
3.
ii) γmax
1< γmax
3γmax
1+γmax
2: In this case the optimal solution of (15) is
given by
ˆγ1=γmax
1,ˆγ2=γmax
3γmax
1,¯γ3=γmax
3.
iii) γmax
3> γmax
1+γmax
2: In this case the optimal solution of (15) is given by
ˆγ1=γmax
1,ˆγ2=γmax
2,¯γ3=γmax
1+γmax
2.
This yields directly (16a) - (16c). Hence, (15) is uniquely solvable.
20 SIMONE G ¨
OTTLICH AND MICHAEL HERTY AND UTE ZIEGLER
Appendix B. Proof of Lemma 2.2.
Proof. With the given distribution parameters α2,3= 0 and α2,4= 1, the optimiza-
tion problem reduces to
max ω·γ1+γ2
such that
γ3=α13γ1
γ4=α14γ1+γ2,
0ˆγkγmax
k,k= 1,...4,
(29)
We can simplify the problem and obtain the equivalent formulation
max ωγ1+γ2
such that
γ1b
α14γ1+γ2γmax
4
γ2γmax
2
γ1, γ20
(30)
where bis a known parameter given by
b:= min{γmax
3
α1,3
, γmax
1}, α1,36= 0.(31)
We introduce slack variables si0, i = 1,...,3 and rewrite (30). This yields
zωγ1γ2= 0
γ1+s1=b
α14γ1+γ2+s2=γmax
4
γ2+s3=γmax
2,
(32)
where zrepresents the objective function value. We apply the SIMPLEX algorithm
to solve the problem and want to have γ1to enter the basis. We have to find the
minimum of band γmax
4
α14 . Hence, we distinguish two cases.
Case i) γmax
4
α14 b.In this case the transformation yields
z+( ω
α14 1)γ2+ω
α14 s2=ω
α14 γmax
4
1
α14 γ2+s11
α14 s2=bγmax
4
α14
γ1+1
α14 γ2+1
α14 s2=γmax
4
α14
γ2+s3=γmax
2
(33)
Since ω > 1 and α14 1, we know that ω
α14 1>0. Hence, all coefficients in the
first row are postive. Thus, the basic solution of (33) is optimal, with s2=γ2= 0
and γ1=1
α14 γmax
4, which corresponds to (18).
Case ii) bγmax
4
α14 .In this case the first SIMPLEX transformation leads to
zγ2+ωs1=ωb
γ1+s1=b
γ2α14s1+s2=γmax
4α14b
γ2+s3=γmax
2
(34)
γ2has a negative coefficient in the first row. Hence, the basic solution is not optimal.
We have to transform the system a second time such that γ2enters the basis as
HAMILTON–JACOBI ON NETWORKS 21
well. In order to pivot the row with minimal ratio between right hand side and
coefficient of the entering variable, again two different cases have to be considered.
Case iia) γmax
4α14bγmax
2.The second SIMPLEX transformation yields
z+(ωα14)s1+s2= (ωα14 )b+γmax
4
γ1+s1=b
γ2α14s1+s2=γmax
4α14b
α14s1s2+s3=γmax
2γmax
4+α14b
(35)
Because ω > 1 and α14 1, all coefficients in the first row are positive. Hence, the
basic solution with s1=s2= 0, γ1=band γ2=γmax
4α14bis optimal and fulfills
(18).
Case iib) γmax
2γmax
4α14b.In this case the next transformation of the system
(34) looks like
z+ωs1+s3=ωb +γmax
2
γ1+s1=b
α14s1+s2s3=γmax
4γmax
2α14b
γ2+s3=γmax
2
(36)
All coefficients of the first row of (36) are positive. Hence, the basic solution is
given by s1=s3= 0, γ1=band γ2=γmax
2.
These cases cover all possibilities and proof the claim.
Received xxxx 20xx; revised xxxx 20xx.
E-mail address:goettlich@uni-mannheim.deu
E-mail address:herty@igpm.rwth-aachen.de
E-mail address:ziegler@igpm.rwth-aachen.de
... We cite also the following works: a convergent semi-Lagrangian scheme is introduced in [33] for equations of eikonal type. In [57], an adapted Lax-Friedrichs scheme is used to solve a traffic model; it is worth mentioning that this discretization implies to pass from the scalar conservation law to the associated Hamilton-Jacobi equation at each time step. ...
... A convergent semi-Lagrangian scheme is introduced in [33] for equations of eikonal type. In [57], an adapted Lax-Friedrichs scheme is used to solve a traffic model; it is worth mentioning that this discretization implies to pass from the scalar conservation law to the associated Hamilton-Jacobi equation at each time step. In [59], Guerand and Koumaiha (see Chapter 2) improved the error estimate for a larger class of Hamiltonians. ...
