The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model - - PowerPoint PPT Presentation

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The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model

Nicolas Laurent-Brouty1,2, Paola Goatin1

1Universit´

e Cˆ

• te d’Azur, Inria, CNRS, LJAD, France

2Ecole des Ponts ParisTech, Champs-sur-Marne, France

May 18th, 2018 nicolas.laurent-brouty@inria.fr

2

Outline

1

Introduction to conservation laws

2

Wave-Front Tracking approximations

3

Convergence of the WFT approximate solutions

4

Convergence of the relaxed ARZ system towards LWR equation

5

Decay estimates of positive waves

3

Outline

1

Introduction to conservation laws

2

Wave-Front Tracking approximations

3

Convergence of the WFT approximate solutions

4

Convergence of the relaxed ARZ system towards LWR equation

5

Decay estimates of positive waves

4

Introduction to conservation laws

Figure: 1D-representation of a stretch of road

t time x space variable ρ(t, x) density of vehicles v(t, x) velocity of the flow f (ρ, v) = ρv Conservation of the number of cars: d dt b

a

ρ(y, t)dy = [flux entering at a] − [flux exiting at b] b

a

∂ ∂t ρ(y, t)dy = − b

a

∂ ∂x [f (ρ, v)](x, t)dx ∂ ∂t ρ + ∂ ∂x f (ρ, v) = 0

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Application to traffic flow

1 The Lighthill, Whitham, Richards (LWR) model

Assume v = v(ρ)

• ∂tρ + ∂x(ρv(ρ)) = 0

ρ(0, x) = ρ0(x) x ∈ R, t > 0,

Figure: Fundamental diagram of traffic flow

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Application to traffic flow

2 The Payne-Whitham model (PW)

Define an anticipation factor Ae(ρ) and a response time from drivers δ

• ∂tρ + ∂x(ρv(ρ)) = 0,

∂tv + v∂xv + 1

ρ∂x(Ae(ρ)) = 0

3 The Aw-Rascle-Zhang (ARZ) model

Assume a pseudo-pressure, strictly increasing, p(ρ) > 0

• ∂tρ + ∂x(ρv) = 0

∂t(v + p(ρ)) + v∂x(v + p(ρ)) = 0

4 The Aw-Rascle-Zhang model with relaxation

• ∂tρ + ∂x(ρv) = 0

∂t(v + p(ρ)) + v∂x(v + p(ρ)) = Veq(ρ)−v

δ

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The ARZ model with relaxation

The ARZ model with relaxation

• ∂tρ + ∂x(ρv) = 0

∂t(v + p(ρ)) + v∂x(v + p(ρ)) = Veq(ρ)−v

δ

The model can be put under conservative form:

• ∂tρ + ∂x(ρv) = 0

∂t(ρ(v + p(ρ))) + ∂x(ρv(v + p(ρ))) = ρ Veq(ρ)−v

δ

Eigenvalues: λ1 = v − ρp′(ρ), λ2 = v To ensure strict hyperbolicity, we assume: ρ > 0, p(ρ) ≥ 0, p′(ρ) > 0.

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Define w := v + p(ρ)

• ∂tρ + ∂x(ρv) = 0

∂t(ρw) + ∂x(ρvw) = ρ Veq(ρ)−v

δ

convert into Lagrangian coordinates (T, X) with τ = 1

ρ.

˜ p(τ) = p 1 τ

• ,

˜ V (τ) = Veq 1 τ

• ,

˜ p′(τ) = − 1 τ 2 p′ 1 τ

• < 0,
• ∂Tτ − ∂Xv = 0

∂Tw =

˜ V (τ)−v δ

with initial data

• τ(0, ·) = τ0

w(0, ·) = w0

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Definition of solutions

• Ut + [F(U)]X = G δ(U)

U(0, x) = U0(x) x ∈ R, t > 0 (1.1) U = τ w

• ,

F(U) = −(w − ˜ p(τ))

• ,

G δ(U) =

• ˜

V (τ)−(w−˜ p(τ)) δ

• .

