Numerical approach for Hamilton-Jacobi equations on a network: application to traffic

Numerical approach for Hamilton-Jacobi equations
on a network: application to traﬃc
Guillaume Costeseque
(PhD with supervisors R. Monneau & J-P. Lebacque)
Universit´e Paris Est, Ecole des Ponts ParisTech & IFSTTAR
NETCO Conference - Tours,
June 24, 2014
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks Tours, June 24, 2014 1 / 51

Flows on a network

Flows on a network
A network is like a (oriented) graph
made of edges and vertices
Examples:
traﬃc ﬂow,
gas pipelines,
blood vessels,
shallow water,
internet communications...

Outline
1 Introduction
2 Numerical scheme
3 Traﬃc interpretation
4 Numerical simulation
5 Recent developments

Introduction
Motivation
Classical approaches (see A. Bressan’s lectures):
Macroscopic modeling on (homogeneous) sections
Coupling conditions at (pointwise) junction
For instance, consider
⎧
⎪⎨
⎪⎩
ρt + (Q(ρ))x = 0, scalar conservation law,
ρ(., t = 0) = ρ0(.), initial conditions,
ψ(ρ(x = 0−, t), ρ(x = 0+, t)) = 0, coupling condition.
(1)
See Garavello, Piccoli [3], Lebacque, Khoshyaran [6] and Bressan et al. [1]

Introduction
Qmax
ρcrit ρmax
Density ρ
Flow Q(ρ)
Q(ρ) = ρV (ρ) with V (ρ) = velocity function
DFs

Introduction HJ junction model
Star-shaped junction
JN
J1
J2
branch Jα
x
x
0
x
x

Junction model
Proposition (Junction model [IMZ, ’13])
That leads to the following junction model (see [5])
⎧
⎪⎨
⎪⎩
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
(2)
with initial condition uα(0, x) = uα
0 (x) and
H(u1
x , . . . , uN
x ) = max
α=1,...,N
H−
α (uα
x )
from optimal control
.

Basic 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

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)
For Hamilton-Jacobi equations on networks:
[G¨ottlich, Ziegler, Herty ’13]: Lax-Freidrichs scheme outside the
junction + coupling conditions (density) at the junction
[Han, Piccoli, Friesz, Yao ’12]: Lax-Hopf formula for HJ equation
coupled with a Riemann solver at junction
[Camilli, Festa, Schieborn ’13]: semi-Lagrangian scheme only
designed for Eikonal equations

Numerical scheme
Outline
1 Introduction
2 Numerical scheme

Numerical scheme Numerical scheme
Presentation of the scheme
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)
With the initial condition Uα,0
i := uα
0 (i∆x).
∆x and ∆t = space and time steps satisfying a CFL condition

Numerical 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 )| (4)

Numerical scheme Mathematical results
Gradient estimates
Theorem (Time and Space Gradient estimates)
Assume (A0)-(A1). If the CFL condition (4) 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
−m0
pα
p
Hα(p)
pα
As for any α = 1, . . . , N, we have that:
pα
≤ pα,n
i ≤ pα for all i, n ≥ 0
Then the CFL condition becomes:
∆x
∆t
≥ sup
α=1,...,N
pα∈[pα
,pα]
|H′
α(pα)| (5)

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, ’13])
Under (A0)-(A1)-(A2), there exists a unique viscosity solution u of (2) 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 (5) are satisﬁed.
Then the numerical solution converges uniformly to u the unique viscosity
solution of (2) 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 Introduction
2 Numerical scheme

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
∂x Uα
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
Setting

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

Links with “classical” approach
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
(6)
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

Proposition (Scheme for vehicles densities)
The scheme deduced from (3) 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

Numerical simulation
Outline
1 Introduction
2 Numerical scheme

Example of a Diverge
An oﬀ-ramp:
J1
ρ1
J2
ρ2
ρ3
J3
with ⎧
⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
γe = 1,
γl = 0.75,
γr = 0.25

Fundamental Diagrams
0 50 100 150 200 250 300 350
0
500
1000
1500
2000
2500
3000
3500
4000
(ρ
c
,f
max
)
(ρ
c
,f
max
)
Density (veh/km)
Flow(veh/h)

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)

Numerical solution: densities

Numerical solution: Hamilton-Jacobi

Trajectories
1
2
3
4
56
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
56
6
7
7
8
8
9
10
11
12
Position (m)
Time(s)
0 50 100 150 200
0
5
10
15

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)

Recent developments
Outline
1 Introduction
2 Numerical scheme

Recent developments
New junction model
Proposition (Junction model [IM, ’14])
From [4], we have
⎧
⎪⎨
⎪⎩
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
(7)
with initial condition uα(0, x) = uα
0 (x) and
H(u1
x , . . . , uN
x ) = max
ﬂux limiter
L , max
α=1,...,N
H−
α (uα
x )
minimum between
demand and supply
.

