Traffic flow models with non-local flux and extensions to networks - - PowerPoint PPT Presentation

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Traffic flow models with non-local flux and extensions to networks

Simone G¨

• ttlich

joint work with Jan Friedrich, Oliver Kolb (University of Mannheim) and Felisia Chiarello, Paola Goatin (INRIA Sophia Antipolis)

Universit¨ at Mannheim CROWDS 2019 Models and Control, CIRM Marseille, France

June 3-7, 2019

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Motivation of non-local models

• In general non-local systems of N conservation laws have the form:

∂tU + divxF(t, x, U, w ∗ U) = 0 (1) with t ∈ R+, x ∈ Rd, U ∈ RN, F ∈ RN×d

[ACG15] Aggarwal, Colombo, Goatin ”Nonlocal systems of conservation laws in several space dimensions”, 2015

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Motivation of non-local models

• In general non-local systems of N conservation laws have the form:

∂tU + divxF(t, x, U, w ∗ U) = 0 (1) with t ∈ R+, x ∈ Rd, U ∈ RN, F ∈ RN×d

• ”The nonlocal nature of (1) is suitable in describing the behavior of

crowds, where each member moves according to her/his evaluation of the crowd density and its variations within her/his horizon” [ACG15]

[ACG15] Aggarwal, Colombo, Goatin ”Nonlocal systems of conservation laws in several space dimensions”, 2015

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Motivation of non-local models

• In general non-local systems of N conservation laws have the form:

∂tU + divxF(t, x, U, w ∗ U) = 0 (1) with t ∈ R+, x ∈ Rd, U ∈ RN, F ∈ RN×d

• ”The nonlocal nature of (1) is suitable in describing the behavior of

crowds, where each member moves according to her/his evaluation of the crowd density and its variations within her/his horizon” [ACG15]

• Many applications related to moving crowds (since 2010):

− crowd dynamics (R. Colombo, Garavello, L´

ecureux-Mercier [2012]) − supply chains (R. Colombo, Herty, Mercier [2010]) − material flow on conveyor belts (G¨

• ttlich et al. [2014])

− traffic flow → see next slide

[ACG15] Aggarwal, Colombo, Goatin ”Nonlocal systems of conservation laws in several space dimensions”, 2015

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Motivation of non-local traffic models

• In case of non-local traffic scalar conservation laws are of interest:

∂tρ + ∂xf (t, x, ρ, w ∗ ρ) = 0 t ∈ R+, x ∈ R, ρ ∈ R, f ∈ R

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Motivation of non-local traffic models

• In case of non-local traffic scalar conservation laws are of interest:

∂tρ + ∂xf (t, x, ρ, w ∗ ρ) = 0 t ∈ R+, x ∈ R, ρ ∈ R, f ∈ R

• Various research directions (since 2015):

− existence and uniqueness (Amorim, R. Colombo, Teixeira [2015])

− modeling and numerical analysis (Blandin, Goatin [2016]) − higher order numerical schemes (Chalons, Goatin, Villada [2018]; Friedrich, Kolb [2019]) − micro to macro limits (Goatin, Rossi [2017]; Ridder, Shen [2018]) − convergence to the classical LWR model (M. Colombo, Crippa, Graff, Spinolo [2019]; Keimer, Pflug [2019])

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Idea of non-local traffic flow models ρmax ρ x

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Idea of non-local traffic flow models ρmax η > 0 ρ x

• Drivers adapt the velocity to the downstream traffic.

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Idea of non-local traffic flow models

!

!

!

!

ρmax η > 0 ρ x

• Drivers adapt the velocity to the downstream traffic.
• Closer vehicles are more important.

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Idea of non-local traffic flow models

!

!

!

!

ρmax η > 0 ρ x

• Drivers adapt the velocity to the downstream traffic.
• Closer vehicles are more important.
• Common approach: velocity of mean downstream traffic.

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Idea of non-local traffic flow models

!

!

!

!

