Convergence of the Follow-The-Leader scheme for scalar conservation - PowerPoint PPT Presentation

Macroscopic model Microscopic models Convergence Convergence of the Follow-The-Leader scheme for scalar conservation laws with space dependent flux Graziano Stivaletta University of L’Aquila Work in collaboration with M. Di Francesco (University of L’Aquila) CROWDS: models and control CIRM Marseille, June 3-7, 2019

Macroscopic model Microscopic models Convergence Macroscopic model 1 A general overview Assumptions Microscopic models 2 The FTL approximation Main properties Convergence 3 The uniform control of TV and W 1 Convergence to the entropy solution

Macroscopic model Microscopic models Convergence A general overview Let us consider the Cauchy problem � ρ t + � ρv ( ρ ) φ ( x ) � x = 0 , x ∈ R , t > 0 , (1) ρ ( x, 0) = ρ ( x ) , x ∈ R . ρ ( x, t ) is a density; v ( ρ ) is a velocity map; φ ( x ) is a given external potential; ρ ( x ) is the initial datum.

Macroscopic model Microscopic models Convergence A general overview Let us consider the Cauchy problem � ρ t + � ρv ( ρ ) φ ( x ) � x = 0 , x ∈ R , t > 0 , (1) ρ ( x, 0) = ρ ( x ) , x ∈ R . ρ ( x, t ) is a density; v ( ρ ) is a velocity map; φ ( x ) is a given external potential; ρ ( x ) is the initial datum. Possible applications: Traffic flows; Sedimentation processes; Flow of glaciers; Formation of Bose-Einstein condensates.

Macroscopic model Microscopic models Convergence A general overview Goal: We want to derive a weak (eventually entropic) solution to (1) from systems of deterministic particles of Follow-The-Leader type.

Macroscopic model Microscopic models Convergence A general overview Goal: We want to derive a weak (eventually entropic) solution to (1) from systems of deterministic particles of Follow-The-Leader type. The FTL particle approach has been applied for other models, as: LWR model. M. Di Francesco and M.D. Rosini (2015). M. Di Francesco, S. Fagioli and M.D. Rosini (2017). H. Holden and N.H. Risebro (2018). ARZ model. A. Aw, A. Klar, T. Materne and M. Rascle (2002). F. Berthelin, P. Degond, M. Delitala and M. Rascle (2008). M. Di Francesco S. Fagioli and M.D. Rosini (2016). F. Berthelin and P. Goatin (2017). Hughes model. M. Di Francesco, S. Fagioli, M.D. Rosini and G. Russo (2017). IBVP with Dirichlet type conditions. M. Di Francesco, S. Fagioli, M.D. Rosini and G. Russo (2017). Nonlocal transport equations. S. Fagioli and E. Radici (2018). M. Di Francesco, S. Fagioli and E. Radici (2019).

Macroscopic model Microscopic models Convergence A general overview We now introduce our concept of entropy solution to (1). Definition Let ρ ∈ BV ( R ) . We say that ρ ∈ L ∞ � [0 , + ∞ ) ; BV ( R ) � is an entropy solution to (1) if: ρ ( · , t ) → ρ strongly in L 1 ( R ) as t ց 0 . ρ satisfies the entropy condition, that is � T � � | ρ ( x, t ) − k | ϕ t ( x, t ) + sign( ρ ( x, t ) − k ) � f ( ρ ( x, t )) − f ( k ) � φ ( x ) ϕ x ( x, t ) 0 R � − sign( ρ ( x, t ) − k ) f ( k ) φ ′ ( x ) ϕ ( x, t ) dxdt ≥ 0 , for all k ≥ 0 and for all ϕ ∈ C c ( R × (0 , + ∞ )) with ϕ ≥ 0 .

Macroscopic model Microscopic models Convergence Assumptions Assumptions on v and ρ : (V) v ∈ Lip ( R + ) is a non-negative and non-increasing function with v (0) := v max < + ∞ .

Macroscopic model Microscopic models Convergence Assumptions Assumptions on v and ρ : (V) v ∈ Lip ( R + ) is a non-negative and non-increasing function with v (0) := v max < + ∞ . (I) ρ ∈ L ∞ ( R ) ∩ BV ( R ) is a non-negative compactly supported function with [ x min , x max ] := Conv(supp( ρ )) .

Macroscopic model Microscopic models Convergence Assumptions Assumptions on v and ρ : (V) v ∈ Lip ( R + ) is a non-negative and non-increasing function with v (0) := v max < + ∞ . (I) ρ ∈ L ∞ ( R ) ∩ BV ( R ) is a non-negative compactly supported function with [ x min , x max ] := Conv(supp( ρ )) . Concerning φ , we deal with four different cases: (P1) φ ( x ) ≥ 0 for all x ∈ R (forward movement). (P2) φ ( x ) ≤ 0 for all x ∈ R (backward movement). (P3) xφ ( x ) ≥ 0 for all x ∈ R (repulsive movement). (P4) xφ ( x ) ≤ 0 for all x ∈ R (attractive movement).

Macroscopic model Microscopic models Convergence Assumptions Assumptions on v and ρ : (V) v ∈ Lip ( R + ) is a non-negative and non-increasing function with v (0) := v max < + ∞ . (I) ρ ∈ L ∞ ( R ) ∩ BV ( R ) is a non-negative compactly supported function with [ x min , x max ] := Conv(supp( ρ )) . Concerning φ , we deal with four different cases: (P1) φ ( x ) ≥ 0 for all x ∈ R (forward movement). (P2) φ ( x ) ≤ 0 for all x ∈ R (backward movement). (P3) xφ ( x ) ≥ 0 for all x ∈ R (repulsive movement). (P4) xφ ( x ) ≤ 0 for all x ∈ R (attractive movement). Assumption on φ : (P) φ ∈ W 2 , ∞ ( R ) .

