From discrete microscopic models to macroscopic models and applications to traffic flow

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Submitted on 17 Jul 2014

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From discrete microscopic models to macroscopic models and applications to traffic flow

Nicolas Forcadel, Wilfredo Salazar

To cite this version:

Nicolas Forcadel, Wilfredo Salazar. From discrete microscopic models to macroscopic models and

applications to traffic flow. NETCO 2014, 2014, Tours, France. �hal-01025232�

and applications to traffic flow.

Nicolas Forcadel

Laboratoire de Math´ ematique de L’INSA de Rouen Joint work with W. Salazar

Conference on ”New Trends in Optimal Control”

23-27 June 2014, Tours

1 Motivations

2 Homogenization of traffic flows models

Plan

1 Motivations

2 Homogenization of traffic flows models

Bando model

Second order discrete model with n 0 ∈ N types of vehicles :

U ¨ j (t) = a j

V j

U j+1 (t) − U j (t) − l j+1 + l j

2

− U ˙ j (t)

. (1)

U _j : position of the vehicle j.

a _j : sensibility of the driver j (a _j + n

= a _j ).

V j : Optimal velocity function (OVF) of the driver j (V j+n

= V j ).

Passing from micro to macro

Goal : Describe the traffic in term of density of vehicles, i.e. passing from the microscopic model to a macroscopic one.

LWR macroscopic model:

ρ t + (ρv(ρ)) x = 0

where v is the average speed of vehicles.

Existing results

for the Frenkel-Kontorova model

Existing results for the Frenkel-Kontorova model

i − 2 i − 1 i i + 1 i + 2

m d ² U _j

dt ² + γ dU _j

dt = (U j +1 − U j ) + (U j− 1 − U j ) + sin(2πU j ) + f

Existing results for the Frenkel-Kontorova model

i − 2 i − 1 i i + 1 i + 2

m d ² U j

dt ² + γ dU j

dt = F _j (U _j−m , . . . , U _j + m )

Homogenization

u(t, x) = U _⌊x⌋ (t)

4 u(t, x)

0 x

U1 U2 U3 U4

1 2 3

Rescalling

u ^ε (t, x) = εu t

ε , x ε

ε u^ε(t, x)

0 x

u ^ε →?

Rescalling

u ^ε (t, x) = εu t

ε , x ε

u⁰(t, x)

0 x

u ^ε → u ⁰

Homogenization results

For ε > 0, we define

u ^ε (t, x) = εu _ε ^t , ^x _ε , u ^ε (0, x) = u 0 (x) Theorem (F., Imbert, Monneau)

Under certain assumptions on u 0 , we have u ^ε → u ⁰ with u ⁰ _t = ¯ F (u ⁰ _x ),

u ⁰ (0, x) = u 0 (x)

The density of particles ρ = _u ¹

satisfies formally a conservation law

ρ _t = ( ¯ H(ρ)) _x with H(ρ) = ¯ −ρ F ¯ (1/ρ)

Ingredients of the proof

Main idea: inject the system of particles into a (system of) PDE.

If m is small, the system of PDE satisfies a comparison principle Order of the particles (link between the F j ).

Notion of hull functions

Plan

1 Motivations

2 Homogenization of traffic flows models

Change of variables

U ¨ j (t) = a j

V j (U j +1 (t) − U j (t)) − U ˙ j (t) .

We set

Ξ _j (t) = U _j (t) + 1

α U ˙ _j (t) with α = 1 2 min

j∈{ 1 ,...,n

} (a _j ).

Then

( U ˙ j (t) = α (Ξ j (t) − U j (t))

˙Ξ j (t) = (a _j − α)(U _j (t) − Ξ _j (t)) + a j

α V _j (U _j +1 (t) − U _j (t)) .

Change of variables

U ¨ j (t) = a j

V j (U j +1 (t) − U j (t)) − U ˙ j (t) .

We set

Ξ _j (t) = U _j (t) + 1

α U ˙ _j (t) with α = 1 2 min

j∈{ 1 ,...,n

} (a _j ).

Then

( U ˙ j (t) = α (Ξ j (t) − U j (t))

˙Ξ j (t) = (a _j − α)(U _j (t) − Ξ _j (t)) + a j

α V _j (U _j +1 (t) − U _j (t)) .

This system is monotone if a j ≥ 4kV ^′ k _∞

Injection in a system of PDE

For j ∈ Z , pour tout (t, x) ∈ (0, +∞) × R ,

u j (t, x) = U _j + n

⌊x⌋ (t) and ξ j (t, x) = Ξ _j + n

⌊x⌋ (t).