ρmax η > 0 ρ x

• Drivers adapt the velocity to the downstream traffic.
• Closer vehicles are more important.
• Common approach: velocity of mean downstream traffic.
• New approach: mean downstream velocity.

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Outline

1 Traffic flow models with non-local flux 2 A one-to-one junction model

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Outline

1 Traffic flow models with non-local flux 2 A one-to-one junction model

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Traffic flow model with mean downstream density [CG17]

• ρ(t, x), x ∈ R, t ≥ 0: density of cars with 0 ≤ ρ ≤ ρmax.

∂tρ(t, x) + ∂x ( ρ v( ρ)) = 0, x ∈ R, t > 0, (2)

[CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017

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Traffic flow model with mean downstream density [CG17]

• ρ(t, x), x ∈ R, t ≥ 0: density of cars with 0 ≤ ρ ≤ ρmax.
• Assuming that drivers adapt their speed based on a mean

downstream density (common approach in literature) results in ∂tρ(t, x) + ∂x ( ρ v(wη ∗ ρ)) = 0, x ∈ R, t > 0, wη ∗ ρ(t, x) := x+η

x

ρ(t, y)wη(y − x)dy, η > 0. (2)

[CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017

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Traffic flow model with mean downstream density [CG17]

• ρ(t, x), x ∈ R, t ≥ 0: density of cars with 0 ≤ ρ ≤ ρmax.
• Assuming that drivers adapt their speed based on a mean

downstream density (common approach in literature) results in ∂tρ(t, x) + ∂x (g(ρ)v(wη ∗ ρ)) = 0, x ∈ R, t > 0, wη ∗ ρ(t, x) := x+η

x

ρ(t, y)wη(y − x)dy, η > 0. (2)

[CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017

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Traffic flow model with mean downstream density [CG17]

• ρ(t, x), x ∈ R, t ≥ 0: density of cars with 0 ≤ ρ ≤ ρmax.
• Assuming that drivers adapt their speed based on a mean

downstream density (common approach in literature) results in ∂tρ(t, x) + ∂x (g(ρ)v(wη ∗ ρ)) = 0, x ∈ R, t > 0, wη ∗ ρ(t, x) := x+η

x

ρ(t, y)wη(y − x)dy, η > 0. (2)

• Under the following hypotheses existence and uniqueness of weak

entropy solutions is given (see [CG17, Theorem 1]):

ρ(0, x) = ρ0(x) ∈ BV(R, I), I = [a, b] ⊆ R+,

[CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017

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Traffic flow model with mean downstream density [CG17]

• ρ(t, x), x ∈ R, t ≥ 0: density of cars with 0 ≤ ρ ≤ ρmax.
• Assuming that drivers adapt their speed based on a mean

downstream density (common approach in literature) results in ∂tρ(t, x) + ∂x (g(ρ)v(wη ∗ ρ)) = 0, x ∈ R, t > 0, wη ∗ ρ(t, x) := x+η

x

ρ(t, y)wη(y − x)dy, η > 0. (2)

• Under the following hypotheses existence and uniqueness of weak

entropy solutions is given (see [CG17, Theorem 1]):

ρ(0, x) = ρ0(x) ∈ BV(R, I), I = [a, b] ⊆ R+, g ∈ C 1(I; R+), v ∈ C 2(I; R+) with v ′ ≤ 0, (H1)

[CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017

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Traffic flow model with mean downstream density [CG17]

• ρ(t, x), x ∈ R, t ≥ 0: density of cars with 0 ≤ ρ ≤ ρmax.
• Assuming that drivers adapt their speed based on a mean

downstream density (common approach in literature) results in ∂tρ(t, x) + ∂x (g(ρ)v(wη ∗ ρ)) = 0, x ∈ R, t > 0, wη ∗ ρ(t, x) := x+η

x

ρ(t, y)wη(y − x)dy, η > 0. (2)

• Under the following hypotheses existence and uniqueness of weak

entropy solutions is given (see [CG17, Theorem 1]):

ρ(0, x) = ρ0(x) ∈ BV(R, I), I = [a, b] ⊆ R+, g ∈ C 1(I; R+), v ∈ C 2(I; R+) with v ′ ≤ 0, (H1) wη ∈ C 1([0, η]; R+) with w ′

η ≤ 0,

η wη(x)dx = W0, lim

η→∞ wη(0) = 0.

[CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017

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Traffic flow model with a mean downstream velocity [FKG18]

• We consider a new approach and assume that drivers adapt their

speed based on a mean downstream velocity, i.e. (3)

[FKG18] Friedrich, Kolb and G¨

• ttlich ”A Godunov type scheme for a class
• f LWR traffic flow models with non-local flux”, 2018

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Traffic flow model with a mean downstream velocity [FKG18]

• We consider a new approach and assume that drivers adapt their

speed based on a mean downstream velocity, i.e. ∂tρ(t, x) + ∂x (g(ρ) (wη ∗ v(ρ))) = 0, x ∈ R, t > 0, (3)

[FKG18] Friedrich, Kolb and G¨

• ttlich ”A Godunov type scheme for a class
• f LWR traffic flow models with non-local flux”, 2018

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Traffic flow model with a mean downstream velocity [FKG18]

• We consider a new approach and assume that drivers adapt their

speed based on a mean downstream velocity, i.e. ∂tρ(t, x) + ∂x (g(ρ) (wη ∗ v(ρ))) = 0, x ∈ R, t > 0, wη ∗ v(ρ)(t, x) := x+η

x

v(ρ(t, y))wη(y − x)dy, η > 0. (3)

[FKG18] Friedrich, Kolb and G¨

• ttlich ”A Godunov type scheme for a class
• f LWR traffic flow models with non-local flux”, 2018

7 / 24

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Traffic flow model with a mean downstream velocity [FKG18]

• We consider a new approach and assume that drivers adapt their

speed based on a mean downstream velocity, i.e. ∂tρ(t, x) + ∂x (g(ρ) (wη ∗ v(ρ))) = 0, x ∈ R, t > 0, wη ∗ v(ρ)(t, x) := x+η

x

v(ρ(t, y))wη(y − x)dy, η > 0. (3)

• The necessary conditions for uniqueness and existence are

(H1) with I = [0, ρmax], (H2)

[FKG18] Friedrich, Kolb and G¨

• ttlich ”A Godunov type scheme for a class
• f LWR traffic flow models with non-local flux”, 2018

7 / 24

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Traffic flow model with a mean downstream velocity [FKG18]

• We consider a new approach and assume that drivers adapt their

speed based on a mean downstream velocity, i.e. ∂tρ(t, x) + ∂x (g(ρ) (wη ∗ v(ρ))) = 0, x ∈ R, t > 0, wη ∗ v(ρ)(t, x) := x+η

x

v(ρ(t, y))wη(y − x)dy, η > 0. (3)

• The necessary conditions for uniqueness and existence are

(H1) with I = [0, ρmax], g′ ≥ 0. (H2)

[FKG18] Friedrich, Kolb and G¨

• ttlich ”A Godunov type scheme for a class
• f LWR traffic flow models with non-local flux”, 2018

7 / 24

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Traffic flow model with a mean downstream velocity [FKG18]

• We consider a new approach and assume that drivers adapt their

speed based on a mean downstream velocity, i.e. ∂tρ(t, x) + ∂x (g(ρ) (wη ∗ v(ρ))) = 0, x ∈ R, t > 0, wη ∗ v(ρ)(t, x) := x+η

x

v(ρ(t, y))wη(y − x)dy, η > 0. (3)

• The necessary conditions for uniqueness and existence are

(H1) with I = [0, ρmax], g′ ≥ 0. (H2)

• Note that in the case of a linear velocity function v(ρ), the model

given by (3) coincides with (2).

