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

The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model Nicolas Laurent-Brouty 1 , 2 , Paola Goatin 1 1 Universit´ e Cˆ ote d’Azur, Inria, CNRS, LJAD, France 2 Ecole des Ponts ParisTech, Champs-sur-Marne, France May 18th, 2018 nicolas.laurent-brouty@inria.fr 1

Outline Introduction to conservation laws 1 Wave-Front Tracking approximations 2 Convergence of the WFT approximate solutions 3 Convergence of the relaxed ARZ system towards LWR equation 4 Decay estimates of positive waves 5 2

Outline Introduction to conservation laws 1 Wave-Front Tracking approximations 2 Convergence of the WFT approximate solutions 3 Convergence of the relaxed ARZ system towards LWR equation 4 Decay estimates of positive waves 5 3

Introduction to conservation laws t time x space variable ρ ( t , x ) density of vehicles v ( t , x ) velocity of the flow f ( ρ, v ) = ρ v Figure: 1D-representation of a stretch of road Conservation of the number of cars: � b d ρ ( y , t ) dy = [flux entering at a] − [flux exiting at b] dt a � b � b ∂ ∂ ∂ t ρ ( y , t ) dy = − ∂ x [ f ( ρ, v )]( x , t ) dx a a ∂ t ρ + ∂ ∂ ∂ x f ( ρ, v ) = 0 4

Application to traffic flow 1 The Lighthill, Whitham, Richards (LWR) model Assume v = v ( ρ ) � ∂ t ρ + ∂ x ( ρ v ( ρ )) = 0 x ∈ R , t > 0 , ρ (0 , x ) = ρ 0 ( x ) Figure: Fundamental diagram of traffic flow 5

Application to traffic flow 2 The Payne-Whitham model (PW) Define an anticipation factor A e ( ρ ) and a response time from drivers δ � ∂ t ρ + ∂ x ( ρ v ( ρ )) = 0 , ∂ t v + v ∂ x v + 1 ρ ∂ x ( A e ( ρ )) = 0 3 The Aw-Rascle-Zhang (ARZ) model Assume a pseudo-pressure, strictly increasing, p ( ρ ) > 0 � ∂ t ρ + ∂ x ( ρ v ) = 0 ∂ t ( v + p ( ρ )) + v ∂ x ( v + p ( ρ )) = 0 4 The Aw-Rascle-Zhang model with relaxation � ∂ t ρ + ∂ x ( ρ v ) = 0 ∂ t ( v + p ( ρ )) + v ∂ x ( v + p ( ρ )) = V eq ( ρ ) − v δ 6

The ARZ model with relaxation The ARZ model with relaxation � ∂ t ρ + ∂ x ( ρ v ) = 0 ∂ t ( v + p ( ρ )) + v ∂ x ( v + p ( ρ )) = V eq ( ρ ) − v δ The model can be put under conservative form: � ∂ t ρ + ∂ x ( ρ v ) = 0 ∂ t ( ρ ( v + p ( ρ ))) + ∂ x ( ρ v ( v + p ( ρ ))) = ρ V eq ( ρ ) − v δ Eigenvalues: λ 1 = v − ρ p ′ ( ρ ) , λ 2 = v To ensure strict hyperbolicity, we assume: p ′ ( ρ ) > 0 . ρ > 0 , p ( ρ ) ≥ 0 , 7

Define w := v + p ( ρ ) � ∂ t ρ + ∂ x ( ρ v ) = 0 ∂ t ( ρ w ) + ∂ x ( ρ vw ) = ρ V eq ( ρ ) − v δ convert into Lagrangian coordinates ( T , X ) with τ = 1 ρ . � 1 � � 1 � p ′ ( τ ) = − 1 � 1 � ˜ τ 2 p ′ p ( τ ) = p ˜ , V ( τ ) = V eq , ˜ < 0 , τ τ τ � ∂ T τ − ∂ X v = 0 ˜ V ( τ ) − v ∂ T w = δ with initial data � τ (0 , · ) = τ 0 w (0 , · ) = w 0 8

