a On the Management of Vehicular Traffic HYP2012 Massimiliano D. Rosini mrosini@icm.edu.pl Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 1 / 77
Table of contents Introduction to Vehicular Traffic 1 Mathematics 2 Applications to LWR 3 Numerical Examples 4 Crowd Accidents 5 The Model 6 Corridor with One Exit 7 Corridor with Two Exits 8 Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 2 / 77
a 1 Introduction to Vehicular Traffic Mathematics 2 Applications to LWR 3 Numerical Examples 4 Crowd Accidents 5 The Model 6 Corridor with One Exit 7 Corridor with Two Exits 8 Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 3 / 77
The fundamental traffic variables Along a road it can be measured: the traffic density ρ : number of vehicles per unit space the velocity v : distance covered by vehicles per unit time the traffic flow f : number of vehicles per unit time Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 4 / 77
The fundamental traffic variables Along a road it can be measured: the traffic density ρ : number of vehicles per unit space the velocity v : distance covered by vehicles per unit time the traffic flow f : number of vehicles per unit time What are the relations between ρ , v and f ? Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 4 / 77
Relations between ρ , v , f Vehicles with the same length L and velocity v move equally spaced observer in x R L d v τ The distance between vehicles and the density do not change. The number of vehicles passing the observer in τ hours is the number of vehicles in [ x − τ v , x ] at time t − τ and therefore f = ρ [ x − ( x − τ v )] = ρ v τ Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 5 / 77
Relations between ρ , v , f If no entries or exits are present in [ a , b ] , then � b � b � T � T − ρ ( T , y ) dy = ρ ( t o , y ) dy + f ( t , a ) dt f ( t , b ) dt a a t o t o � �� � � �� � � �� � � �� � cars in [ a , b ] cars in [ a , b ] cars entering cars exiting at time t = T at time t = t o in [ a , b ] from [ a , b ] or equivalently � T � b [ ∂ t ρ ( t , x ) − ∂ x f ( t , x )] dx dt = 0 . t o a Since a , b , T and t o are arbitrary we deduce scalar conservation law: ∂ t ρ + ∂ x f = 0 Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 6 / 77
Relations between ρ , v , f f = ρ v and ∂ t ρ + ∂ x f = 0 � 2 equations = ⇒ necessary a further independent equation 3 unknown variables Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 7 / 77
Relations between ρ , v , f f = ρ v and ∂ t ρ + ∂ x f = 0 � 2 equations = ⇒ necessary a further independent equation 3 unknown variables v = v ( ρ ) LWR: with v : [ 0 , ρ m ] → [ 0 , v m ] decreasing, v ( 0 ) = v m and v ( ρ m ) = 0 v f v m f free max Greenshields: ρ free ρ m ρ ρ m ρ max Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 7 / 77