Then 

 

 

 

 

 

 

 

 

 

 

∂u j

∂t = α(ξ j (t, x) − u j (t, x))

∂ξ j

∂t = (a j − α)(u j − ξ j ) + a j

α V j (u j+1 − u j ) u j + n

(t, x) = u j (t, x + 1)

ξ j+n

(t, x) = ξ j (t, x + 1).

(2)

Ordering the vehicles

We assume that V _j (h) = 0 for h ≤ 2h 0 and we consider the worst case

Using the solution of this ode as barrier, we get U j+1 ≥ U j + h 0 for a good choice of h 0

This implies that Ξ _j +1 ≥ Ξ _j

Notion of hull function

Hull function h



 

 

 

 

∂u _j

∂t = α(ξ _j (t, x) − u _j (t, x))

∂ξ _j

∂t = (a _j − α)(u _j − ξ _j ) + a _j

α V _j (u _j +1 − u _j ) We search particular solutions of the form

u _j (t, x) = h _j (λt + px), ξ _j (t, x) = g _j (λt + px) (h _j , g _j ) = hull functions

λ = mean velocity

n

p = mean density

U _i + n

l − U i

l → p as l → +∞

Equation of the hull functions

( λ = α(g _j (z) − h _j (z))

λ = (a _j − α)(h _j (z) − g _j (z)) + a _j

α V _j (h _j +1 (z) − h _j (z))

Existence of hull functions

Theorem (F., Salazar)

For every p ∈ (0, +∞), there exists a unique λ := F (p) for which there exists hull functions (h j , g j ). Moreover, (h j , g j ) can be constructed such that

h _j (z) = z + h _j (0), g _j (z) = z + g _j (0).

Properties of λ

Theorem (F., Salazar)

Monotonicity : λ is non-decreasing

Upper boundary : lim _p→ + ∞ λ(p) = min _j∈{ 1 ,...,n

} ||V _j || _∞ zero velocity : If p ≤ 2h 0 n 0 , then λ = 0

Figure : Schematic representation of the effective Hamiltonian.

Construction of the hull functions

Existence of hull functions

Theorem (F., Salazar)

For every p > 0, there exists some (¯ u ^∞ _j , ξ ¯ _j ^∞ ) such that there exists a unique λ =: ¯ F (q) such that

u ^∞ _j (t, x)

j , ξ _j ^∞ (t, x)

j

=

px + λt + ¯ u ^∞ _j

j , px + λt + ¯ ξ _j ^∞

j

,

is a solution of (2). Moreover, if we define (h _j , g _j ) such that u ^∞ _j (t, x) = h _j (λt + qx), and ξ _j ^∞ (t, x) = g _j (λt + qx) then (h j , g j ) is a hull function and satisfies

h j (z) = z + h j (0) and g j (z) = z + g j (0).

Ideas to construct h

1

(Initial data) u(0, x) = ξ(0, x) = px

Ideas to construct h

1

(Initial data) u(0, x) = ξ(0, x) = px

2

(Particular form of the solutions)

u j (t, x) = px + u j (t, 0) and ξ j (t, x) = px + ξ j (t, 0)

Ideas to construct h

1

(Initial data) u(0, x) = ξ(0, x) = px

2

(Particular form of the solutions)

u j (t, x) = px + u j (t, 0) and ξ j (t, x) = px + ξ j (t, 0)

3

(Long time behavior) u j (t, x)

t , ξ j (t, x)

t → λ as t → +∞

Ideas to construct h

1

(Initial data) u(0, x) = ξ(0, x) = px

2

(Particular form of the solutions)

u j (t, x) = px + u j (t, 0) and ξ j (t, x) = px + ξ j (t, 0)

3

(Long time behavior) u j (t, x)

t , ξ j (t, x)

t → λ as t → +∞

4

(Global control of the solution)

|u j (t, 0) − λt| ≤ C and |ξ j (t, 0) − λt| ≤ C

Ideas to construct h

1

(Initial data) u(0, x) = ξ(0, x) = px

2

(Particular form of the solutions)

u j (t, x) = px + u j (t, 0) and ξ j (t, x) = px + ξ j (t, 0)

3

(Long time behavior) u j (t, x)

t , ξ j (t, x)

t → λ as t → +∞

4

(Global control of the solution)

|u j (t, 0) − λt| ≤ C and |ξ j (t, 0) − λt| ≤ C

5

(Translation at infinity) By considering

(u j (t + n, y) − λn, ξ j (t + n, y) − λn) as n −→ +∞, one can construct

global in time solution (u ^∞ _j , ξ _j ^∞ )

Homogenization results

Homogenization

u _j (t, x) = U _j + n

⌊x⌋ (t)

0 4 x

U1 u1(t, x)

U3n0+1

U2n0+1 Un0+1

1 2 3

Rescalling

u ^ε _j (t, x) = εu j

t ε , x

ε

ε u^ε_j(t, x)

0 x

u ^ε _j →?

