From heterogeneous microscopic traffic flow models tomacroscopic models

Pierre CardaliaguetUniversité Paris Dauphine

Joint work with N. Forcadel (INSA Rouen)

May 18, 2020

The problem of modelling traffic flow

§ By which laws do vehicles interact with each other?§ Temporal evolution of traffic density?

What we address here

§ Traffic on a single line§ No overtaking

Two classical models of traffic flow

We study traffic flow models on a single straight road (without overtake).

Two kinds of models:

1) Microscopic models: e.g., the follow-the-leader model is a system ofODEs

ddt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z.

2) Macroscopic models: e.g., the Lighthill-Whitham-Richards (LWR)model is the scalar conservation law

Btρ` pρvpρqqx “ 0 in Rˆ p0,`8q,

(M. J. Lighthill and G. B. Whitham (1955), P. I. Richards (1956))

The follow-the-leader model

i=0 i=1 i=2i=-1i=-2

U-2(t)U

-2(t) U

-1(t) U

0(t) U

1(t) U

2(t)

ddt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z,

where§ Uiptq denotes the position of car i P Z at time t ě 0,§ Cars are ordered: Uiptq ď Ui`1ptq for all t, i ,§ The velocity V “ V ppq ě 0 of car i depends in an increasing way onthe distance p of car i to car i ` 1.

Vz(h)

hz0

0

V zmax

h

Figure: Typical shape of the optimal velocity function V .

Properties of the follow-the-leader model

ddt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z,

§ Compute trajectories of each vehicle

§ Can be extended to multiple lanes

§ At the core of most micro-simulators

§ Good for simulation

(cf. Seibold, B. (2015). A mathematical introduction to traffic flow theory.IPAM Tutorials.)

The LWR Model

The Lighthill-Whitham-Richards (LWR) model is the scalar conservation law

Btρ` pρvpρqqx “ 0 in Rˆ p0,`8q,

where§ ρ is the density of vehicles on the road,§ v : R` Ñ R`. The map f pρq “ ρvpρq is the so-called “fundamentaldiagram"

Properties of the LWR model:§ Describe aggregate quantities via PDE§ Natural framework for traveling waves and shocks

The fundamental diagram f pρq “ ρvpρq

(after Seo, T., Kawasaki, Y., Kusakabe, T., & Asakura, Y. (2019))

Goal of the talk

§ Discuss how to derive the LWR model

Btρ` pρvpρqqx “ 0 in Rˆ p0,`8q,

from the follow-the-leader modelddt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z.

§ Well-known when all the vehicles are identical, i.e., V does not dependon i . Then f pρq “ ρvpρq “ ρV p1ρq (Aw, Klar, Materne, and Rascle (2002))

§ The fact that the vehicles are identical is a very restrictive (andunatural) assumption.

§ Main contribution: we address the case where the vehicles are different:ddt Uiptq “ VipUi`1ptq ´ Uiptqq, t ě 0,@i P Z,

where the distribution of the Vi is “well distributed".

From an homogeneous traffic flow...ddt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z, f pρq “ ρV p1ρq

i=0 i=1 i=2i=-1i=-2

U-2(t)U

-2(t) U

-1(t) U

0(t) U

1(t) U

2(t)

... to an heterogenous one:ddt Uiptq “ VipUi`1ptq ´ Uiptqq, t ě 0,@i P Z,

i=0 i=1 i=2i=-1i=-2

U-2(t)U

-2(t) U

-1(t) U

0(t) U

1(t) U

2(t)

Outline

Heuristic arguments in the homogeneous case

Main results

Ideas of proof

Outline

Heuristic arguments in the homogeneous case

Main results

Ideas of proof

Outline

Heuristic arguments in the homogeneous case

Main results

Ideas of proof

Outline

Heuristic arguments in the homogeneous case

Main results

Ideas of proof

Heuristic derivation of the macro model from the micro oneThe microscopic model: d

dt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z,The macroscopic model (LWR): Btρ` pρvpρqqx “ 0 in Rˆ p0,`8q

§ Consider the distribution of vehicles Rptq “ÿ

iPZδUi ptq.

§ After an hyperbolic scaling px , tq Ñ pε´1x , ε´1tq, we obtainρεptq “ ε

ÿ

iPZδεUi pε´1tq.

§ Then, for any test function φ P C8c pRq,

ddt

ż

Rφpxqρεpdx , tq “ d

dt εÿ

iPZφpεUipε

´1tqq

“ εÿ

iPZφx pεUipε

´1tqq ddt Uipε

´1tq

“ εÿ

iPZφx pεUipε

´1tqq V`

Upi`1qpε´1tq ´ Uipε

´1tq˘

§ Next we show that Upi`1qpε´1tq ´ Uipε

´1tq » 1`

ρεpUipε´1tq

˘

.

§ Proof that Upi`1qpε´1tq ´ Uipε

´1tq » 1`

ρεpUipε´1tq

˘

:

Indeed, if x “ εUipε´1tqq and dx “ εpUpi`1qpε

´1tq ´ Uipε´1tqq, then,

ρεprx , x ` dxq, tq “ ε cardtj P Z, εUjpε´1tq P rx , x ` dxqq “ ε,

so that

ρεpx , tq » ρεprx , x ` dxq, tq pdxq´1 “ ε pdxq´1

“ pUpi`1qpε´1tq ´ Uipε

´1tqq´1.