[FKG18] Friedrich, Kolb and G¨

• ttlich ”A Godunov type scheme for a class
• f LWR traffic flow models with non-local flux”, 2018

7 / 24

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Traffic flow model with a mean downstream velocity [FKG18]

• We consider a new approach and assume that drivers adapt their

speed based on a mean downstream velocity, i.e. ∂tρ(t, x) + ∂x (g(ρ) (wη ∗ v(ρ))) = 0, x ∈ R, t > 0, wη ∗ v(ρ)(t, x) := x+η

x

v(ρ(t, y))wη(y − x)dy, η > 0. (3)

• The necessary conditions for uniqueness and existence are

(H1) with I = [0, ρmax], g′ ≥ 0. (H2)

• Note that in the case of a linear velocity function v(ρ), the model

given by (3) coincides with (2).

• Advantage: With this new approach changes in the velocity function

can be captured!

[FKG18] Friedrich, Kolb and G¨

• ttlich ”A Godunov type scheme for a class
• f LWR traffic flow models with non-local flux”, 2018

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Theorem [FKG18, Theorem 2.2] Let hypotheses (H2) hold. Then, the Cauchy problem

• ∂tρ(t, x) + ∂x (g(ρ(t, x))(v(ρ) ∗ wη)) = 0,

x ∈ R, t > 0, ρ(0, x) = ρ0(x), x ∈ R, admits a unique weak entropy solution and inf

R {ρ0} ≤ ρ(t, x) ≤ sup R

{ρ0} for a.e. x ∈ R, t > 0.

[FKG18] Friedrich, Kolb and G¨

• ttlich ”A Godunov type scheme for a class
• f LWR traffic flow models with non-local flux”, 2018

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Theorem [FKG18, Theorem 2.2] Let hypotheses (H2) hold. Then, the Cauchy problem

• ∂tρ(t, x) + ∂x (g(ρ(t, x))(v(ρ) ∗ wη)) = 0,

x ∈ R, t > 0, ρ(0, x) = ρ0(x), x ∈ R, admits a unique weak entropy solution and inf

R {ρ0} ≤ ρ(t, x) ≤ sup R

{ρ0} for a.e. x ∈ R, t > 0.

• Approximate the solution by a Godunov-type numerical scheme (instead of a

Lax-Friedrichs-type scheme as in other works).

• Derive L∞ and bounded variation estimates to the sequence of approximate

solutions and apply Helly’s Theorem. [FKG18] Friedrich, Kolb and G¨

• ttlich ”A Godunov type scheme for a class
• f LWR traffic flow models with non-local flux”, 2018

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The convolution term

ρj−1 ρj ρj+1 ρj+2 ρj+3

xj−1 xj xj+1 xj+2 xj+3 xj+ 1

2

• Grid: xj cell centers j ∈ Z; spatial step size h with Nh = η and

N ∈ N; time step size τ; λ = τ/h; ρj finite volume approximations.

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The convolution term

ρj−1 ρj ρj+1 ρj+2 ρj+3

xj−1 xj xj+1 xj+2 xj+3 xj+ 1

2

η = 2h

• Grid: xj cell centers j ∈ Z; spatial step size h with Nh = η and

N ∈ N; time step size τ; λ = τ/h; ρj finite volume approximations.

• The convolution term at xj+ 1

2 is calculated by

V n

j+ 1

2 =

N−1

• k=0

xj+k+ 3

2

xj+k+ 1

2

wη(y − xj+ 1

2 )v(ρ(t, y))dy ≈

N−1

• k=0

γkv(ρn

j+k+1)

with γk = (k+1)h

kh

wη(y)dy ∀ k ∈ {0, . . . , N − 1}.

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A Godunov type scheme

• Based on the approximate convolution term V n

j+ 1

2 , we adapt the

numerical flux function of the Godunov scheme for local conservation laws as follows:

F(ρn

j , . . . , ρn j+N) =

     min

ρ∈[ρn

j ,ρn j+1] V n

j+ 1

2 g(ρ),

if ρn

j ≤ ρn j+1

max

ρ∈[ρn

j+1,ρn j ] V n

j+ 1

2 g(ρ),

if ρn

j ≥ ρn j+1

     = V n

j+ 1

2 g(ρn

j ).