Definition (definition of solutions) Assume U0 ∈ BV (R) and T > 0. We say that a function Uδ : [0, T] × R → R2 is a weak solution to the Cauchy problem (1.1) if the map t → Uδ(t, ·) ∈ L1

loc(R) is

continuous, Uδ(t = 0, ·) = U0(·) and if for any φ ∈ C 1

c ([0, T] × R)

+∞

−∞

φ(0, X)Uδ

0(X)dX +

T +∞

−∞

[φtUδ(t, X) + φXF(Uδ(t, X))]dXdt + T +∞

−∞

φG δ(Uδ(t, X))dXdt = 0

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Main results

Theorem For each relaxation parameter δ, the ARZ model with relaxation admits a weak entropy solution Uδ = (τ δ, wδ). Theorem The subsequence of weak entropy solutions Uδ = (τ δ, wδ) of the relaxed ARZ model converges to ¯ U = (¯ τ, ¯ w) as δ → 0. Then ¯ w = ˜ V (¯ τ) + ˜ p(¯ τ) and ¯ τ is a weak solution of the scalar Cauchy problem:

• ∂tτ − ∂X ˜

V (τ) = 0, τ(0, ·) = τ0(·), X ∈ R, t > 0. (1.2)

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Outline

1

Introduction to conservation laws

2

Wave-Front Tracking approximations

3

Convergence of the WFT approximate solutions

4

Convergence of the relaxed ARZ system towards LWR equation

5

Decay estimates of positive waves

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What is Wave-Front Tracking?

General idea of wave front tracking for a system of conservation laws:

• Ut + [F(U)]X = 0

U(0, x) = U0(x) x ∈ R, t > 0

1 approximate the initial datum U0 by a piecewise constant function U0

ǫ such

that U0 − U0

ǫ L∞ ≤ ǫ

2 for t = 0+ solve for each discontinuity of U0

ǫ the associated Riemann

• problem. the solution is piecewise constant.

3 the solution can be propagated along the wavefront until two wave fronts

interact.

4 At this time, treat the solution as initial condition and restart the process.

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WFT approximations

How to treat the relaxation term? → two step process

1 solve the Cauchy problem associated to the homogeneous system via WFT

• n a time interval [t0, t0 + ∆t]

2 at t = t0 + ∆t, integrate the source term following wt =

˜ V (τ)−v δ

Definition (BV space) Let Ω an open set. We say that a function u ∈ L1

loc(Ω; R) belongs to BV (Ω; R) if

its total variation TV (u) < ∞, where for every n-tuple {x1, .., xn} ∈ Ω: TV (u) = sup

n−1

• i=1

|u(xi+1) − u(xi)|

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Algorithm

Let T > 0 and a sequence ∆tν > 0 s.t. ∆tν − − − →

ν→∞ 0.

1 Approximate the initial value U0 ∈ BV (R+ × R) by a piecewise constant

function Uν

0 = (τ ν 0 , wν 0 )

2 Solve the homogeneous system via WFT and name Uν(t, ·), t ∈ [0, ∆tν) the

solution

• ∂tτ − ∂Xv = 0,

∂tw = 0,

15 3 At t = ∆tν integrate the source term wt =

˜ V (τ)−v δ

, i.e. define τ ν(∆tν, ·) = τ ν(∆tν−, ·), wν(∆tν, ·) = wν(∆tν−, ·) + ∆tν ˜ V (τ ν(∆tν, ·)) − v(Uν(∆tν−, ·)) δ ∆tν− ∆tν+ τl, w−

l

τr, w−

r

τl, w+

l

τr, w+

r

Figure: Notations used in step 3.

4 Treat Uν(∆tν, ·) as a new piecewise constant initial condition and iterate

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Existence of an invariant domain

Definition (invariant domain) E :=

• u = (τ, w): V min

eq

≤ w − ˜ p(τ) < w ≤ V max

eq

+ max

τ

˜ p(τ)

• Let M > 0.

D(M) := {u : R → E: TV (w(u)) + TV (v(u)) ≤ M} Lemma For ∆t ≤ δ, the set E is an invariant domain for the proposed WFT scheme.

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Decreasing TV and Lipschitz estimates

Lemma For ∆t ≤ δ, the total variation of the Riemann invariants of the constructed approximation Uν is non-increasing in time: TV (wν(t, ·)) + TV (v(Uν(t, ·))) ≤ TV (wν

0 ) + TV (v(Uν 0 )),

for a.e. t > 0. Lemma Let ν ∈ N and Uν

0 ∈ D(M). Then ∀ a < b, ∀ 0 ≤ s < t:

b

a

|τ ν(t, X) − τ ν(s, X)|dX ≤ CM(t − s), b

a

|wν(t, X) − wν(s, X)|dX ≤ (CM + Lδ)(t − s + ∆t).