Rescalling

u ^ε _j (t, x) = εu j

t ε , x

ε

u⁰(t, x)

0 x

u ^ε _j → u ⁰

Homogenization results

For ε > 0, we define

( u ^ε _j (t, x) = εu _{j ε} ^t , ^x _ε

, ξ _j ^ε (t, x) = εξ _{j ε} ^t , ^x _ε u ^ε _j (0, x) = u 0

x + _n ^jε

0

, ξ _j ^ε (0, x) = u 0

x + _n ^jε

0

Theorem (F., Imbert, Monneau)

Under certain assumptions on u 0 , we have u ^ε _j → u ⁰ and ξ _j ^ε → u ⁰ with u ⁰ _t = ¯ F (u ⁰ _x ),

u ⁰ (0, x) = u 0 (x)

Ansatz for the proof : u ^ε _j (t, x) ≃ εh _j

u ⁰ (t, x) ε

, ξ _j ^ε (t, x) ≃ εg _j

u ⁰ (t, x) ε

Homogenization results

For ε > 0, we define

( u ^ε _j (t, x) = εu j t ε , ^x _ε

, ξ _j ^ε (t, x) = εξ j t ε , ^x _ε u ^ε _j (0, x) = u 0

x + _n ^jε

0

, ξ _j ^ε (0, x) = u 0

x + _n ^jε

0

Theorem (F., Imbert, Monneau)

Under certain assumptions on u 0 , we have u ^ε _j → u ⁰ and ξ _j ^ε → u ⁰ with u ⁰ _t = ¯ F (u ⁰ _x ),

u ⁰ (0, x) = u 0 (x)

Ansatz for the proof :

u ^ε _j (t, x) ≃ u ⁰ (t, x) + εh _j (0), ξ _j ^ε (t, x) ≃ u ⁰ (t, x) + εg _j (0)

Formal idea of the proof

v _j ^ε (t, x) := u ⁰ (t, x) + εh j (0), w _j ^ε (t, x) := u ⁰ (t, x) + εg j (0)

Formal idea of the proof

v _j ^ε (t, x) := u ⁰ (t, x) + εh j (0), w _j ^ε (t, x) := u ⁰ (t, x) + εg j (0) We then have very formally

(v _j ^ε ) t (t, x) =u ⁰ _t = ¯ F (u ⁰ _x ) = α(g j (0) − h j (0)) = α

ε (v _j ^ε − w _j ^ε ) (w _j ^ε ) t (t, x) =u ⁰ _t = ¯ F (u ⁰ _x )

=(a _j − α)(h _j (0) − g _j (0)) + a j

α V _j (h _j +1 (0) − h _j (0))

=(a j − α) v ^ε _j − w ^ε _j ε + a j

α V j

v ^ε _{j +1} − v ^ε _j ε

Formal idea of the proof

v _j ^ε (t, x) := u ⁰ (t, x) + εh j (0), w _j ^ε (t, x) := u ⁰ (t, x) + εg j (0) We then have very formally

(v _j ^ε ) t (t, x) =u ⁰ _t = ¯ F (u ⁰ _x ) = α(g j (0) − h j (0)) = α

ε (v _j ^ε − w _j ^ε ) (w _j ^ε ) t (t, x) =u ⁰ _t = ¯ F (u ⁰ _x )

=(a _j − α)(h _j (0) − g _j (0)) + a j

α V _j (h _j +1 (0) − h _j (0))

=(a j − α) v ^ε _j − w ^ε _j ε + a j

α V j

v ^ε _{j +1} − v ^ε _j ε

(v _j ^ε , w ^ε _j ) and (u ^ε _j , ξ _j ^ε ) satisfy (almost) the same equation, so u ^ε _j ≃ v _j ^ε and

ξ _j ^ε ≃ w _j ^ε .

An example of computation of F (W. Salazar)

Conclusions and Perspectives

Conclusions :

Homogenization results for discrete traffic flow models

This allows to add microscopic phenomena in the modeling (red light, car crashes,....)

Perspectives :

Study of car crashes, red light

Homogenization on networks

Numerical computation of F

Homogenization in random media