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A Godunov type scheme

• Based on the approximate convolution term V n

j+ 1

2 , we adapt the

numerical flux function of the Godunov scheme for local conservation laws as follows:

F(ρn

j , . . . , ρn j+N) =

     min

ρ∈[ρn

j ,ρn j+1] V n

j+ 1

2 g(ρ),

if ρn

j ≤ ρn j+1

max

ρ∈[ρn

j+1,ρn j ] V n

j+ 1

2 g(ρ),

if ρn

j ≥ ρn j+1

     = V n

j+ 1

2 g(ρn

j ).

• From the initial data ρ0 we get

ρ0

j = 1

h xj+ 1

2

xj− 1

2

ρ0(x)dx (4) and then we have the scheme ρn+1

j

= ρn

j − λ

• V n

j+ 1

2 g(ρn

j ) − V n j− 1

2 g(ρn

j−1)

• .

(5)

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Properties of the Godunov type scheme Let hypotheses (H1) hold, ρ0

j be given by (4) with ρM = supj∈Z ρ0 j and

ρm = infj∈Z ρ0

j and let the following CFL condition hold

λ ≤ 1 γ0v′g + vg′, (6) then:

• the approximate solutions constructed by (5) satisfy the bounds

ρm ≤ ρn

j ≤ ρM

∀j ∈ Z, n ∈ N,

• with ¯

ρ given by (5) for every T > 0 the following discrete space BV estimate is satisfied: TV (¯ ρ(T, ·)) ≤ exp(C(wη, v, g)T)TV (ρ0) with C(wη, v, g) = wη(0)(v′g + vg′).

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Numerical examples

• Consider

traffic flow models (g(ρ) = ρ) with

v(ρ) = 1 − ρ5, wη(x) = 1/η.

• Initial conditions:

ρ0(x) =

• 1,

if x ∈ [1/3, 2/3], 1/3, else.

0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 x ρ

• η = 0.1; τ given by the CFL condition; final time T = 0.05.

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Numerical examples

• Consider

traffic flow models (g(ρ) = ρ) with

v(ρ) = 1 − ρ5, wη(x) = 1/η.

• Initial conditions:

ρ0(x) =

• 1,

if x ∈ [1/3, 2/3], 1/3, else.

0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 x ρ

• η = 0.1; τ given by the CFL condition; final time T = 0.05.
• We are interested in the following points:

− Comparison of the new model (3) with the common one (2).

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Numerical examples

• Consider

traffic flow models (g(ρ) = ρ) with

v(ρ) = 1 − ρ5, wη(x) = 1/η.

• Initial conditions:

ρ0(x) =

• 1,

if x ∈ [1/3, 2/3], 1/3, else.

0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 x ρ

• η = 0.1; τ given by the CFL condition; final time T = 0.05.
• We are interested in the following points:

− Comparison of the new model (3) with the common one (2). − What happens to our model for η → 0?

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Comparison of the models (2) and (3) 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 x ρ initial conditions model (2)

• Model (2) contains rather large oscillations while resolving the traffic

jam.

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Comparison of the models (2) and (3) 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 x ρ initial conditions model (2) model (3)

• Model (2) contains rather large oscillations while resolving the traffic

jam.

• Model (3) resolves the traffic jam in a more monotone way.