§ As ρεptq “ εř

iPZ δεUi pε´1tq, we have

ddt

ż

Rφpxqρεpdx , tq » ε

ÿ

iPZφx pεUipε

´1tqqVˆ

1ρεpεUipε´1tq, tq

˙

“

ż

Rφx pxqV

ˆ

1ρεpx , tq

˙

ρεpdx , tq.

§ So ρε solves pLWRq Btρε ` pρεvpρεqqx “ 0 in the sense of

distribution with vpsq “ V p1sq.

Some references

Rigorous derivation of the macroscopic model from the microscopic one:§ For one type of vehicles:

§ Argall, Cheleshkin, Greenberg, Hinde, and Lin (2002)§ Aw, Klar, Materne, and Rascle (2002)§ Di Francesco and Rosini (2015)§ Goatin and Rossi (2017)§ Holden and Risebro (2018)

§ For several types of cars:§ Chiabaut, Leclercq, and Buisson (2010)

(random model, heuristic derivation)§ Forcadel and Salazar (2015)

(periodic setting)

Another heuristic derivation (through Hamilton-Jacobi)

The microscopic model: ddt Uiptq “ VipUi`1ptq ´ Uiptqq, t ě 0,@i P Z,

The macroscopic model (LWR): Btρ` pρvpρqqx “ 0 in Rˆ p0,`8q

§ Let up¨, tq be the piecewise affine map such that upi , tq “ Uiptq for alli P Z.

§ We consider the hyperbolic scaling: uεpx , tq “ εupε´1x , ε´1tq.§ If ρεptq “ ε

ÿ

iPZδεUi pε´1tq, one can show that ρεptq “ Bx puεq´1p¨, tq.

§ If x “ εi ,

Btuεpx , tq “ddt

`

εUipε´1tq

˘

“ddt Uipε

´1tqq “ VipUi`1pε´1tq ´ Uipε

´1tqq

“ Vrxεs`

ε´1puεpx ` ε, tq ´ uεpx , tqq˘

» VrxεspBxuεpx , tqq.

§ So uε “solves” the HJ equation: Btuεpx , tq “ VrxεspBxuεpx , tqqfrom which one expect to derive (LWR).

Some references (cont’d)

The proof based on Hamilton-Jacobi is related to the analysis of theFrenkel-Kontorova models:

§ Aubry (1983), Aubry and Le Daeron (1983),§ Forcadel, Imbert, and Monneau (2009)

Our work is within the framework of (stochastic) homogenization of HJequations:

§ Lions, Papanicolaou, and Varadhan (1987): periodic setting§ Souganidis (1999), Rezakhanlou and Tarver (2000): convergence§ Armstrong, C., Souganidis: convergence rate§ Subsequent works by Armstrong, Ciomaga, Davini, Feldman, Kosygina,Lin, Lions, C., Nolen, Novikov, Schwab, Seeger, Smart, Souganidis,Tran, Varadhan, Yilmaz, Zeitouni...

Outline

Heuristic arguments in the homogeneous case

Main results

Ideas of proof

A random microscopic model

We consider a random version of the follow-the-leader model:

ddt Uiptq “ VZi pUi`1ptq ´ Uiptqq, t ě 0,@i P Z,

where§ Uiptq denotes the position of car i at time t,§ Cars are ordered: Uiptq ď Ui`1ptq for all t, i ,§ The velocity V “ VZi ppq of car i depends on the distance p of car i tocar i ` 1 and on the “type” Zi of car i

§ The types are pZiq are I.I.D. random variables.

(cf. N. Chiabaut, L. Leclercq, and C. Buisson (2010))

Assumptions

On the optimal velocity map V : Z ˆ R` Ñ R`, we assume the following:pH1q The map pz , pq Ñ Vzppq is uniformly continuous on Z ˆ R` and

p Ñ Vzppq is Lipschitz continuous, uniformly with respect to z P Z;pH2q For any z P Z, there exists hz

0 ą 0 (depending in a measurable way onz) such that Vzppq “ 0 for all p P r0, hz

0s;pH3q For any z P Z, p Ñ Vzppq is increasing in rhz

0,`8q;pH4q There exists Vmax ą 0 and, for any z P Z, there exists V z

max ď Vmax,such that limpÑ`8 Vzppq “ V z

max.

pH5q If we set Vmax :“ infzPZ V zmax, then lim

θÑVmax´E”

V´1Z0pθq

ı

“ `8.

Vz(h)

hz0

0

V zmax

h

Figure: Schematic representation of the optimal velocity functions.

Main result (1)

Scaling: For ε ą 0, we consider an initial condition pUε,0i q such that there

exists a Lipschitz continuous function u0 : RÑ R with

limεÑ0, εiÑx

εUε,0i “ u0pxq,

locally uniformly with respect to x . Let pUεi q be the solution of

ddt Uε

i ptq “ VZi pUεi`1ptq ´ Uε

i ptqq, t ě 0,@i P Z.

with initial condition pUε,0i q.

