CHAIR OF SCIENTIFIC COMPUTING Traffic flow models with non-local flux and extensions to networks Simone G¨ ottlich 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 1 / 24
CHAIR OF SCIENTIFIC COMPUTING Motivation of non-local models • In general non-local systems of N conservation laws have the form: ∂ t U + div x F ( t , x , U , w ∗ U ) = 0 (1) with t ∈ R + , x ∈ R d , U ∈ R N , F ∈ R N × d [ACG15] Aggarwal, Colombo, Goatin ”Nonlocal systems of conservation laws in several space dimensions”, 2015 2 / 24
CHAIR OF SCIENTIFIC COMPUTING Motivation of non-local models • In general non-local systems of N conservation laws have the form: ∂ t U + div x F ( t , x , U , w ∗ U ) = 0 (1) with t ∈ R + , x ∈ R d , U ∈ R N , F ∈ R N × 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 2 / 24
CHAIR OF SCIENTIFIC COMPUTING Motivation of non-local models • In general non-local systems of N conservation laws have the form: ∂ t U + div x F ( t , x , U , w ∗ U ) = 0 (1) with t ∈ R + , x ∈ R d , U ∈ R N , F ∈ R N × 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¨ ottlich et al. [2014]) − traffic flow → see next slide [ACG15] Aggarwal, Colombo, Goatin ”Nonlocal systems of conservation laws in several space dimensions”, 2015 2 / 24
CHAIR OF SCIENTIFIC COMPUTING Motivation of non-local traffic models • In case of non-local traffic scalar conservation laws are of interest: ∂ t ρ + ∂ x f ( t , x , ρ, w ∗ ρ ) = 0 t ∈ R + , x ∈ R , ρ ∈ R , f ∈ R 3 / 24
CHAIR OF SCIENTIFIC COMPUTING Motivation of non-local traffic models • In case of non-local traffic scalar conservation laws are of interest: ∂ t ρ + ∂ x f ( 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]) 3 / 24
CHAIR OF SCIENTIFIC COMPUTING Idea of non-local traffic flow models ρ ρ max x 4 / 24
CHAIR OF SCIENTIFIC COMPUTING Idea of non-local traffic flow models ρ ρ max x η > 0 • Drivers adapt the velocity to the downstream traffic . 4 / 24
CHAIR OF SCIENTIFIC COMPUTING Idea of non-local traffic flow models ρ ρ max ! ! ! ! x η > 0 • Drivers adapt the velocity to the downstream traffic . • Closer vehicles are more important . 4 / 24
CHAIR OF SCIENTIFIC COMPUTING Idea of non-local traffic flow models ρ ρ max ! ! ! ! x η > 0 • Drivers adapt the velocity to the downstream traffic . • Closer vehicles are more important . • Common approach: velocity of mean downstream traffic . 4 / 24
CHAIR OF SCIENTIFIC COMPUTING Idea of non-local traffic flow models ρ ρ max ! ! ! ! x η > 0 • 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 . 4 / 24
CHAIR OF SCIENTIFIC COMPUTING Outline 1 Traffic flow models with non-local flux 2 A one-to-one junction model 5 / 24
CHAIR OF SCIENTIFIC COMPUTING Outline 1 Traffic flow models with non-local flux 2 A one-to-one junction model 5 / 24
CHAIR OF SCIENTIFIC COMPUTING Traffic flow model with mean downstream density [CG17] • ρ ( t , x ) , x ∈ R , t ≥ 0: density of cars with 0 ≤ ρ ≤ ρ max . ∂ t ρ ( 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 6 / 24
CHAIR OF SCIENTIFIC COMPUTING Traffic flow model with mean downstream density [CG17] • ρ ( t , x ) , x ∈ R , t ≥ 0: density of cars with 0 ≤ ρ ≤ ρ max . • Assuming that drivers adapt their speed based on a mean downstream density (common approach in literature) results in ∂ t ρ ( t , x ) + ∂ x ( ρ v ( w η ∗ ρ )) = 0 , x ∈ R , t > 0 , � x + η (2) w η ∗ ρ ( t , x ) := ρ ( t , y ) w η ( y − x ) dy , η > 0 . x [CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017 6 / 24
