Numerical Simulation Of A Mathematical Traffic Flow Model Based On A Nonlinear Velocity-density Function

Numerical Simulation of a Mathematical Traffic Flow Model based on a Nonlinear Velocity-Density Function M H Kabir, M O Gani & L S Andallah

Department of Mathematics Jahangirnagar University Savar Dhaka 1

Traffic problems amenable to scientific analysis  

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How to develop traffic light system, Whether to change a two-way street to a one way street, Where to construct entrances, exits, overpasses How many lanes to be build for a highway Where to build a highway or to develop alternative forms of transportation like trains or trams, etc. 2

Mathematical models of traffic flow There are three types of mathematical models of traffic flow, such as   

Microscopic model Macroscopic (Fluid-dynamic) model Kinetic (Boltzmann) model

We deal with Macroscopic model. 3

Outlines of the work 

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We consider a classical mathematical model which has been approximated by density-velocity relationship We find the exact solution of the model by method of characteristics. The solution is in implicit form and not suitable to incorporate data. Way out: Numerical solution of the model. We develop finite difference scheme and establish the well-posedness and stability condition. Implement the scheme for error estimation. We provide convergent test numerically. We study some qualitative behavior with respect to various parameters.

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Macroscopic Traffic Variables 

Car Density :

in 1 KM ρ (t ,:=x#(Cars) )

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Car Velocity :

v(t , x) := dx / dt

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Car Flow

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Relationship:

:

:= Average #(Cars) q (t , x)passing per hour

q(t , x) = ρ (t , x)v(t , x) 5

Macroscopic model of traffic flow The model is based on continuum hypothesis & law of conservation of mass and fomulates a nonlinear hyperbolic PDE.

∂ρ ∂q ( ρ ) + =0 ∂t ∂x This model was developed by Lighthill, Whitham and Richards (LWR) in 1955. This is also called LWR model. 6

Qualitative density-velocity relationship v( ρ = 0) = vmax , dv ≤ 0, dρ v( ρ = ρ max ) = 0.

v( ρ ) ρ

ρ

  ρ2  ρ3 Non - linear : v( ρ ) = vmax 1 − 2  ⇒ q ( ρ ) = vmax  ρ − 2 ρmax  ρ max  

   7

Qualitative flux-density relationship 

Qualitative Analysis

q( ρ = 0) = 0, v( ρ max ) = 0

⇒ q( ρ max ) = ρ maxv( ρ max ) = 0. q ≥ 0 in 0 < ρ < ρ max dq dv Slope: = v( ρ ) + ρ dρ dρ

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Fundamental diagram

q ( ρ)

ρ 8

Nonlinear PDE as an IVP 

Non-linear

v( ρ )

2 ∂ρ   ∂ ρ  + ρ ⋅ vmax (1 − 2 )  =0    ∂x  ρ max   ∂t ρ(t , x ) = ρ ( x ) 0  0

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Exact Solution of the IVPs 

By the Method of Characteristics 2   3ρ   ρ ( x, t ) = ρ 0 ( x0 ,0) = ρ 0 ( x0 ) = ρ 0  x − Vm ax 1 − 2  t   ρ m ax   

Solution in Implicit Form

Very difficult to formulate ρ ( x ) from data 0 Way Out : Numerical solution of IBVP Simplified exact sol can be used for err est of num sol 10

Numerical solution of IBVP 

Finite Difference Discretization of the IBVP

 ∂ρ ∂q ( ρ )  ∂t + ∂x = 0  with I.C. ρ (t0 , x) = ρ 0 ( x) and B.C. ρ (t, a) = ρ (t ) a  

 ρ Non - linear q ( ρ ) = vmax  ρ − 2 ρmax  3

   11

Explicit Finite Difference Scheme 

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Discretizaton of

∂ρ ∂t

by forward difference

Discretizaton of

∂q ∂x

by backward difference

⇒ρ 

n+ 1 i

[

]

∆t n n = ρ − qi − qi − 1 ; i = 0,, M − 1; n = 1,, N ∆x ρ n i

n +1 i

Stencil ρ in−1

ρ in

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Well-posed-ness & Convergence 

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Well-posed-ness for non-linear case:

Stability condition: ⇒ (*) can be ensured via (**) by

3ρ 2 q′( ρ ) = vmax (1 − 2 ) ≥ 0 ρ max 2 ⇒ ρ max ≥ 3ρ 2 (*) ⇒ q′( ρ ) ≤ vmax

vmax ∆t γ= ≤ 1 (**) ∆x

ρmax =k max ρ0 ( x ), k ≥ 3 x

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Error Estimation

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Convergence

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Numerical Simulation 

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v max= 60.12 km/hour, ρ max =155/km

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Initial density

For 3 Minutes in 1800 time steps Stability condition :

∆x ∆t ≤ v max

c=5

Well-posed condition: 5 KM Highway in 101 grid pts, 1 step =50 m Boundary value density =150/ km

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Computed Traffic density 

Nonlinear

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Computed Traffic Velocity

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Computed Traffic Flow

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Larger vmax & ρ max :Faster Traffic

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Future Interest 

Simulation of Multi-lane Traffic model

∂ρ1 ∂( ρ1v1 ) ρ 2 ρ1 + = 1− 2 ∂t ∂x T2 T1

∂ρ 2 ∂( ρ 2 v2 ) ρ1 ρ 2 + = 2− 1 ∂t ∂x T1 T2 21

References 

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Randall J. LeVeque, “Numerical Methods for Conservation Laws”, second Edition, 1992, Springer. Nicholas Linesch, Michael Perez, “A Nonlinear Traffic Model Dynamics on a One Dimensional lane”, June 2007 Arpad Takaci, “Mathematical and Simulation models of Traffic Flow”, PAMM. Proc. Appl. Math. Mech. 5, 633-634 (2005)/ DOI 10.002/pamm. 200510293, 2005 WILEY- VCH Verlag GmbH & Co. KGaA, Weinheim. Dirk Helbeing, Andreas Greiner, “Modeling and Simulation of Multilane Traffic Flow”, Phys. Rev. E 55, 5498-5508(1997)

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THANK YOU

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