Resulting system If there is an entry , say sited at x = 0, we have to add the equation f ( ρ ( t , 0 )) = q b ( t ) If there is a restriction (traffic lights, toll gates, construction sites, etc.), say sited at x = x c , we have to add the equation f ( ρ ( t , x c )) ≤ q c ( t ) The resulting system is then conservation ∂ t ρ + ∂ x f ( ρ ) = 0 ( t , x ) ∈ R × ] 0 , + ∞ [ x ∈ ] 0 , + ∞ [ initial datum ρ ( 0 , x ) = ρ o ( x ) entry f ( ρ ( t , 0 ))= q b ( t ) t ∈ ] 0 , + ∞ [ f ( ρ ( t , x c )) ≤ q c ( t ) t ∈ ] 0 , + ∞ [ constraint Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 8 / 77
Resulting system If there is an entry , say sited at x = 0, we have to add the equation f ( ρ ( t , 0 )) = q b ( t ) If there is a restriction (traffic lights, toll gates, construction sites, etc.), say sited at x = x c , we have to add the equation f ( ρ ( t , x c )) ≤ q c ( t ) The resulting system is then ( t , x ) ∈ ] 0 , + ∞ [ 2 conservation ∂ t ρ + ∂ x f ( ρ ) = 0 x ∈ ] 0 , + ∞ [ initial datum ρ ( 0 , x ) = ρ o ( x ) entry f ( ρ ( t , 0 )) = q b ( t ) t ∈ ] 0 , + ∞ [ f ( ρ ( t , x c )) ≤ q c ( t ) t ∈ ] 0 , + ∞ [ constraint Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 8 / 77
Resulting system If there is an entry , say sited at x = 0, we have to add the equation f ( ρ ( t , 0 )) = q b ( t ) If there is a restriction (traffic lights, toll gates, construction sites, etc.), say sited at x = x c , we have to add the equation f ( ρ ( t , x c )) ≤ q c ( t ) The resulting system is then ( t , x ) ∈ ] 0 , + ∞ [ 2 conservation ∂ t ρ + ∂ x f ( ρ ) = 0 x ∈ ] 0 , + ∞ [ initial datum ρ ( 0 , x ) = ρ o ( x ) entry f ( ρ ( t , 0 )) = q b ( t ) t ∈ ] 0 , + ∞ [ f ( ρ ( t , x c )) ≤ q c ( t ) t ∈ ] 0 , + ∞ [ constraint Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 8 / 77
a 1 Introduction to Vehicular Traffic Mathematics 2 Applications to LWR 3 Numerical Examples 4 Crowd Accidents 5 The Model 6 Corridor with One Exit 7 Corridor with Two Exits 8 Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 9 / 77
Conservation law+unilateral constraint ∂ t ρ + ∂ x f ( ρ ) = 0 x ∈ R , t ∈ R + ( CCP ) ρ ( 0 , x ) = ρ o ( x ) x ∈ R f ( ρ ( t , 0 )) ≤ F ( t ) t ∈ R + • f ∈ L ip ([ 0 , R ]; [ 0 , + ∞ [) , f ( 0 ) = f ( R ) = 0, ∃ ρ s.t. f ′ ( ρ ) ( ρ − ρ ) > 0 • ρ o ∈ L ∞ ( R ; [ 0 , R ]) • F ∈ L ∞ ( R + ; [ 0 , f ( ρ )]) f f F F ρ F ˇ ρ ρ F ˆ R ρ ρ F ˇ ρ ρ F ˆ R ρ Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 10 / 77
The Riemann solver R F � ρ l ∂ t ρ + ∂ x f ( ρ ) = 0 if x < 0 ( CRP ) ρ ( 0 , x ) = ρ o ( x ) ρ o ( x ) = ρ r if x > 0 f ( ρ ( t , 0 )) ≤ F Definition (Colombo–Goatin ’07) f If f ( R ( ρ l , ρ r ))( 0 )) ≤ F , then R F ( ρ l , ρ r ) = R ( ρ l , ρ r ) . F Otherwise � R ( ρ l , ˆ ρ F ) if x < 0 R F ( ρ l , ρ r ) = ˇ ρ F ρ ρ F ˆ R ρ ρ F , ρ r ) R (ˇ if x > 0 . ⇒ non classical shock at x = 0 = Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 11 / 77