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Numerical convergence for η → 0 Convergence of the approximate solutions towards the solution of the (local) LWR traffic flow model: ∂tρ + ∂x(ρv(ρ)) = 0. 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 x ρ h = 10−4

LWR η = 10−1

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Numerical convergence for η → 0 Convergence of the approximate solutions towards the solution of the (local) LWR traffic flow model: ∂tρ + ∂x(ρv(ρ)) = 0. 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 x ρ h = 10−4

LWR η = 10−1 η = 10−2

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Numerical convergence for η → 0 Convergence of the approximate solutions towards the solution of the (local) LWR traffic flow model: ∂tρ + ∂x(ρv(ρ)) = 0. 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 x ρ h = 10−4

LWR η = 10−1 η = 10−2 η = 10−3 η = 10−4

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Numerical convergence for η → 0 Convergence of the approximate solutions towards the solution of the (local) LWR traffic flow model: ∂tρ + ∂x(ρv(ρ)) = 0. 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 x ρ h = 10−4

LWR η = 10−1 η = 10−2 η = 10−3 η = 10−4

η 10−1 10−2 10−3 10−4 L1 distance 4.46e-02 6.85e-03 9.90e-04 1.60e-04

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Outline

1 Traffic flow models with non-local flux 2 A one-to-one junction model

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A one-to-one junction 50 30

• Typical approach: consider a directed graph G = (V , E) with roads

E = {1, 2} and the density ρi(x, t), i ∈ {1, 2} with the dynamics:

∂tρi(x, t) + ∂xfi(ρi(x, t)) = 0, ∀i ∈ {1, 2}, x ∈ (0, Li), t ∈ [0, T]

with fi(ρ) being the flow function of the considered traffic flow model.

[CGP05] Coclite, Garavello, Picolli, ”Traffic flow on a road network”, 2005

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A one-to-one junction 50 30

• Typical approach: consider a directed graph G = (V , E) with roads

E = {1, 2} and the density ρi(x, t), i ∈ {1, 2} with the dynamics:

∂tρi(x, t) + ∂xfi(ρi(x, t)) = 0, ∀i ∈ {1, 2}, x ∈ (0, Li), t ∈ [0, T]

with fi(ρ) being the flow function of the considered traffic flow model.

• Solutions for the LWR traffic flow model (f (ρ) = ρv(ρ)) are given by

e.g. [CGP05].

[CGP05] Coclite, Garavello, Picolli, ”Traffic flow on a road network”, 2005

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Non-local situation for the 1-to-1 junction [CFG+19]

• Roads with different velocity functions might be of interest.

[CFG+19] Chiarello, Friedrich, Goatin, G¨

• ttlich, Kolb ”A non-local

traffic flow model for 1-to-1 junctions”, 2019

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Non-local situation for the 1-to-1 junction [CFG+19]

• Roads with different velocity functions might be of interest.
• The flux with mean downstream velocity is then given by:

f (t, x, ρ) := ρ(t, x)V1(t, x) + g(ρ(t, x))V2(t, x), (7) with g(ρ) := min{ρ, ρ2

max},

V1(t, x) := min{x+η,0}

min{x,0}

v1(ρ(t, y))ωη(y − x)dy, V2(t, x) := max{x+η,0}

max{x,0}

v2(ρ(t, y))ωη(y − x)dy, for any η > 0 representing the look-ahead distance of the drivers.

[CFG+19] Chiarello, Friedrich, Goatin, G¨

• ttlich, Kolb ”A non-local

traffic flow model for 1-to-1 junctions”, 2019

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Some remarks

• The model assumes that drivers adapt their speed based on a

weighted mean of downstream velocities.

• For v1 ≡ v2 and ρ1

max ≡ ρ2 max the model coincides with (3)

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Some remarks

• The model assumes that drivers adapt their speed based on a

weighted mean of downstream velocities.

• For v1 ≡ v2 and ρ1

max ≡ ρ2 max the model coincides with (3)

• Note that the flux also accounts for the maximum capacity of the

second road segment. ρ1

max

ρ2

max

η > 0 ρ x v1 v2

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Non-local situation for the 1-to-1 junction

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Non-local situation for the 1-to-1 junction

• We couple (7) with the initial data

ρ(0, x) = ρ0(x) ∈ BV(R), s.t. ρ0(x) ∈ [0, ρ1

max] for x < 0 and ρ0(x) ∈ [0, ρ2 max] for x > 0.

