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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 t∈R+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 f−1

l:f7→ f−1

l(f)∈[0, ρ∗] and f−1

r:f7→ f−1

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 ·ρ∗−ρ) 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 Li∈R+. 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

i∈δin

v

f(ˆρi(t)) = X

j∈δout

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

i∈δout

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 f−1

il (ˆγi) = ρi(Li)

f−1

ir (ˆγi) else ,(7)

and

¯ρj=ρj(0) if f−1

jr (¯γj) = ρj(0)

f−1

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)>ρ∗

i:

ρi

f(ρi)

ρ∗

iρmax

i

ρi(L)

ρj(0)≤ρ∗

j:

ρj

f(ρj)

ρ∗

jρmax

j

ρj(0)

ρj(0)>ρ∗

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=γ1,0≤γ1≤γmax

1,and 0 ≤γ2≤γmax

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=α1,3γ1,0≤γk≤γmax

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+γ2,0≤γk≤γmax

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= ¯γ3−ˆγ1,(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+α2,4γ2,0≤ˆγk≤γmax

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

4−α1,4ˆγ1},(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 ρ≥0, 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

∀e∈E

∂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:

∆t≤∆x

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((j−1

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

j−∆t

2"fMt

j+1 −Mt

j

∆x+fMt

j−Mt

j−1

∆x#

+∆t

2∆xat

j(Mt

j+1 −2Mt

j+Mt

j−1),(23)

and due to the CFL condition

an

j≥max

x∈[(j−1)∆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 (j−1

2)∆x=i∆x. We have to add an additional index efor all e∈Eto 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

i−1)−∆t

2∆x(f(ρt

i+1)−f(ρt

i−1)).(26)

Proof. Scheme (23) and equation (24) allow for the following calculation:

ρt+1

i−ρt

i

∆t

(24)

=Mt+1

j+1 −Mt

j+1

∆t·∆x−Mt+1

j−Mt

j

∆t·∆x

(23)

=−1

2∆x"fMt

j+2 −Mt

j+1

∆x−fMt

j−Mt

j−1

∆x#

+a

2∆xMt

j+2 −Mt

j+1

∆x−2·Mt

j+1 −Mt

j

∆x+Mt

j−Mt

j−1

∆x

(24)

=−1

2∆xhf(ρt

i+1)−f(ρt

i−1)i+a

2∆xhρt

i+1 −2ρt

i+ρt

i−1i

(25)&(22)

⇐⇒ ρt+1

i=1

2ρt

i+1 +ρt

i−1−∆t

2∆x·hf(ρt

i+1)−f(ρt

i−1)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∈[0,ρmax]|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

i−1)−∆t

2∆x(f(ρt

i+1)−f(ρt

i−1))

(27)

=1

2(ρt

i+1 +ρt

i−1)−1

2λ(f(ρt

i+1)−f(ρt

i−1)).(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

i−1, l

i, r

i+1, r

i+2,...,r).

The scheme preserves the density values inside the constant regions, because (28)

yields

if ρt

j−1=ρt

j+1 ⇒ρt+1

j=ρt

j−1

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

i−1, 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

i−1, 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

i−1, m

i, m

i+1, r

i+2,...,r),

with m=l+r−ρ∗∈[l, r].

ii) Given the densities

ρ¯

t= (l, . . . , l

i−1, 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+r−ρ∗.Due to the assumption made for Case 3, we

have

ˆm=m+r−ρ∗≥m≥l

Furthermore, we have

ˆm=m

|{z}

a) ≤ρ∗

−ρ∗+r≤r.

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

i−2,˚m

i−1,˚m

i, r

i+1,...,r),

with ˚m=l+m−ρ∗.We have

˚m=l+m

|{z}

b)≥ρ∗

−ρ∗≥l

and

˚m=l

|{z}

(Case 3)≤ρ∗

+m−ρ∗≤m≤r

⇒˚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

i−1, 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∗

i−1

, 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,nx∀e∈δin

vand ρt+1

e,0∀e∈δout

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 ˆγe∀e∈δin

vand

¯γe∀e∈δout

vusing equations (12), (14), (16) or (18), respectively. The density

boundary values ˆρe∀e∈δin

vand ¯ρe∀e∈δout

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 e∈δin

v:

ρt+1

e,nx

(9)γmax

e

(12), (14), (16) or (18)ˆγt+1

e

(7)ˆρt+1

e

∀outgoing roads e∈δout

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,∀e∈E;

3timegridsize: ∆t=∆x

maxe∈E{maxρ|f′

e(ρ)|} ;

4number of timesteps: nt=⌈T

∆t⌉+ 1;

/* Transfer initial values from ρto M. */

5forall the e∈Edo

6ˆ

M0

e= 0; /* right boundary value */

7M0

e,rng=ˆ

M0

e−∆xˆρ0

e;/* rightmost ghost-cell */

8M0

e,j =Me,j+1 −∆xρ0

e,i ∀gridpoints j(including ghost-cells);

9¯

M0

e=M0

e,0−∆x¯ρ0

e;/* left boundary value */

10 for t= 0,...,nt−1do

/* Compute next time iteration for each road e*/

11 forall the e∈Edo

12 Compute Mt+1

e,j by (23)∀gridpoints j(including ghost-cells)

/* see Figure 6(a) */

/* Transfer Mto ρ*/

13 forall the e∈Edo

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 v∈Vdo

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 e∈Edo

18 left: ¯

Mt+1

e=˜

Mlng−¯ρt+1

e∆x;

19 right: ˆ

Mt+1

e=˜

Mrng+ ˆρt+1

e∆x/* 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.

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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

3≤γmax

1: In this case the optimal solution of (15) is given by

ˆγ1=γmax

3,ˆγ2= 0,¯γ3=γmax

3.

ii) γmax

1< γmax

3≤γmax

1+γmax

2: In this case the optimal solution of (15) is

given by

ˆγ1=γmax

1,ˆγ2=γmax

3−γmax

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,

0≤ˆγk≤γmax

k,∀k= 1,...4,

(29)

We can simplify the problem and obtain the equivalent formulation

max ωγ1+γ2

such that

γ1≤b

α14γ1+γ2≤γmax

4

γ2≤γmax

2

γ1, γ2≥0

(30)

where bis a known parameter given by

b:= min{γmax

3

α1,3

, γmax

1}, α1,36= 0.(31)

We introduce slack variables si≥0, i = 1,...,3 and rewrite (30). This yields

z−ωγ1−γ2= 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+s1−1

α14 s2=b−γmax

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) b≤γmax

4

α14 .In this case the ﬁrst SIMPLEX transformation leads to

z−γ2+ωs1=ωb

γ1+s1=b

γ2−α14s1+s2=γmax

4−α14b

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

4−α14b≤γmax

2.The second SIMPLEX transformation yields

z+(ω−α14)s1+s2= (ω−α14 )b+γmax

4

γ1+s1=b

γ2−α14s1+s2=γmax

4−α14b

α14s1−s2+s3=γmax

2−γmax

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

4−α14bis optimal and fulﬁlls

(18).

Case iib) γmax

2≤γmax

4−α14b.In this case the next transformation of the system

(34) looks like

z+ωs1+s3=ωb +γmax

2

γ1+s1=b

−α14s1+s2−s3=γmax

4−γmax

2−α14b

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