CHAIR OF SCIENTIFIC COMPUTING Traffic flow model with mean downstream density [CG17] • ρ ( t , x ) , x ∈ R , t ≥ 0: density of cars with 0 ≤ ρ ≤ ρ max . • Assuming that drivers adapt their speed based on a mean downstream density (common approach in literature) results in ∂ t ρ ( t , x ) + ∂ x ( g ( ρ ) v ( w η ∗ ρ )) = 0 , x ∈ R , t > 0 , � x + η (2) w η ∗ ρ ( t , x ) := ρ ( t , y ) w η ( y − x ) dy , η > 0 . x [CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017 6 / 24
CHAIR OF SCIENTIFIC COMPUTING Traffic flow model with mean downstream density [CG17] • ρ ( t , x ) , x ∈ R , t ≥ 0: density of cars with 0 ≤ ρ ≤ ρ max . • Assuming that drivers adapt their speed based on a mean downstream density (common approach in literature) results in ∂ t ρ ( t , x ) + ∂ x ( g ( ρ ) v ( w η ∗ ρ )) = 0 , x ∈ R , t > 0 , � x + η (2) w η ∗ ρ ( t , x ) := ρ ( t , y ) w η ( y − x ) dy , η > 0 . x • Under the following hypotheses existence and uniqueness of weak entropy solutions is given (see [CG17, Theorem 1]): I = [ a , b ] ⊆ R + , ρ (0 , x ) = ρ 0 ( x ) ∈ BV( R , I ) , [CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017 6 / 24
CHAIR OF SCIENTIFIC COMPUTING Traffic flow model with mean downstream density [CG17] • ρ ( t , x ) , x ∈ R , t ≥ 0: density of cars with 0 ≤ ρ ≤ ρ max . • Assuming that drivers adapt their speed based on a mean downstream density (common approach in literature) results in ∂ t ρ ( t , x ) + ∂ x ( g ( ρ ) v ( w η ∗ ρ )) = 0 , x ∈ R , t > 0 , � x + η (2) w η ∗ ρ ( t , x ) := ρ ( t , y ) w η ( y − x ) dy , η > 0 . x • Under the following hypotheses existence and uniqueness of weak entropy solutions is given (see [CG17, Theorem 1]): I = [ a , b ] ⊆ R + , ρ (0 , x ) = ρ 0 ( x ) ∈ BV( R , I ) , v ∈ C 2 ( I ; R + ) with v ′ ≤ 0 , g ∈ C 1 ( I ; R + ) , (H1) [CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017 6 / 24
CHAIR OF SCIENTIFIC COMPUTING Traffic flow model with mean downstream density [CG17] • ρ ( t , x ) , x ∈ R , t ≥ 0: density of cars with 0 ≤ ρ ≤ ρ max . • Assuming that drivers adapt their speed based on a mean downstream density (common approach in literature) results in ∂ t ρ ( t , x ) + ∂ x ( g ( ρ ) v ( w η ∗ ρ )) = 0 , x ∈ R , t > 0 , � x + η (2) w η ∗ ρ ( t , x ) := ρ ( t , y ) w η ( y − x ) dy , η > 0 . x • Under the following hypotheses existence and uniqueness of weak entropy solutions is given (see [CG17, Theorem 1]): I = [ a , b ] ⊆ R + , ρ (0 , x ) = ρ 0 ( x ) ∈ BV( R , I ) , v ∈ C 2 ( I ; R + ) with v ′ ≤ 0 , g ∈ C 1 ( I ; R + ) , (H1) � η w η ∈ C 1 ([0 , η ]; R + ) with w ′ η ≤ 0, w η ( x ) dx = W 0 , lim η →∞ w η (0) = 0 . 0 [CG17] Chiarello and Goatin ”Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel”, 2017 6 / 24
CHAIR OF SCIENTIFIC COMPUTING 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¨ ottlich ”A Godunov type scheme for a class of LWR traffic flow models with non-local flux”, 2018 7 / 24
CHAIR OF SCIENTIFIC COMPUTING 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. ∂ t ρ ( t , x ) + ∂ x ( g ( ρ ) ( w η ∗ v ( ρ ))) = 0 , x ∈ R , t > 0 , (3) [FKG18] Friedrich, Kolb and G¨ ottlich ”A Godunov type scheme for a class of LWR traffic flow models with non-local flux”, 2018 7 / 24
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