Road junction modeling using a scheme based on Hamilton-Jacobi equations

Road junction modelling using a scheme based on
Hamilton-Jacobi equation
Guillaume Costeseque
(PhD with supervisors R. Monneau & J-P. Lebacque)
Ecole des Ponts ParisTech, CERMICS & IFSTTAR, GRETTIA
Workshop on Traffic Modeling and Management
March 20-22, 2013 - Sophia-Antipolis
G. Costeseque (Université ParisEst) Numerical scheme for traffic junction Sophia, March 2013 1 / 42

Introduction
A junction
JN
J1
J2
branch Jα
x
x
0
x
x

Introduction
Ideas of the model
JN
J1
J2
branch Jα
x
x
0
x
x



uα
t + Hα(uα
x ) = 0, x > 0, α = 1, . . . , N
uα = uβ =: u, x = 0,
ut + H(u1
x, . . . , uN
x ) = 0, x = 0

Introduction
Outline
1 Motivation
2 Junction model and adapted scheme
3 Traﬃc interpretation
4 Numerical simulation
5 Conclusion

Motivation
Outline
1 Motivation
5 Conclusion

Motivation
The simple divergent road
x > 0
x > 0γl
γrx < 0
Il
Ir
γe
Ie



γe = 1,
0 ≤ γl, γr ≤ 1,
γl + γr = 1
LWR model [Lighthill, Whitham ’55; Richards ’56]:
ρt + (Q(ρ))x = 0

Motivation
Qmax
ρcrit ρmax
Density ρ
Flow Q(ρ)
Q(ρ) = ρV (ρ) with V (ρ) = speed of the equilibrium ﬂow

Junction model and adapted scheme
Outline
1 Motivation
Getting of the HJ equations
Junction model
Numerical scheme
Mathematical results
5 Conclusion

Junction model and adapted scheme Getting of the HJ equations
ρα
t + (Qα
(ρα
))x = 0 on branch α
Primitive:
Uα
(x, t) = Uα
(0, t) +
1
γα
x
0
ρα
(y, t)dy
and
Uα
(0, t) = g(t) = index of the single car at the junction point
x > 0
x > 09
11
8
10
12
6420 1 3 5
7
−1
x < 0

Junction model and adapted scheme Getting of the HJ equations
ρα
t + (Qα
(ρα
))x = 0 on branch α
Primitive:
Uα
(x, t) = Uα
(0, t) +
1
γα
x
0
ρα
(y, t)dy
and
Uα
(0, t) = g(t) = index of the single car at the junction point
x > 0
x > 09
11
8
10
12
6420 1 3 5
7
−1
x < 0
Uα
t +
1
γα
Qα
(γα
Uα
x ) = g′
(t) +
1
γα
Qα
(ρα
(0, t))
= 0 for a good choice of g

Junction model and adapted scheme Junction model
Junction model
New functions uα:
uα(x, t) = −Uα(x, t), x > 0, for outgoing roads
uα(x, t) = −Uα(−x, t), x > 0
J3
e3
J2
e2
e1 J1
JN
eN
Proposition (Junction model)



uα
t + Hα(uα
x ) = 0, x > 0,
uα(0, t) = u(0, t), x = 0, α = 1, ..., N
ut + max
α=1,...,N
H−
α (uα
x ) = 0, x = 0.
(3.1)
with the initial condition uα(0, x) = uα
0 (x).

Junction model and adapted scheme Junction model
Assumptions
For all α = 1, . . . , N,
(A0) The initial condition uα
0 is Lipschitz continuous.
(A1) The Hamiltonians Hα are C1(R) and convex such that:
p
H−
α (p) H+
α (p)
pα
0

Junction model and adapted scheme Numerical scheme
Presentation of the NS
Proposition (Numerical Scheme)
Let us consider the discrete space and time derivatives:
pα,n
i :=
Uα,n
i+1 − Uα,n
i
∆x
and (DtU)α,n
i :=
Uα,n+1
i − Uα,n
i
∆t
Then we have the following numerical scheme:



(DtU)α,n
i + max{H+
α (pα,n
i−1), H−
α (pα,n
i )} = 0, i ≥ 1
Un
0 := Uα,n
0 , i = 0, α = 1, ..., N
(DtU)n
0 + max
α=1,...,N
H−
α (pα,n
0 ) = 0, i = 0
(3.2)
With the initial condition Uα,0
i := uα
0 (i∆x).
∆x and ∆t = space and time steps satisfying a CFL condition