Entropy conditions Definition (Colombo–Goatin ’07) ρ ∈ L ∞ is a weak entropy solution to (CCP) if ∀ ϕ ∈ C 1 c , ϕ ≥ 0, and ∀ k ∈ [ 0 , R ] � � � ( | ρ − k | ∂ t + Φ( ρ, k ) ∂ x ) ϕ dx dt + | ρ o − k | ϕ ( 0 , x ) dx R + R R � � � 1 − F ( t ) + 2 f ( k ) ϕ ( t , 0 ) dt ≥ 0 f ( ρ ) R + f ( ρ ( t , 0 − )) = f ( ρ ( t , 0 +)) ≤ F ( t ) for a.e. t > 0 where Φ( a , b ) = sgn ( a − b ) ( f ( a ) − f ( b )) and ρ ( t , 0 ± ) the measure theoretic traces implicitly defined by � + ∞ � x c + ε 1 | ρ ( t , x ) − ρ ( t , 0 +) | ϕ ( t , x ) d x d t = 0 ∀ ϕ ∈ C 1 c ( R 2 ; R ) lim ε ε → 0 + 0 x c Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 12 / 77
Entropy conditions Definition (Colombo–Goatin ’07) ρ ∈ L ∞ is a weak entropy solution to (CCP) if ∀ ϕ ∈ C 1 c , ϕ ≥ 0, and ∀ k ∈ [ 0 , R ] � � � ( | ρ − k | ∂ t + Φ( ρ, k ) ∂ x ) ϕ dx dt + | ρ o − k | ϕ ( 0 , x ) dx R + R R � � � 1 − F ( t ) + 2 f ( k ) ϕ ( t , 0 ) dt ≥ 0 f ( ρ ) R + f ( ρ ( t , 0 − )) = f ( ρ ( t , 0 +)) ≤ F ( t ) for a.e. t > 0 where Φ( a , b ) = sgn ( a − b ) ( f ( a ) − f ( b )) and ρ ( t , 0 ± ) the measure theoretic traces implicitly defined by � + ∞ � x c 1 | ρ ( t , x ) − ρ ( t , 0 − ) | ϕ ( t , x ) d x d t = 0 ∀ ϕ ∈ C 1 c ( R 2 ; R ) lim ε ε → 0 + 0 x c − ε Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 12 / 77
Entropy conditions Definition (Colombo–Goatin ’07) ρ ∈ L ∞ is a weak entropy solution to (CCP) if ∀ ϕ ∈ C 1 c , ϕ ≥ 0, and ∀ k ∈ [ 0 , R ] � � � ( | ρ − k | ∂ t + Φ( ρ, k ) ∂ x ) ϕ dx dt + | ρ o − k | ϕ ( 0 , x ) dx R + R R � � � 1 − F ( t ) + 2 f ( k ) ϕ ( t , 0 ) dt ≥ 0 f ( ρ ) R + f ( ρ ( t , 0 − )) = f ( ρ ( t , 0 +)) ≤ F ( t ) for a.e. t > 0 (Cfr. conservation laws with discontinuous flux function: Baiti–Jenssen ’97, Karlsen–Risebro–Towers ’03, Karlsen–Towers ’04, Coclite–Risebro ’05, Andreianov–Goatin–Seguin ’10...) Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 12 / 77
Well–posedness in BV ⇒ TV ( ρ ) explosion constraint = Example ρ x < ( f (ˆ ρ F ) − f ( ρ )) / (ˆ ρ F − ρ ) ρ F ) − f ( ρ )) / (ˆ ρ F − ρ ) < x < 0 ρ F ˆ ( f (ˆ ρ o ( x ) ≡ ρ = ⇒ ρ ( t , x ) = ρ F ˇ 0 < x < ( f (ˇ ρ F ) − f ( ρ )) / (ˇ ρ F − ρ ) ρ F ) − f ( ρ )) / (ˇ ρ F − ρ ) ρ x > ( f (ˇ t ρ F ˆ ρ F ˇ f F ρ ρ ρ ρ x ρ F ˇ ρ ρ F ˆ R ρ Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 13 / 77
Well–posedness in BV ⇒ TV ( ρ ) explosion constraint = We consider the set � � ρ ∈ L 1 : Ψ( ρ ) ∈ BV Ψ( ρ ) = sgn ( ρ − ρ ) ( f ( ρ ) − f ( ρ )) f f ρ ρ ρ ρ R R − f − f (cfr. Temple ’82, Coclite–Risebro ’05...) Padua 28.06.2012 M.D. Rosini (ICM, Warsaw University) 13 / 77
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