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Non-local situation for the 1-to-1 junction

• We couple (7) with the initial data

ρ(0, x) = ρ0(x) ∈ BV(R), s.t. ρ0(x) ∈ [0, ρ1

max] for x < 0 and ρ0(x) ∈ [0, ρ2 max] for x > 0.

• We impose the following hypotheses on vi, i ∈ {1, 2} and wη:

vi ∈ C2([0, ρi

max]; R+): v′ i ≤ 0, vi(ρi max) = 0,

wη ∈ C1([0, η]; R+): w′

η ≤ 0,

η wη(x)dx = 1 ∀η > 0. (H3)

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Non-local situation for the 1-to-1 junction Theorem [CFG+19, Theorem 1] Let hypotheses (H3) hold. Then, the Cauchy problem

• ∂tρ(t, x) + ∂xf (t, x, ρ) = 0,

x ∈ R, t > 0, ρ(0, x) = ρ0(x), x ∈ R, admits a unique entropy weak solution and 0 ≤ ρ(t, x) ≤ ρ1

max

for a.e. x < 0, t > 0, 0 ≤ ρ(t, x) ≤ ρ2

max

for a.e. x ≥ 0, t > 0.

[CFG+19] Chiarello, Friedrich, Goatin, G¨

• ttlich, Kolb ”A non-local

traffic flow model for 1-to-1 junctions”, 2019

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Numerical examples

• We consider two roads with the initial data:

ρ1

0 = ρL,

ρ2

0 = ρR

with ρL = 0.75 and ρR = 0.5 at final time T = 1.

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Numerical examples

• We consider two roads with the initial data:

ρ1

0 = ρL,

ρ2

0 = ρR

with ρL = 0.75 and ρR = 0.5 at final time T = 1.

• Linear kernel function

wη(x) = 2(η − x)/η2 with η = 0.1.

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Numerical examples

• We consider two roads with the initial data:

ρ1

0 = ρL,

ρ2

0 = ρR

with ρL = 0.75 and ρR = 0.5 at final time T = 1.

• Linear kernel function

wη(x) = 2(η − x)/η2 with η = 0.1.

• Linear velocity function

vi(ρ) = vi

max

• 1 −

ρ ρi

max

• .

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Numerical examples

• We consider two roads with the initial data:

ρ1

0 = ρL,

ρ2

0 = ρR

with ρL = 0.75 and ρR = 0.5 at final time T = 1.

• Linear kernel function

wη(x) = 2(η − x)/η2 with η = 0.1.

• Linear velocity function

vi(ρ) = vi

max

• 1 −

ρ ρi

max

• .
• Parameters for each road:

ρi

max = 1,

vi

max ∈ {1, 2} ∀ i ∈ {1, 2}.

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Numerical results −4 −2 2 4 0.2 0.4 0.6 0.8 1 x ρ v1

max < v2 max

numerical solution initial conditions x = 0

−4 −2 2 4 0.2 0.4 0.6 0.8 1 x ρ v1

max > v2 max

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Numerical convergence for η → 0

• So far, the convergence to the classical LWR traffic flow model can
• nly be proven for monotone initial data, see [CCS18] for a recent

discussion.

[CCS19] Colombo, Crippa, Spinolo, ”Blow-up of the total variation in the local limit of a nonlocal traffic model”, 2019 [KT17] Karlsen and Towers, ”Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition”, 2017

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Numerical convergence for η → 0

• So far, the convergence to the classical LWR traffic flow model can
• nly be proven for monotone initial data, see [CCS18] for a recent

discussion.

• The model (7) is in general not monotonicity preserving for v1 = v2,
• even for constant initial data.

[CCS19] Colombo, Crippa, Spinolo, ”Blow-up of the total variation in the local limit of a nonlocal traffic model”, 2019 [KT17] Karlsen and Towers, ”Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition”, 2017

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Numerical convergence for η → 0

• So far, the convergence to the classical LWR traffic flow model can
• nly be proven for monotone initial data, see [CCS18] for a recent

discussion.

