On stability of special solutions for quasi-linear equations of - PowerPoint PPT Presentation

On stability of special solutions for quasi-linear equations of traffic flow Podoroga Anastasia Vladimirovna, Tikhonov Ivan Vladimirovich CMC MSU department of Mathematical Physics Dolgoprudny, 12 Sep. 2015 – 15 Sep. 2016 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 1 / 46

Main parameters ρ ( x, t ) — flow density, v ( x, t ) — flow velocity, q ( x, t ) — flow rate . x Moscow, MRR, north. Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 2 / 46

Interface of the program “Cars” Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 3 / 46

Ring road Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 4 / 46

Interface of the program “Cars” Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 5 / 46

Jam movement on ring road ( N = 140 cars, L = 5 km, T = 3 hours) Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 6 / 46

Tendency to stable condition L = 20 km, T = 6 h. Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 7 / 46

Tendency to stable condition L = 20 km, T = 12 h. Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 8 / 46

Fundamental diagram q = Q ( ρ ) , Q is concave function , Q ( ρ ) > 0 , 0 < ρ < ρ max , Q (0) = 0 , Q ( ρ max ) = 0 . Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 9 / 46

Quasi-linear differential equation Main equation of road traffic: ∂t + ∂Q ( ρ ) ∂ρ = 0 , ρ = ρ ( x, t ) . ∂x Integral identity: β d � ρ ( x, t ) dx = Q ( ρ ( α, t )) − Q ( ρ ( α, t )) , dt α for a.e. α, β ∈ R , t � 0 . Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 10 / 46

Cauchy problem  ∂ρ ∂t + ∂Q ( ρ ) = 0 , t > 0 , x ∈ R ,  ∂x ρ ( x, 0) = µ ( x ) ,  with initial function 0 � µ ( x ) � ρ max , x ∈ R . Technical solution of this problem is ρ = µ ( x − k ( ρ ) t ) , where k = k ( ρ ) = Q ′ ( ρ ) is a slope of the characteristic. Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 11 / 46

Nagel–Schreckenberg diagram K. Nagel, M. Schreckenberg. A cellular automaton model for freeway traffic . Journal de Physique I France. 1992. Vol. 2. No 12. P. 2221–2229. Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 12 / 46

Analytical representation � 0 � ρ � ρ ∗ , k 1 ρ, Q ( ρ ) = ρ ∗ � ρ � ρ max . k 2 ( ρ max − ρ ) , k 1 = q max q max ρ ∗ , k 2 = ρ max − ρ ∗ . Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 13 / 46

Typical solutions 1 If µ ( x ) � ρ ∗ on R , then ρ ( x, t ) = µ ( x − k 1 t ) (forward wave). 2 If µ ( x ) � ρ ∗ on R , then ρ ( x, t ) = µ ( x + k 2 t ) (backward wave). Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 14 / 46

Forward wave for t = 0 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 15 / 46

Forward wave for t = 0 . 3 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 16 / 46

Forward wave for t = 0 . 6 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 17 / 46

Backward wave for t = 0 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 18 / 46

Backward wave for t = 0 . 7 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 19 / 46

Backward wave for t = 1 . 4 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 20 / 46

Backward wave for t = 2 . 1 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 21 / 46

Combined solution: depression wave for t = 0 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 22 / 46

Combined solution: depression wave for t = 0 . 2 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 23 / 46

Combined solution: depression wave for t = 0 . 4 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 24 / 46

Combined solution: depression wave for t = 0 . 6 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 25 / 46

Combined solution: shock wave for t = 0 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 26 / 46

Combined solution: shock wave for t = 0 . 4 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 27 / 46

Combined solution: shock wave for t = 0 . 8 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 28 / 46

Combined solution: shock wave for t = 1 . 2 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 29 / 46

Hugoniot condition Speed of the shock wave is ψ ′ ( t ) = Q ( ρ right ) − Q ( ρ left ) , ρ right − ρ left where ψ ( t ) is a boundary of the shock wave. Integral identity: β d � ρ ( x, t ) dx = Q ( ρ ( α, t )) − Q ( ρ ( α, t )) . dt α Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 30 / 46

Ring road idea Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 31 / 46

Specifics of the ring road L is length of the ring road. All functions are L -periodic for x . The amount of cars on the road is main value L � µ ( x ) dx. M ≡ 0 Two cases: M < ρ ∗ L (almost free movement); 1 M > ρ ∗ L (crowded traffic). 2 Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 32 / 46

Stability of solutions on t → ∞ L � µ ( x ) dx. M ≡ 0 Theorem There is an alternative. 1 If M < Lρ ∗ , then ∃ t ∗ > 0 such that ρ ( x, t ) = f ( x − k 1 t ) , ∀ t � t ∗ , where 0 � f ( s ) � ρ ∗ for ∀ s ∈ R . 2 If M > Lρ ∗ , then ∃ t ∗ > 0 such that ρ ( x, t ) = g ( x + k 2 t ) , ∀ t � t ∗ , where ρ ∗ � g ( s ) � ρ max for ∀ s ∈ R . Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 33 / 46

Combinatorial result Lemma In every set of real numbers a 1 , a 2 , . . . , a n , with a 1 + a 2 + . . . + a n > 0 , there exists a dominant element a k , such as a k > 0 , a k + a k +1 > 0 , a k + a k +1 + a k +2 > 0 , . . . . . . . . . a k + a k +1 + a k +2 + . . . + a k + n − 1 > 0 . Index changes cyclically. Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 34 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 35 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 36 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 37 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 38 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 39 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 40 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 41 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 42 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 43 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 44 / 46

Example Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 45 / 46

Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 46 / 46