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

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On stability of special solutions for quasi-linear equations

f 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

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Main parameters ρ(x, t) — flow density, v(x, t) — flow velocity, q(x, t) — flow rate. x

Moscow, MRR, north.

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Interface of the program “Cars”

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Ring road

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Interface of the program “Cars”

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Jam movement on ring road (N = 140 cars, L = 5 km, T = 3 hours)

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Tendency to stable condition

L = 20 km, T = 6 h.

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Tendency to stable condition

L = 20 km, T = 12 h.

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Fundamental diagram q = Q(ρ), Q is concave function, Q(ρ) > 0, 0 < ρ < ρmax, Q(0) = 0, Q(ρmax) = 0.

Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 9 / 46

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Quasi-linear differential equation Main equation of road traffic: ∂ρ ∂t + ∂Q(ρ) ∂x = 0, ρ = ρ(x, t). Integral identity: d dt

β

α

ρ(x, t)dx = Q(ρ(α, t)) − Q(ρ(α, t)), for a.e. α, β ∈ R, t 0.

Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 10 / 46

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Cauchy problem    ∂ρ ∂t + ∂Q(ρ) ∂x = 0, x ∈ R, t > 0, ρ(x, 0) = µ(x), with initial function 0 µ(x) ρmax, 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

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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

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Analytical representation Q(ρ) =

k1 ρ,

0 ρ ρ∗, k2 (ρmax − ρ), ρ∗ ρ ρmax. k1 = qmax ρ∗ , k2 = qmax ρmax − ρ∗.

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Typical solutions

1 If µ(x) ρ∗ on R, then

ρ(x, t) = µ(x − k1t) (forward wave).

2 If µ(x) ρ∗ on R, then

ρ(x, t) = µ(x + k2t) (backward wave).

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Forward wave

for t = 0

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Forward wave

for t = 0.3

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Forward wave

for t = 0.6

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Backward wave

for t = 0

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Backward wave

for t = 0.7

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Backward wave

for t = 1.4

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Backward wave

for t = 2.1

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Combined solution: depression wave

for t = 0

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Combined solution: depression wave

for t = 0.2

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Combined solution: depression wave

for t = 0.4

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Combined solution: depression wave

for t = 0.6

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Combined solution: shock wave

for t = 0

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Combined solution: shock wave

for t = 0.4

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Combined solution: shock wave

for t = 0.8

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Combined solution: shock wave

for t = 1.2

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Hugoniot condition Speed of the shock wave is ψ′(t) = Q(ρright) − Q(ρleft) ρright − ρleft , where ψ(t) is a boundary of the shock wave. Integral identity: d dt

β

α

ρ(x, t)dx = Q(ρ(α, t)) − Q(ρ(α, t)).

Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 30 / 46

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Ring road idea

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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 M ≡

L

µ(x)dx.

Two cases:

1

M < ρ∗L (almost free movement);

2

M > ρ∗L (crowded traffic).

Podoroga A. V., Tikhonov I. V. (MSU) Simulation of traffic flow Dolgoprudny, 2016 32 / 46

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Stability of solutions on t → ∞

M ≡

L

µ(x)dx.

Theorem There is an alternative.

1 If M < Lρ∗, then ∃ t∗ > 0 such that

ρ(x, t) = f(x − k1 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 + k2 t), ∀ t t∗, where ρ∗ g(s) ρmax for ∀s ∈ R.

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Combinatorial result

Lemma In every set of real numbers a1, a2, . . . , an, with a1 + a2 + . . . + an > 0, there exists a dominant element ak, such as ak > 0, ak + ak+1 > 0, ak + ak+1 + ak+2 > 0, . . . . . . . . . ak + ak+1 + ak+2 + . . . + ak+n−1 > 0. Index changes cyclically.

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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