We want to study the limit

upx , tq :“ limεÑ0, εpi ,sqÑpx ,tq

εUεi psq

Main result (2)

Theorem (C.-Forcadel)Under assumptions pH1q ´ pH5q, the limit

upx , tq :“ limεÑ0, εpi ,sqÑpx ,tq

εUεi psq

exists a.s., locally uniformly in px , tq, and u is the unique (deterministic)viscosity solution of

"

Btu “ F pBxuq in Rˆs0,`8rupx , 0q “ u0pxq in R

where the effective velocity F : r0,`8q Ñ r0,Vmaxq is the continuous andincreasing map defined by

§ F ppq “ 0 if p ď h0 where h0 :“ ErhZ00 s,

§ and F ppq is the unique solution to ErV´1Z0pF ppqqs “ p if p ą h0.

Link with the Lighthill-Whitham-Richards (LWR) modelWe consider the (rescaled) empirical density of cars:

ρεptq “ εÿ

iPZδεUε

i ptεq, t ě 0.

Corollary [Convergence to the LWR model]As εÑ 0, ρεptq converges, a.s., in distribution and locally uniformly in time,to the density of cars

ρptq :“ up¨, tq7dx ,

where u is the solution of the limit HJ equation. If, in addition, there existsC ą 0 such that

C´1 ď Bxu0pxq ď C ,

then ρ has an absolutely continuous density which is locally bounded and isthe entropy solution of the LWR model

pLWRq Btρ` Bx pρvpρqq “ 0 in Rˆ R`,

with vpρq “ F p1ρq.

Sketch of proof of the corollary

§ Let ϕ P C0c pRq. Then, for any t 1 ě 0,

ż

Rϕpxqρεpdx , t 1q “ ε

ÿ

iPZϕpεUε

i pt 1εqq “ż

RϕpεUε

rxεspt1εqqdx .

§ As εprxεs, t 1εq Ñ px , tq as εÑ 0 and t 1 Ñ t, the main Theoremimplies:

limεÑ0, t1Ñt

ż

Rϕpxqρεpdx , tq “

ż

Rϕpupx , tqqdx

“

ż

Rϕpxqdpup¨, tq7dxq “

ż

Rϕpxqρpdx , tq.

This proves that ρεptq converges locally uniformly in time and in thesense of measures to ρptq :“ up¨, tq7dx .

§ This implies that ρ is an entropy solution to (LWR) (Caselles (1992)).

l

Outline

Heuristic arguments in the homogeneous case

Main results

Ideas of proof

Preliminary results on the micro model

Lemma [Uniform bounds]Let Ui be a solution of

pMicroq ddt Uiptq “ VZi pUi`1ptq ´ Uiptqq, t ě 0,@i P Z.

Then, for all t ě 0,0 ď Uiptq ´ Uip0q ď Vmaxt.

We also have the following comparison principle:Proposition [Comparison]Let Ui and Ui be two solutions of (Micro) such that there exists i0 P Z with

Uip0q ď Uip0q @i ě i0.

ThenUiptq ď Uiptq @t ě 0 and i ě i0.

Construction of the effective velocity

Recall that h0 :“ ErhZ00 s. Given p ą h0, we consider the solution Up to the

problem with linear initial condition:

ddt Up

i ptq “ VZi pUpi`1ptq ´ Up

i ptqq, t ě 0, Upi p0q “ p i @i ě 0.

Proposition [Convergence for linear initial conditions]There exists Ω0 P F with PpΩ0q “ 1 such that for every p ě 0, i P N andω P Ω0

limtÑ`8

Upi ptqt “ F ppq @i ě 0,

where the continuous and non-decreasing map F : R` Ñ R` is defined by§ F ppq “ 0 if p ď h0 where h0 :“ ErhZ0

0 s,§ ErV´1

Z0pF ppqqs “ p if p ą h0.

Main argument of the proof of the proposition: CorrectorsGiven θ P p0,Vmaxq, we consider the random sequence pcθi qiě0 defined by

cθ0 “ 0, cθi`1 “ cθi ` V´1Zipθq i ě 0.

In other words,VZi pcθi`1 ´ cθi q “ θ @i ě 0.

Thus, if we set Uθi ptq “ cθi ` tθ, we have

ddt Uθ

i ptq “ θ “ VZi pcθi`1 ´ cθi q “ VZi pUθi`1ptq ´ Uθ

i ptqq.

So the pUθi q are the correctors of the problem.

By the law of large numbers, there exists Ω0 with PpΩ0q “ 1 such that forevery ω P Ω0, we have

cθii “

1i

i´1ÿ

j“0V´1

Zjpθq Ñ E

”

V´1Z0pθq

ı

as i Ñ `8.

Open problems

‚ Convergence rate

‚ Models with local perturbations(cf. Forcadel-Salazar-Zaydan (2017),deterministic setting)

‚ Models with several roads(cf. Forcadel-Salazar (2019), twooutgoing roads)