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Outline

1

Introduction to conservation laws

2

Wave-Front Tracking approximations

3

Convergence of the WFT approximate solutions

4

Convergence of the relaxed ARZ system towards LWR equation

5

Decay estimates of positive waves

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Convergence of the WFT approximations

Theorem Let U0 = (τ0, w0) ∈ D(M) for some M > 0, and denote by Uδ = (τ δ, wδ) the limit of a subsequence Uν = (τ ν, wν) of WFT approximate solutions as ν → ∞. Then Uδ is the weak entropic solution of            ∂Tτ − ∂Xv = 0, ∂Tw =

˜ V (τ)−v δ

, τ(0, ·) = τ0, w(0, ·) = w0. (3.1)

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Sketch of proof

The existence of the limit Uδ and the convergence in L1

loc([0, +∞[ ×R) is

guaranteed by Helly’s theorem. Theorem (Helly’s theorem) Consider a sequence of functions Uν s.t.: TV (Uν(t, ·)) ≤ C, |Uν(t, x)| ≤ M for all t, x, ∞

−∞

|Uν(t, X) − Uν(s, X)|dX ≤ L(t − s) for all t, s ≥ 0, Then there exists a subsequence Uµ which converges to some function U in L1

loc([0, +∞[ ×R). The limit satisfies

∞

−∞

|U(t, X) − U(s, X)|dX ≤ L(t − s) for all t, s ≥ 0,

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Sketch of proof

Ut + [F(U)]X = G δ(U), U = τ w

• ,

F(U) = −(w − ˜ p(τ))

• ,

G δ(U) =

• ˜

V (τ)−(w−˜ p(τ)) δ

• .

Definition (definition of solutions) Assume U0 ∈ BV (R) and T > 0. We say that a function Uδ : [0, T] × R → R2 is a weak solution to the Cauchy problem (1.1) if the map t → Uδ(t, ·) ∈ L1

loc(R) is

continuous, Uδ(t = 0, ·) = U0(·) and if for any φ ∈ C 1

c ([0, T] × R)

+∞

−∞

φ(0, X)Uδ

0(X)dX +

T +∞

−∞

[φtUδ(t, X) + φXF(Uδ(t, X))]dXdt + T +∞

−∞

φG(Uδ(t, X))dXdt = 0

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Let Nν ∈ N such that T = Nν∆tν + βν, βν ∈ [0, ∆tν[. Decompose the integral

• n intervals [k∆tν, (k + 1)∆tν]

(k+1)∆tν

k∆tν

+∞

−∞

[φtτ ν − φX(wν − ˜ p(τ ν))]dXdt = +∞

−∞

φτ ν((k + 1)∆tν−, X)dX − +∞

−∞

φτ ν(k∆tν+, X)dX Then T +∞

−∞

[φtτ ν − φX(wν − ˜ p(τ ν))]dXdt =

Nν−1

• k=0

(k+1)∆tν

k∆tν

· · · + T

Nν∆tν . . .

= +∞

−∞

φ(Nν∆tν, X)τ ν(Nν∆tν−, X)dX − +∞

−∞

φ(0, X)τ ν(0+, X)dX + T

Nν∆tν

+∞

−∞

[φtτ ν − φX(wν − ˜ p(τ ν))]dXdt. Thus +∞

−∞

φ(0, X)τ0(X)dX + T +∞

−∞

[φtτ ν − φX(wν − ˜ p(τ ν))]dXdt → 0

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Now let us consider wν (k+1)∆tν

k∆tν

+∞

−∞

φtwνdXdt = +∞

−∞

[φwν((k + 1)∆t−, X) − φwν(k∆tν+, X)]dX = +∞

−∞

φwν((k + 1)∆tν−, X)dX − +∞

−∞

φ

• wν(k∆tν−, X) + ∆tν ˜

V (τ ν) − vν δ (k∆tν−, X)

• dX.

where we use that wν(k∆tν+, X) = wν(k∆tν−, X) + ∆tν ˜ V (τ ν) − vν δ (k∆tν−, X) Finally use that τ ν → τ δ and vν → vδ in L1

loc and conclude with the Dominated

Convergence Theorem.

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For the entropy solution aspect, fix a smooth convex entropy η with associated entropy-flux q s.t. : ∇ηT(z)DF(z) = ∇Tq(z), ∇ηT(z)G(z) ≤ 0, for z ∈ R2

+.

The same way, we show that T +∞

−∞

• η(Uδ(t, X))φt(t, X) + q(Uδ(t, X))φx(t, X)

+ ∇ηT(Uδ(t, X))G(t, X, Uδ(t, X))φ(t, X)

• dXdt +

+∞

−∞

φ(0, X)η(Uδ

0(X))dX ≥ 0.