Definition of solutions � U t + [ F ( U )] X = G δ ( U ) x ∈ R , t > 0 (1.1) U (0 , x ) = U 0 ( x ) � � 0 � τ � � − ( w − ˜ � p ( τ )) G δ ( U ) = U = , F ( U ) = , . ˜ V ( τ ) − ( w − ˜ p ( τ )) w 0 δ Definition (definition of solutions) Assume U 0 ∈ BV ( R ) and T > 0. We say that a function U δ : [0 , T ] × R → R 2 is a weak solution to the Cauchy problem (1.1) if the map t → U δ ( t , · ) ∈ L 1 loc ( R ) is continuous, U δ ( t = 0 , · ) = U 0 ( · ) and if for any φ ∈ C 1 c ([0 , T ] × R ) � + ∞ � T � + ∞ φ (0 , X ) U δ [ φ t U δ ( t , X ) + φ X F ( U δ ( t , X ))] dXdt 0 ( X ) dX + 0 −∞ −∞ � T � + ∞ φ G δ ( U δ ( t , X )) dXdt = 0 + 0 −∞ 9

Main results Theorem For each relaxation parameter δ , the ARZ model with relaxation admits a weak entropy solution U δ = ( τ δ , w δ ) . Theorem The subsequence of weak entropy solutions U δ = ( τ δ , w δ ) of the relaxed ARZ model converges to ¯ w = ˜ U = (¯ τ, ¯ w ) as δ → 0 . Then ¯ V (¯ τ ) + ˜ p (¯ τ ) and ¯ τ is a weak solution of the scalar Cauchy problem: � ∂ t τ − ∂ X ˜ V ( τ ) = 0 , X ∈ R , t > 0 . (1.2) τ (0 , · ) = τ 0 ( · ) , 10

Outline Introduction to conservation laws 1 Wave-Front Tracking approximations 2 Convergence of the WFT approximate solutions 3 Convergence of the relaxed ARZ system towards LWR equation 4 Decay estimates of positive waves 5 11

What is Wave-Front Tracking? General idea of wave front tracking for a system of conservation laws: � U t + [ F ( U )] X = 0 x ∈ R , t > 0 U (0 , x ) = U 0 ( x ) 1 approximate the initial datum U 0 by a piecewise constant function U 0 ǫ such that � U 0 − U 0 ǫ � L ∞ ≤ ǫ 2 for t = 0 + solve for each discontinuity of U 0 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. 12

WFT approximations How to treat the relaxation term? → two step process 1 solve the Cauchy problem associated to the homogeneous system via WFT on a time interval [ t 0 , t 0 + ∆ t ] ˜ V ( τ ) − v 2 at t = t 0 + ∆ t , integrate the source term following w t = δ Definition ( BV space) Let Ω an open set. We say that a function u ∈ L 1 loc (Ω; R ) belongs to BV (Ω; R ) if its total variation TV ( u ) < ∞ , where for every n-tuple { x 1 , .., x n } ∈ Ω: n − 1 � TV ( u ) = sup | u ( x i +1 ) − u ( x i ) | i =1 13

Algorithm Let T > 0 and a sequence ∆ t ν > 0 s.t. ∆ t ν − ν →∞ 0. − − → 1 Approximate the initial value U 0 ∈ BV ( R + × R ) by a piecewise constant function U ν 0 = ( τ ν 0 , w ν 0 ) 2 Solve the homogeneous system via WFT and name U ν ( t , · ), t ∈ [0 , ∆ t ν ) the solution � ∂ t τ − ∂ X v = 0 , ∂ t w = 0 , 14