Junction model and adapted scheme Numerical scheme
CFL condition
The natural CFL condition is given by:
∆x
∆t
≥ sup
α=1,...,N
i≥0, 0≤n≤nT
|H′
α(pα,n
i )| (3.3)

Junction model and adapted scheme Mathematical results
Gradient estimates
Theorem (Time and Space Gradient estimates)
Assume (A0)-(A1). If the CFL condition (3.3) is satisﬁed, then we have
that:
(i) Considering Mn = sup
α,i
(DtU)α,n
i and mn = inf
α,i
(DtU)α,n
i , we have
the following time derivative estimate:
m0
≤ mn
≤ mn+1
≤ Mn+1
≤ Mn
≤ M0
(ii) Considering pα
= (H−
α )−1(−m0) and pα = (H+
α )−1(−m0), we have
the following gradient estimate:
pα
≤ pα,n
i ≤ pα, for all i ≥ 0, n ≥ 0 and α = 1, ..., N
Proof

Stronger CFL condition
As for any α = 1, . . . , N, we have that:
pα
≤ pα,n
i ≤ pα for all i, n ≥ 0
−m0
pα
p
Hα(p)
pα
Then the CFL condition becomes:
∆x
∆t
≥ sup
α=1,...,N
pα∈[pα
,pα]
|H′
α(pα)| (3.4)

Existence and uniqueness
(A2) Technical assumption (Legendre-Fenchel transform)
Hα(p) = sup
q∈R
(pq − Lα(q)) with L′′
α ≥ δ > 0, for all index α

Existence and uniqueness
(A2) Technical assumption (Legendre-Fenchel transform)
Hα(p) = sup
q∈R
(pq − Lα(q)) with L′′
α ≥ δ > 0, for all index α
Theorem (Existence and uniqueness [IMZ, ’11])
Under (A0)-(A1)-(A2), there exists a unique viscosity solution u of (3.1)
on the junction, satisfying for some constant CT > 0
|u(t, y) − u0(y)| ≤ CT for all (t, y) ∈ JT .
Moreover the function u is Lipschitz continuous with respect to (t, y).

Convergence
Theorem (Convergence from discrete to continuous [CML, ’13])
Assume that (A0)-(A1)-(A2) and the CFL condition (3.4) are satisﬁed.
Then the numerical solution converges uniformly to u the unique viscosity
solution of (3.1) when ε → 0, locally uniformly on any compact set K:
lim sup
ε→0
sup
(n∆t,i∆x)∈K
|uα
(n∆t, i∆x) − Uα,n
i | = 0
Proof

Traﬃc interpretation
Outline
1 Motivation
Traﬃc notations
Links with “classical” approach
Literature review
5 Conclusion

Traﬃc interpretation Traﬃc notations
Setting
J1
JNI
JNI+1
JNI+NO
x < 0 x = 0 x > 0
Jβ
γβ Jλ
γλ
NI incoming and NO outgoing roads

Car densities
The car density ρα solves the LWR equation on branch α:
ρα
t + (Qα
(ρα
))x = 0
By deﬁnition
ρα
= γα
∂xUα
on branch α
And
uα(x, t) = −Uα(−x, t), x > 0, for incoming roads
uα(x, t) = −Uα(x, t), x > 0, for outgoing roads
where the car index uα solves the HJ equation on branch α:
uα
t + Hα
(uα
x ) = 0, for x > 0

Flow
Hα(p) :=



−
1
γα
Qα(γαp) for α = 1, ..., NI
−
1
γα
Qα(−γαp) for α = NI + 1, ..., NI + NO
Incoming roads Outgoing roads
ρcrit
γα
ρmax
γα
p
−
Qmax
γα
p
−
Qmax
γα
HαHα
H−
α H−
α H+
αH+
α
−
ρmax
γα
−
ρcrit
γα

Traﬃc interpretation Links with “classical” approach
Discrete car densities
Deﬁnition (Discrete car density)
The discrete car density ρα,n
i with n ≥ 0 and i ∈ Z is given by:
ρα,n
i :=



γαpα,n
|i|−1 for α = 1, ..., NI , i ≤ −1
−γαpα,n
i for α = NI + 1, ..., NI + NO, i ≥ 0
(4.5)
J1
JNI
JNI+1
JNI+NO
x < 0 x > 0
−2
−1
2
1
0
−2
−2
−1
−1
1
1
2
2
Jβ
Jλ
ρλ,n
1