Recent developments
Weaker assumptions on the Hamiltonians
For all α = 1, . . . , N,
(A0) The initial condition uα
0 is Lipschitz continuous.
(A1) The Hamiltonians Hα are continuous and quasi-convex i.e.
there exists points pα
0 such that
⎧
⎪⎨
⎪⎩
Hα is non-increasing on (−∞, pα
0 ],
Hα is non-decreasing on [pα
0 , +∞).

Recent developments
Homogenization on a network
Proposition (Homogenization on a periodic network [IM’14])
Assume (A0)-(A1). Consider a periodic network.
If uε satisﬁes (oscillating) HJ equation on network,
then uε converges uniformly towards u0 when ε → 0,
with u0 solution of
u0
t + H ∇x u0
= 0, t > 0, x ∈ Rd
(8)
See Prof. R. Monneau’s lecture and [4]

Recent developments
Numerical homogenization on a network
Numerical scheme adapted to the cell problem
Traffic
Traffic eH
γH
eV
γH
γV
γV
i = 0
i =
N
2
i = −
N
2

Recent developments
First example
Proposition (Effective Hamiltonian for fixed coefficients [IM’14])
If (γH , γV ) are fixed, then the
(Hamiltonian) effective Hamiltonian H is given by
H(uH,x , uV ,x ) = max L, max
i={H,V }
H(ui,x ) ,
(traffic flow) effective flow Q is given by
Q(ρH , ρV ) = min −L,
Q(ρH)
γH
,
Q(ρV )
γV
.

Recent developments
First example
Numerics: assume Q(ρ) = 4ρ(1 − ρ) and L = −1.5,

Recent developments
Second example
Two consecutive traﬃc signals on a 1D road
ﬂow
l LL
x1 x2xE
E
xS
S
Homogenization theory by [G. Galise, C. Imbert, R. Monneau, ’14]

Recent developments
Second example
Eﬀective ﬂux limiter −L (numerics only)
0 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Offset (s)
Fluxlimiter
l=0 m
l=5 m
l=10 m
l=20 m

Recent developments
Thanks for your attention
guillaume.costeseque@cermics.enpc.fr
guillaume.costeseque@ifsttar.fr

Complements References
Some references I
A. Bressan, S. Canic, M. Garavello, M. Herty, and B. Piccoli, Flows
on networks: recent results and perspectives, EMS Surveys in Mathematical
Sciences, (2014).
G. Costeseque, J.-P. Lebacque, and R. Monneau, A convergent scheme for
hamilton-jacobi equations on a junction: application to traffic, arXiv preprint
arXiv:1306.0329, (2013).
M. Garavello and B. Piccoli, Traffic flow on networks, American institute of
mathematical sciences Springfield, MO, USA, 2006.
C. Imbert and R. Monneau, Level-set convex hamilton-jacobi equations on
networks, (2014).
C. Imbert, R. Monneau, and H. Zidani, A hamilton-jacobi approach to
junction problems and application to traffic flows, ESAIM: Control, Optimisation
and Calculus of Variations, 19 (2013), pp. 129–166.

Some references II
J.-P. Lebacque and M. M. Khoshyaran, First-order macroscopic traffic flow
models: Intersection modeling, network modeling, in Transportation and Traffic
Theory. Flow, Dynamics and Human Interaction. 16th International Symposium on
Transportation and Traffic Theory, 2005.

Fundamental diagram
Fundamental diagram: multi-valued in congested case
[S. Fan, M. Herty, B. Seibold, 2013], NGSIM dataset
Back

Complements Hamilton-Jacobi model
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] on each branch α:
ρα
t + (Qα
(ρα
))x = 0

Getting the Hamilton-Jacobi equation
LWR model on each branch (outside the junction point)
ρα
t + (Qα
(ρα
))x = 0 on branch α
Primitive:
⎧
⎨
⎩
Uα(x, t) = Uα(0, t) +
1
γα
x
0
ρα
(y, t)dy,
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

Getting the Hamilton-Jacobi equation
LWR model on each branch (outside the junction point)
ρα
t + (Qα
(ρα
))x = 0 on branch α
Primitive:
⎧
⎨
⎩
Uα(x, t) = Uα(0, t) +
1
γα
x
0
ρα
(y, t)dy,
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
Back

Complements 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]
See [2]
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 (2) (contradiction on
Deﬁnition inequality with a test function ϕ)
2. Similarly, uα is a supersolution of (2)
3. Conclusion: uα = uα viscosity solution of (2)
See [2]

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 (2)
See [2]
Back

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