Macroscopic model Microscopic models Convergence Assumptions Assumptions on v and ρ : (V) v ∈ Lip ( R + ) is a non-negative and non-increasing function with v (0) := v max < + ∞ . (I) ρ ∈ L ∞ ( R ) ∩ BV ( R ) is a non-negative compactly supported function with [ x min , x max ] := Conv(supp( ρ )) . Concerning φ , we deal with four different cases: (P1) φ ( x ) ≥ 0 for all x ∈ R (forward movement). (P2) φ ( x ) ≤ 0 for all x ∈ R (backward movement). (P3) xφ ( x ) ≥ 0 for all x ∈ R (repulsive movement). (P4) xφ ( x ) ≤ 0 for all x ∈ R (attractive movement). Assumption on φ : (P) φ ∈ W 2 , ∞ ( R ) . Extra assumption on v in case (P4): (V ∗ ) There exists R max > 0 such that R := || ρ || L ∞ ( R ) ≤ R max , v ( ρ ) > 0 for ρ < R max and v ( ρ ) ≡ 0 for ρ ≥ R max .

Macroscopic model Microscopic models Convergence The FTL approximation Let ρ a fixed initial datum satisfying (I) and such that || ρ || L 1 ( R ) = 1 . We consider, for a fixed n ∈ N , n + 1 particles having equal mass ℓ n := 1 /n .

Macroscopic model Microscopic models Convergence The FTL approximation Let ρ a fixed initial datum satisfying (I) and such that || ρ || L 1 ( R ) = 1 . We consider, for a fixed n ∈ N , n + 1 particles having equal mass ℓ n := 1 /n . Initial conditions of the particles Suitable atomization of ρ : we split [ x min , x max ] into n sub-intervals such that the mass of ρ is equal to ℓ n on each of them and we set x n x n 0 := x min , n := x max , � x � � x n i := sup x ∈ R : ρ ( x ) dx < l n for i ∈ { 1 , . . . , n − 1 } . x n i − 1

Macroscopic model Microscopic models Convergence The FTL approximation Evolution of the particles ℓ n i ( t ) and k n := max � i ≤ 0 � Let us define R n i ∈ { 0 , . . . , n } : x n i ( t ) := . x n i +1 ( t ) − x n

Macroscopic model Microscopic models Convergence The FTL approximation Evolution of the particles ℓ n i ( t ) and k n := max � i ≤ 0 � Let us define R n i ∈ { 0 , . . . , n } : x n i ( t ) := . x n i +1 ( t ) − x n The microscopic models we study are � ˙ x n i ( t ) = v ( R n i ( t )) φ ( x n i ( t )) , i ∈ { 0 , . . . , n − 1 } , x n n ( t ) = v max φ ( x n ( Case P1 ) ˙ n ( t )) , (2) x n i (0) = x n i , i ∈ { 0 , . . . , n } ,

Macroscopic model Microscopic models Convergence The FTL approximation Evolution of the particles ℓ n i ( t ) and k n := max � i ≤ 0 � Let us define R n i ∈ { 0 , . . . , n } : x n i ( t ) := . x n i +1 ( t ) − x n The microscopic models we study are � ˙ x n i ( t ) = v ( R n i ( t )) φ ( x n i ( t )) , i ∈ { 0 , . . . , n − 1 } , x n n ( t ) = v max φ ( x n ( Case P1 ) ˙ n ( t )) , (2) x n i (0) = x n i , i ∈ { 0 , . . . , n } , � ˙ x n 0 ( t ) = v max φ ( x n 0 ( t )) , x n i ( t ) = v ( R n i − 1 ( t )) φ ( x n ( Case P2 ) ˙ i ( t )) , i ∈ { 1 , . . . , n } , (3) i (0) = x n x n i , i ∈ { 0 , . . . , n } ,

Macroscopic model Microscopic models Convergence The FTL approximation Evolution of the particles ℓ n i ( t ) and k n := max � i ≤ 0 � Let us define R n i ∈ { 0 , . . . , n } : x n i ( t ) := . x n i +1 ( t ) − x n The microscopic models we study are � ˙ x n i ( t ) = v ( R n i ( t )) φ ( x n i ( t )) , i ∈ { 0 , . . . , n − 1 } , x n n ( t ) = v max φ ( x n ( Case P1 ) ˙ n ( t )) , (2) x n i (0) = x n i , i ∈ { 0 , . . . , n } , � ˙ x n 0 ( t ) = v max φ ( x n 0 ( t )) , x n i ( t ) = v ( R n i − 1 ( t )) φ ( x n ( Case P2 ) ˙ i ( t )) , i ∈ { 1 , . . . , n } , (3) i (0) = x n x n i , i ∈ { 0 , . . . , n } ,  x n 0 ( t ) = v max φ ( x n ˙ 0 ( t )) ,   x n i ( t ) = v ( R n i − 1 ( t )) φ ( x n ˙ i ( t )) , i ∈ { 1 , . . . , k n } ,   x n i ( t ) = v ( R n i ( t )) φ ( x n ( Case P3 ) ˙ i ( t )) , i ∈ { k n + 1 , . . . , n − 1 } , (4) x n n ( t ) = v max φ ( x n ˙ n ( t )) ,    i (0) = x n  x n i ∈ { 0 , . . . , n } , i