Traﬃc interpretation
Proposition (Scheme for vehicles densities)
The scheme deduced from (3.2) for the discrete densities is given by:
∆x
∆t
{ρα,n+1
i − ρα,n
i } =



Fα(ρα,n
i−1, ρα,n
i ) − Fα(ρα,n
i , ρα,n
i+1) for i = 0, −1
Fα
0 (ρ·,n
0 ) − Fα(ρα,n
i , ρα,n
i+1) for i = 0
Fα(ρα,n
i−1, ρα,n
i ) − Fα
0 (ρ·,n
0 ) for i = −1
With



Fα(ρα,n
i−1, ρα,n
i ) := min Qα
D(ρα,n
i−1), Qα
S(ρα,n
i )
Fα
0 (ρ·,n
0 ) := γα min min
β≤NI
1
γβ
Qβ
D(ρβ,n
0 ), min
λ>NI
1
γλ
Qλ
S(ρλ,n
0 )
incoming outgoing
ρλ,n
0ρβ,n
−1ρβ,n
−2 ρλ,n
1
x
x = 0

Supply and demand functions
Remark
It recovers the seminal Godunov scheme with passing ﬂow = minimum
between upstream demand QD and downstream supply QS.
Density ρ
ρcrit ρmax
Supply QS
Qmax
Density ρ
ρcrit ρmax
Flow Q
Qmax
Density ρ
ρcrit
Demand QD
Qmax
From [Lebacque ’93, ’96]

Supply and demand VS Hamiltonian
H−
α (p) =



−
1
γα
Qα
D(γαp) for α = 1, ..., NI
−
1
γα
Qα
S(−γαp) for α = NI + 1, ..., NI + NO
And
H+
α (p) =



−
1
γα
Qα
S(γαp) for α = 1, ..., NI
−
1
γα
Qα
D(−γαp) for α = NI + 1, ..., NI + NO

Traﬃc interpretation Literature review
Some references for conservation laws
ρt + (Q(x, ρ))x = 0 with Q(x, p) = 1{x<0}Qin
(p) + 1{x≥0}Qout
(p)
Uniqueness results only for restricted conﬁgurations:
See [Garavello, Natalini, Piccoli, Terracina ’07]
and [Andreianov, Karlsen, Risebro ’11]
Book of [Garavello, Piccoli ’06] for conservation laws on networks:
Construction of a solution using the “wave front tracking method”
No proof of the uniqueness of the solution on a general network

Numerics on networks
Godunov scheme mainly used for conservation laws:
[Bretti, Natalini, Piccoli ’06, ’07]: Godunov scheme compared to
kinetic schemes / fast algorithms
[Blandin, Bretti, Cutolo, Piccoli ’09]: Godunov scheme adapted for
Colombo model (only tested for 1 × 1 junctions)

Numerics on networks
Godunov scheme mainly used for conservation laws:
[Bretti, Natalini, Piccoli ’06, ’07]: Godunov scheme compared to
kinetic schemes / fast algorithms
[Blandin, Bretti, Cutolo, Piccoli ’09]: Godunov scheme adapted for
Colombo model (only tested for 1 × 1 junctions)
[Han, Piccoli, Friesz, Yao ’12]: Lax-Hopf formula for HJ equation coupled
with a Riemann solver at junction

Junction modelling
State-of-the-art review:
[Lebacque, Khoshyaran ’02, ’05, ’09]
[Tampère, Corthout, Cattrysse, Immers ’11],
[Flötteröd, Rohde ’11]
Calibration of γα for realistic models:
[Cassidy and Ahn ’05]
[Bar-Gera and Ahn ’10],
[Ni and Leonard ’05] (small data set)

Numerical simulation
Outline
1 Motivation
5 Conclusion

Example of a Diverge
An oﬀ-ramp:
J1
ρ1
J2
ρ2
ρ3
J3
Branch Number of lanes Maximal speed γα
1 2 90 km/h 1
2 2 90 km/h 0.75
3 1 50 km/h 0.25

Diverge: Fundamental Diagrams
0 50 100 150 200 250 300 350
0
500
1000
1500
2000
2500
3000
3500
4000
(15, 1772)(15, 1772)
(20, 2250)
(11, 1329)
(5, 344)
(7, 443)
Density (veh/km)
Flow(veh/h)
Fundamental diagrams per branch
1
2
3