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Outline

1

Introduction to conservation laws

2

Wave-Front Tracking approximations

3

Convergence of the WFT approximate solutions

4

Convergence of the relaxed ARZ system towards LWR equation

5

Decay estimates of positive waves

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General idea

Numerically, one can observe that the solutions to the relaxed-ARZ system            ∂tρ + ∂x(ρv) = 0 ∂t(v + p(ρ)) + v∂x(v + p(ρ)) = Veq(ρ)−v

δ

ρ(0, x) = ρ0(x) v(0, x) = v0(x) x ∈ R, t > 0, converges to a solution of LWR when δ → 0

• ∂tρ + ∂x(ρVeq(ρ)) = 0

ρ(0, x) = ρ0(x) x ∈ R, t > 0, ”The system forces the velocity towards the equilibrium speed”

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zero-relaxation limit

We recall that the sequence of WFT approximation satisfies: b

a

|τ ν(t, X) − τ ν(s, X)|dX ≤ CM(t − s), b

a

|wν(t, X) − wν(s, X)|dX ≤ (CM + Lδ)(t − s + ∆t). To pass to the limit as δ → 0, we need a stronger estimate on Lδ. Lemma Lδ ≤ 2 δ e− s

δ

b

a

| ˜ V (τ0(X)) − v0(X)|dX.

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zero-relaxation limit

Theorem Let U0 = (τ0, w0) ∈ D(M) for some M > 0, and denote by ¯ U = (¯ τ, ¯ w) the limit

• f a subsequence of weak entropy solutions Uδ = (τ δ, wδ) of (3.1), as δ → 0.

Then ¯ w = ˜ V (¯ τ) + ˜ p(¯ τ) and ¯ τ is a weak solution of the scalar Cauchy problem:

• ∂tτ − ∂X ˜

V (τ) = 0, τ(0, ·) = τ0(·), X ∈ R, t > 0.

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Sketch of proof

Difficulty: prove convergence of the sequence Uδ on the set [0, ∞[×R in 0+, since Lδ ≤ 2 δ e− s

δ

b

a

| ˜ V (τ0(X)) − v0(X)|dX. Idea: consider sets of the type [1/n, ∞[×[−n, n], where you have: Lδ ≤ C δ (b − a)e− 1

nδ → 0

Then on each set isolate a converging subsequence thanks to Helly’s theorem, and construct a converging sequence in L1

loc([0, +∞[×R) with a diagonalization

process. Then pass to the limit in the weak solution definition, to obtain ¯ w(t, ·) = ˜ V (¯ τ(t, ·)) + ˜ p(¯ τ(t, ·)) and then ¯ τt − ˜ V (¯ τ)x = 0.

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Outline

1

Introduction to conservation laws

2

Wave-Front Tracking approximations

3

Convergence of the WFT approximate solutions

4

Convergence of the relaxed ARZ system towards LWR equation

5

Decay estimates of positive waves

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Oleinik estimates - the entropy condition

Proposition (Kruzkov entropy condition) The unique solution u to the scalar conservation law must satisfy, for any k ∈ R:

• R+
• R

(|u − k|φt + sgn(u − k)(f (u) − f (k))φx)dxdt +

• R

|u0 − k|φ(0, x)dx ≥ 0 Proposition A weak solution u of the scalar conservation law is entropic if and only if there exists a constant C > 0 such that: u(x + z, t) − u(x, t) ≤ C t z ∀t > 0, ∀x ∈ R, ∀z > 0.

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Decay estimates of positive waves

Proposition Assume that ∃c0 > 0 such that ∀u, u′ ∈ E: |λ1(u) − λ1(u′)| = |˜ p′(τ) − ˜ p′(τ ′)| ≤ c0, |λ1(u) − λ2(u′)| = |˜ p′(τ)| ≥ 2c0. (5.1) Then, there exists a constant C > 0 such that, for any interval ]a, b[, for any time horizon T > 0, and every initial condition, the measure µ1+

T (]a, b[) of positive

1-waves contained in the solution of (3.1) satisfies µ1+

T (]a, b[) ≤ C b − a

T eC T

δ (TV (w0)+TV (v0)) + C T

δ (TV (w0) + TV (v0)) . (5.2) Remark Unfortunately, the estimate is not sufficient to recover Oleinik’s estimates and proove that the sequence converges towards the entropy solution of LWR.

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Thank you for your attention

◮

Paola Goatin and Nicolas Laurent-Brouty. “The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model”. working paper or preprint. Apr. 2018. url: https://hal.inria.fr/hal-01760930.