• The model (7) is in general not monotonicity preserving for v1 = v2,
• even for constant initial data.
• Again, we investigate the limit numerically and consider

ρt + f (x, ρ)x = 0, with f (x, ρ) := H(−x)ρv1(ρ) + H(x)ρv2(ρ), where H(x) is the Heaviside function.

[CCS19] Colombo, Crippa, Spinolo, ”Blow-up of the total variation in the local limit of a nonlocal traffic model”, 2019 [KT17] Karlsen and Towers, ”Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition”, 2017

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Numerical convergence for η → 0

• So far, the convergence to the classical LWR traffic flow model can
• nly be proven for monotone initial data, see [CCS18] for a recent

discussion.

• The model (7) is in general not monotonicity preserving for v1 = v2,
• even for constant initial data.
• Again, we investigate the limit numerically and consider

ρt + f (x, ρ)x = 0, with f (x, ρ) := H(−x)ρv1(ρ) + H(x)ρv2(ρ), where H(x) is the Heaviside function.

• The vanishing viscosity solution [KT17] defines a weak entropy

solution for x < 0 and x > 0.

[CCS19] Colombo, Crippa, Spinolo, ”Blow-up of the total variation in the local limit of a nonlocal traffic model”, 2019 [KT17] Karlsen and Towers, ”Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition”, 2017

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Numerical convergence for η → 0 −1 1 0.2 0.4 0.6 0.8 x ρ v1

max < v2 max

initial conditions x = 0 vanishing viscosity solution η = 50∆x η = 10∆x η = 2∆x

−2 −1 1 0.4 0.6 0.8 x ρ v1

max > v2 max

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Open points

• Extension to general networks, i.e., coupling conditions for dispersing

and merging junctions, maximum principle, well-posedness

• Optimal control issues, e.g., speed limits and ramp-metering
• Model validation and calibration (if appropriate data will be available)
• ...

This work is supported by the project ”Non-local conservation laws for engineering applications” co-funded by DAAD and PHC Procope.

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Open points

• Extension to general networks, i.e., coupling conditions for dispersing

and merging junctions, maximum principle, well-posedness

• Optimal control issues, e.g., speed limits and ramp-metering
• Model validation and calibration (if appropriate data will be available)
• ...

Thank you for your attention!

This work is supported by the project ”Non-local conservation laws for engineering applications” co-funded by DAAD and PHC Procope.

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[ACG15] Aekta Aggarwal, Rinaldo M. Colombo, and Paola Goatin. Nonlocal systems

• f conservation laws in several space dimensions. SIAM J. Numer. Anal.,

53(2):963–983, 2015. [CCS18] Maria Colombo, Gianluca Crippa, and Laura V. Spinolo. Blow-up of the total variation in the local limit of a nonlocal traffic model. arXiv e-prints, page arXiv:1808.03529, Aug 2018. [CFG+19] Felisia Angela A Chiarello, J Friedrich, Paola Goatin, S G¨

• ttlich, and O Kolb.

A non-local traffic flow model for 1-to-1 junctions. working paper or preprint, May 2019. [CG17] Felisia Angela Chiarello and Paola Goatin. Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel. https://hal.inria.fr/hal-01567575, July 2017. [CGP05] G. Coclite, M. Garavello, and B. Piccoli. Traffic flow on a road network. SIAM journal on mathematical analysis, 36(6):1862–1886, 2005. [FKG18] Jan Friedrich, Oliver Kolb, and Simone G¨

• ttlich. A Godunov type scheme for

a class of LWR traffic flow models with non-local flux. Netw. Heterog. Media, 2018. arXiv:1802.07484. [KT17] K. H. Karlsen and J. D. Towers. Convergence of a godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition. Journal of Hyperbolic Differential Equations, 14(04):671–701, 2017.

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