˜ 3 At t = ∆ t ν integrate the source term w t = V ( τ ) − v , i.e. define δ τ ν (∆ t ν , · ) = τ ν (∆ t ν − , · ) , w ν (∆ t ν , · ) = w ν (∆ t ν − , · ) + ∆ t ν ˜ V ( τ ν (∆ t ν , · )) − v ( U ν (∆ t ν − , · )) δ τ l , w + τ r , w + ∆ t ν + l r ∆ t ν − τ l , w − τ r , w − r l Figure: Notations used in step 3. 4 Treat U ν (∆ t ν , · ) as a new piecewise constant initial condition and iterate 15

Existence of an invariant domain Definition (invariant domain) � � u = ( τ, w ): V min p ( τ ) < w ≤ V max E := ≤ w − ˜ + max p ( τ ) ˜ eq eq 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. 16

Decreasing TV and Lipschitz estimates Lemma For ∆ t ≤ δ , the total variation of the Riemann invariants of the constructed approximation U ν is non-increasing in time: TV ( w ν ( t , · )) + TV ( v ( U ν ( t , · ))) ≤ TV ( w ν 0 ) + TV ( v ( U ν 0 )) , for a.e. t > 0 . Lemma Let ν ∈ N and U ν 0 ∈ D ( M ) . Then ∀ a < b , ∀ 0 ≤ s < t: � b | τ ν ( t , X ) − τ ν ( s , X ) | dX ≤ C M ( t − s ) , a � b | w ν ( t , X ) − w ν ( s , X ) | dX ≤ ( C M + L δ )( t − s + ∆ t ) . a 17

Outline Introduction to conservation laws 1 Wave-Front Tracking approximations 2 Convergence of the WFT approximate solutions 3 Convergence of the relaxed ARZ system towards LWR equation 4 Decay estimates of positive waves 5 18

Convergence of the WFT approximations Theorem Let U 0 = ( τ 0 , w 0 ) ∈ D ( M ) for some M > 0 , and denote by U δ = ( τ δ , w δ ) the limit of a subsequence U ν = ( τ ν , w ν ) of WFT approximate solutions as ν → ∞ . Then U δ is the weak entropic solution of  ∂ T τ − ∂ X v = 0 ,   ˜  V ( τ ) − v  ∂ T w = ,  δ (3.1) τ (0 , · ) = τ 0 ,     w (0 , · ) = w 0 .  19

Sketch of proof The existence of the limit U δ and the convergence in L 1 loc ([0 , + ∞ [ × R ) is guaranteed by Helly’s theorem. Theorem (Helly’s theorem) Consider a sequence of functions U ν s.t.: TV ( U ν ( t , · )) ≤ C , | U ν ( t , x ) | ≤ M for all t , x , � ∞ | U ν ( t , X ) − U ν ( s , X ) | dX ≤ L ( t − s ) for all t , s ≥ 0 , −∞ Then there exists a subsequence U µ which converges to some function U in L 1 loc ([0 , + ∞ [ × R ) . The limit satisfies � ∞ | U ( t , X ) − U ( s , X ) | dX ≤ L ( t − s ) for all t , s ≥ 0 , −∞ 20

Sketch of proof U t + [ F ( U )] X = G δ ( U ) , � � 0 � τ � � − ( w − ˜ � p ( τ )) G δ ( U ) = U = , F ( U ) = , . ˜ V ( τ ) − ( w − ˜ p ( τ )) 0 w δ Definition (definition of solutions) Assume U 0 ∈ BV ( R ) and T > 0. We say that a function U δ : [0 , T ] × R → R 2 is a weak solution to the Cauchy problem (1.1) if the map t → U δ ( t , · ) ∈ L 1 loc ( R ) is continuous, U δ ( t = 0 , · ) = U 0 ( · ) and if for any φ ∈ C 1 c ([0 , T ] × R ) � + ∞ � + ∞ � T φ (0 , X ) U δ [ φ t U δ ( t , X ) + φ X F ( U δ ( t , X ))] dXdt 0 ( X ) dX + 0 −∞ −∞ � + ∞ � T φ G ( U δ ( t , X )) dXdt = 0 + 0 −∞ 21