... Apart from [41], we mention [33], where a convergent semi-Lagrangian scheme is introduced for equations of eikonal type. In [57], an adapted Lax-Friedrichs scheme is used to solve a traffic model. ...
Thesis
The aim of this work is mainly to study on the one hand a numerical approximation of a first order Hamilton-Jacobi equation posed on a junction. On the other hand, we are concerned with the stability and the exact indirect boundary controllability of coupled wave equations in a one-dimensional setting.Firstly, using the Crandall-Lions technique, we establish an error estimate of a finite difference scheme for flux-limited junction conditions, associated to a first order Hamilton-Jacobi equation. We prove afterwards that the scheme can generally be extended to general junction conditions. We prove then the convergence of the numerical solution towards the viscosity solution of the continuous problem. We adopt afterwards a new approach, using the Crandall-Lions technique, in order to improve the error estimates for the finite difference scheme already introduced, for a class of well chosen Hamiltonians. This approach relies on the optimal control interpretation of the Hamilton-Jacobi equation under consideration.Secondly, we study the stabilization and the indirect exact boundary controllability of a system of weakly coupled wave equations in a one-dimensional setting. First, we consider the case of coupling by terms of velocities, and by a spectral method, we show that the system is exactly controllable through one single boundary control. The results depend on the arithmetic property of the ratio of the propagating speeds and on the algebraic property of the coupling parameter. Furthermore, we consider the case of zero coupling parameter and we establish an optimal polynomial energy decay rate. Finally, we prove that the system is exactly controllable through one single boundary control
... Ils ont montré la convergence de ce schéma. Citons également les travaux de Camilli, Festa et Schieborn [27] qui ont étudié un schéma semi-lagrangien pour des équations eikonales, et Göttlich, Ziegler, Herty [59] qui ont utilisé un schéma adapté de Lax-Friedrichs pour modéliser un problème de trafic routier. ...
... A convergent semi-Lagrangian scheme is introduced in [27] for equations of eikonal type. In [59], an adapted Lax-Friedrichs scheme is used to solve a traffic model; it is worth mentioning that this discretization implies to pass from the scalar conservation law to the associated Hamilton-Jacobi equation at each time step. ...
Thesis
Cette thèse est constituée de deux parties. Une première partie est consacrée à l’étude des équations de Hamilton-Jacobi du premier ordre. Ces équations apparaissent en contrôle optimal et permettent de modéliser des problèmes de trafic routier, de supraconductivité et de mouvements d’interface. Le premier chapitre de la thèse présente un résultat d’équivalence de conditions au bord de type contraintes d’état. On obtient l’équivalence de trois formulations de ces conditions au bord. Ceci permet notamment de déduire que les résultats d’existence et d’unicité valables pour l’une des trois formulations sont valables pour les trois. Le second chapitre porte principalement sur un résultat d’équivalence de conditions au bord de type dynamique ; il est complété par un résultat d’unicité pour ce problème. En considérant la relation «avoir les mêmes solutions», on peut regrouper les conditions aux limites (vérifiées en un sens faible) en classe d’équivalence. Nous montrons que dans chaque classe il y a une unique condition vérifiée en un sens fort. Le troisième chapitre est consacrée à l’étude d’un schéma monotone aux différences finies pour une équation de Hamilton-Jacobi posée sur une jonction. Une jonction est un réseau formé d’un seul noeud et d’un nombre fini d’arrêtes infinies. La convergence du schéma vers l’unique solution a été montrée par Costesèque, Lebacque, Monneau dans le cas d’une condition de jonction de type «flux limité minimal». Nous présenterons un résultat de convergence pour une condition de jonction générale, ainsi qu’une estimation d’erreur dans le cas d’une condition de jonction de type «flux limité» (pas forcément minimal). Une seconde partie porte sur la régularité höldérienne pour une large classe d’équations paraboliques à coefficients peu réguliers par la méthode introduite par De Giorgi en 1957. Le quatrième chapitre contient un résultat quantitatif d’une des deux grandes étapes de la méthode de De Giorgi : le lemme des valeurs intermédiaires. Ce lemme permet de quantifier (en mesure) le fait que les solutions de ces équations ne peuvent pas faire de saut entre deux valeurs numériques. Deux versions quantitatives de ce lemme sont présentées.