Initial conditions (t=0s)
−200 −150 −100 −50 0
0
10
20
30
40
50
60
70
Road n° 1 (t= 0s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 2 (t= 0s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 3 (t= 0s)
Position (m)
Density(veh/km)

Trajectories
1
2
3
4
5
6
7
78
8
9
9
10
10
11
11
12
12
13
13
Trajectories on road n° 1
Position (m)
Time(s)
−200 −150 −100 −50 0
0
5
10
15
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
12
Position (m)
Time(s)
0 50 100 150 200
0
5
10
15
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
10
11
12
Position (m)
Time(s)
0 50 100 150 200
0
5
10
15

Cumulative Vehicles Count
0 10 20 30 40
0
5
10
15
20
25
Time (s)
CumulativeNumberofVehicles
CVC on road n°1
Up. st.
Down. st.
0 10 20 30 40
−5
0
5
10
15
20
Time (s)
CVC on road n°2
Up. st.
Down. st.
0 10 20 30 40
−2
0
2
4
6
Time (s)
CVC on road n°3
Up. st.
Down. st.

Gradient estimates
0 10 20 30
0
50
100
150
200
250
Time (s)
Density(veh/km)
Density time evolution on road n° 1
0 10 20 30
0
50
100
150
200
250
300
Time (s)
Density(veh/km)
0 10 20 30
0
50
100
150
Time (s)
Density(veh/km)

Conclusion
Outline
1 Motivation
5 Conclusion

Conclusion
Complementary results [CML ’13]:
Generalization for weaker assumptions on the Hamiltonians
Numerical simulation for other junction conﬁgurations (merge)

Conclusion
Complementary results [CML ’13]:
Generalization for weaker assumptions on the Hamiltonians
Numerical simulation for other junction configurations (merge)
Open questions:
Error estimate
Non-fixed coefficients γα
Other link models (GSOM)
Other junction condition

Conclusion
The End
Thanks for your attention
guillaume.costeseque@cermics.enpc.fr
guillaume.costeseque@ifsttar.fr

Complements
Some references
G. Costeseque, R. Monneau, J-P. Lebacque, A convergent numerical
scheme for Hamilton-Jacobi equations on a junction: application to
traffic, Working paper, (2013).
C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to
junction problems and application to traffic flows, ESAIM: COCV,
(2011), 38 pages.
J.P. Lebacque and M.M. Koshyaran, First-order macroscopic traffic
flow models: intersection modeling, network modeling, 16th ISTTT
(2005), pp. 365-386.
C. Tampère, R. Corthout, D. Cattrysse and L. Immers, A generic class
of first order node models for dynamic macroscopic simulations of
traffic flows, Transp. Res. Part B, 45 (1) (2011), pp. 289-309.

Proofs of the main results
Sketch of the proof (gradient estimates):
Time derivative estimate:
1. Estimate on mα,n = inf
i
(DtU)α,n
i and partial result for mn = inf
α
mα,n
2. Similar estimate for Mn
3. Conclusion
Space derivative estimate:
1. New bounded Hamiltonian ˜Hα(p) for p ≤ pα
and p ≥ pα
2. Time derivative estimate from above
3. Lemma: if for any (i, n, α), (DtU)α,n
i ≥ m0 then
pα
≤ pα,n
i ≤ pα
4. Conclusion as ˜Hα = Hα on [pα
, pα]
Back

Convergence with uniqueness assumption
Sketch of the proof: (Comparison principle very helpful)
1. uα(t, x) := lim sup
ε
Uα,n
i is a subsolution of (3.1) (contradiction on
Deﬁnition inequality with a test function ϕ)
2. Similarly, uα is a supersolution of (3.1)
3. Conclusion: uα = uα viscosity solution of (3.1)
Back

Convergence without uniqueness assumption
Sketch of the proof: (No comparison principle)
1. Discrete Lipschitz bounds on uα
ε (n∆t, i∆x) := Uα,n
i
2. Extension by continuity of uα
ε
3. Ascoli theorem (convergent subsequence on every compact set)
4. The limit of one convergent subsequence (uα
ε )ε is super and
sub-solution of (3.1)

Road junction modeling using a scheme based on Hamilton-Jacobi equations

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