... 9 Typically, state variables in a domain within given boundaries can be computed based on given boundary conditions using the proper numerical schemes. For proper approach to this problem, one needs to consider the appropriate discretization of the corresponding boundary value problem (BVP) (Evans, 1998;Garavello et al., 2016) Standard numerical schemes based on FDMs (c.f., LeVeque, 1992) has been widely used for computing numerical solutions to these problems, such as the Godunov scheme (Lebacque, 1996), an upwind scheme (Leclercq et al., 2007), Lax-Friedrichs scheme (i.e., first order scheme) (Wong and Wong, 2002;Göttlich et al., 2013), and Lax-Wendroff scheme (i.e., second-order scheme) (Michalopoulos et al., 1993). In particular, to compute PW-like models, specific FDMs for the models have been extensively used (Papageorgiou et al., 1989). ...
... Yuan et al. (2012) avoided this shortcoming by using the fact that the Godunov scheme can be differential in Lagrangian coordinates. A notable example of accurate and differential schemes is the Lax-Friedrichs scheme used by, for example, Wong and Wong (2002) and Göttlich et al. (2013). ...
Article
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Traffic state estimation (TSE) refers to the process of the inference of traffic state variables (i.e., flow, density, speed and other equivalent variables) on road segments using partially observed traffic data. It is a key component of traffic control and operations, because traffic variables are measured not everywhere due to technological and financial limitations, and their measurement is noisy. Therefore, numerous studies have proposed TSE methods relying on various approaches, traffic flow models, and input data. In this review article, we conduct a survey of highway TSE methods, a topic which has gained great attention in the recent decades.
... To overcome the limitation of PW and ARZ models, partial differential equations (PDE) are utilized to discretize their model formulations by transferring the road segment and time interval into elements. In summary, the discrete reformulation can be categorized by: (a) the Godunov scheme (Lebacque, 1996); (b) the upwind scheme (Lebacque et al., 2007); (c) the Lax-Wendroff scheme (Michalopoulos et al., 1993); and (d) the Lax-Friedrichs scheme (Wong and Wong, 2002;Göttlich et al., 2013). The cell transmission model (CTM) is a simplified case of the Godunov scheme of the LWR discretized with the Courant-Friedrichs-Lewy (CFL) number equal to 1 (Daganzo, 1994). ...
... To solve them numerically within reasonable computational time, partial differential equations (PDE) are then used to discretize their model formulations, by the road segment and time period, into elements. In summary, such discrete reformulations can be classified into: the Godunov scheme (Lebacque, 1996), the upwind scheme (Lebacque et al., 2007), the LaxFriedrichs scheme (Wong and Wong, 2002;Göttlich et al., 2013), and the LaxWendroff scheme (Michalopoulos et al., 1993). Cell transmission model (CTM) is one special case of Godunov scheme of the LWR where the CourantFriedrichsLewy (CFL) number equals to 1 (Daganzo, 1994). ...
Preprint
Despite the success of classical traffic flow (e.g., second-order macroscopic) models and data-driven (e.g., Machine Learning - ML) approaches in traffic state estimation, those approaches either require great efforts for parameter calibrations or lack theoretical interpretation. To fill this research gap, this study presents a new modeling framework, named physics regularized Gaussian process (PRGP). This novel approach can encode physics models, i.e., classical traffic flow models, into the Gaussian process architecture and so as to regularize the ML training process. Particularly, this study aims to discuss how to develop a PRGP model when the original physics model is with discrete formulations. Then based on the posterior regularization inference framework, an efficient stochastic optimization algorithm is developed to maximize the evidence lowerbound of the system likelihood. To prove the effectiveness of the proposed model, this paper conducts empirical studies on a real-world dataset that is collected from a stretch of I-15 freeway, Utah. Results show the new PRGP model can outperform the previous compatible methods, such as calibrated physics models and pure machine learning methods, in estimation precision and input robustness.
... For the classical Hamilton-Jacobi equations, there are many numerical methods, such as ENO [21] and WENO [22], to compute the discontinuities such as shock waves. Numerical methods are also used in [23], [24] to compute the Hamilton-Jacobi system. The goal of this paper is to propose different types of junctions and provide an effective numerical method for computing junctions. ...
Article
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In this paper, we propose junction conditions for discontinuities due to local perturbation, diverging, merging, and multi-in-multi-out junctions. Traffic flows on junctions can be described by a system of coupled Hamilton-Jacobi equations. At their connection points, it is necessary to propose appropriate junction conditions to close the system. Then, we provide an effective numerical method to compute approximate solutions to these Hamilton-Jacobi equations on junctions. The numerical boundary conditions to close the Hamilton-Jacobi system are also proposed. Numerical tests demonstrate the effectiveness of both the proposed junction conditions and the numerical method.
... A convergent semi-Lagrangian scheme is introduced in [9] for equations of eikonal type and in [13] for Hamilton-Jacobi equations with application to traffic flow models. In [23], an adapted Lax-Friedrichs scheme is used to solve a traffic model; it is worth mentioning that this discretization implies to pass from the scalar conservation law to the associated Hamilton-Jacobi equation at each time step. ...
Article
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This paper is concerned with monotone (time-explicit) finite difference scheme associated with first order Hamilton–Jacobi equations posed on a junction. It extends the scheme introduced by Costeseque et al. (Numer Math 129(3):405–447, 2015) to general junction conditions. On the one hand, we prove the convergence of the numerical solution towards the viscosity solution of the Hamilton–Jacobi equation as the mesh size tends to zero for general junction conditions. On the other hand, we derive some optimal error estimates of in \(L^{\infty }_{\text {loc}}\) for junction conditions of optimal-control type.
... For one dimensional flow models, the LWR model is the most seminal work in macroscopic traffic flow modeling and it is widely used since its appearance in 1950 and many numerical schems have been used to compute numerical solutions to it, such as finite difference methods (FMD) and others based on FMDs, such us the Godunov scheme [24], the upwind scheme [25] and Lax-Friedrichs scheme [19,40]. For more details on how to solve LWR model numericaly, we refer the reader to [39]. ...
... Imbert et al [6] introduced Hamilton-Jacobi equations for modeling junction problems to traffic flow and its application. Several works [1,4,5] had separately introduced an Hamilton-Jacobi formulation for networks of which they need to deal with tedious coupling conditions at each junctions. Given that impulsive differential equation is appearing new in research and is a very important concept to deal with discontinuities, also the context of traffic flow is of macroscopic nature based on partial differential equations for the traffic flow density, and had not been looked into to the best of our knowledge. ...
Article
The main motivation of this work is to assess the validity of a LWR traffic flow model to model measurements obtained from trajectory data, and propose extensions of this model to improve it. A formulation for a discrete dynamical system is proposed aiming at reproducing the evolution in time of the density of vehicles along a road, as observed in the measurements. This system is formulated as a chemical reaction network where road cells are interpreted as compartments, the transfer of vehicles from one cell to the other is seen as a chemical reaction between adjacent compartment and the density of vehicles is seen as a concentration of reactant. Several degrees of flexibility on the parameters of this system, which basically consist of the reaction rates between the compartments, can be considered: a constant value or a function depending on time and/or space. Density measurements coming from trajectory data are then interpreted as observations of the states of this system at consecutive times. Optimal reaction rates for the system are then obtained by minimizing the discrepancy between the output of the system and the state measurements. This approach was tested both on simulated and real data, proved successful in recreating the complexity of traffic flows despite the assumptions on the flux–density relation.
Article
This article describes a new approach to the macroscopic first order modeling and simulation of traffic flow in complex urban road intersections. The framework is theoretically sound, operational, and comprises a large body of models presented so far in the literature. Working within the generic node model class of Tampere et al. (forthcoming), the approach is developed in two steps. First, building on the incremental transfer principle of Daganzo et al. (1997), an incremental node model for general road intersections is developed. A limitation of this model (as of the original incremental transfer principle) is that it does not capture situations where the increase of one flow decreases another flow, e.g., due to conflicts. In a second step, the new model is therefore supplemented with the capability to describe such situations. A fixed-point formulation of the enhanced model is given, solution existence and uniqueness are investigated, and two solution algorithms are developed. The feasibility and realism of the new approach is demonstrated through both a synthetic and a real case study.
Article
If one numbers vehicles consecutively along a roadway and draws the space-time trajectory of each vehicle on the same x − t graph, then this family of curves can be interpreted as the contours of a three-dimensional surface for which the third dimension is vehicle number n. The intersection of the surface with planes of constant x are the cumulative arrival curves, n vs. t, at the location x. If the surface is smoothed, the orientation of the tangent plane at any point determines the flow q, density k, and car velocity v. All commonly observed properties of traffic flow have simple geometric interpretations in this three-dimensional model.
Article
In this article, we propose a computational method for solving the Lighthill–Whitham–Richards (LWR) partial differential equation (PDE) semi-analytically for arbitrary piecewise-constant initial and boundary conditions, and for arbitrary concave fundamental diagrams. With these assumptions, we show that the solution to the LWR PDE at any location and time can be computed exactly and semi-analytically for a very low computational cost using the cumulative number of vehicles formulation of the problem. We implement the proposed computational method on a representative traffic flow scenario to illustrate the exactness of the analytical solution. We also show that the proposed scheme can handle more complex scenarios including traffic lights or moving bottlenecks. The computational cost of the method is very favorable, and is compared with existing algorithms. A toolbox implementation available for public download is briefly described, and posted at http://traffic.berkeley.edu/project/downloads/lwrsolver.