Traffic Flow Problems - Faculteit Wiskunde en … · Traffic Flow Problems Nicodemus Banagaaya...
Transcript of Traffic Flow Problems - Faculteit Wiskunde en … · Traffic Flow Problems Nicodemus Banagaaya...
/centre for analysis, scientific computing and applications
Traffic Flow ProblemsNicodemus Banagaaya
Supervisor : Dr. J.H.M. ten Thije Boonkkamp
October 15, 2009
/centre for analysis, scientific computing and applications
Outline
IntroductionMathematical model derivationGodunov Scheme for the Greenberg Traffic model.Numerical experimentsHigher Order Effects.Shock structureConclusion
/centre for analysis, scientific computing and applications
Outline
IntroductionMathematical model derivationGodunov Scheme for the Greenberg Traffic model.Numerical experimentsHigher Order Effects.Shock structureConclusion
/centre for analysis, scientific computing and applications
Outline
IntroductionMathematical model derivationGodunov Scheme for the Greenberg Traffic model.Numerical experimentsHigher Order Effects.Shock structureConclusion
/centre for analysis, scientific computing and applications
Outline
IntroductionMathematical model derivationGodunov Scheme for the Greenberg Traffic model.Numerical experimentsHigher Order Effects.Shock structureConclusion
/centre for analysis, scientific computing and applications
Outline
IntroductionMathematical model derivationGodunov Scheme for the Greenberg Traffic model.Numerical experimentsHigher Order Effects.Shock structureConclusion
/centre for analysis, scientific computing and applications
Outline
IntroductionMathematical model derivationGodunov Scheme for the Greenberg Traffic model.Numerical experimentsHigher Order Effects.Shock structureConclusion
/centre for analysis, scientific computing and applications
Outline
IntroductionMathematical model derivationGodunov Scheme for the Greenberg Traffic model.Numerical experimentsHigher Order Effects.Shock structureConclusion
/centre for analysis, scientific computing and applications
Outline
IntroductionMathematical model derivationGodunov Scheme for the Greenberg Traffic model.Numerical experimentsHigher Order Effects.Shock structureConclusion
/centre for analysis, scientific computing and applications
IntroductionDefinitionTraffic flow
Figure: copyright British library Board
/centre for analysis, scientific computing and applications
Mathematical model derivationConsider the traffic flow of the cars on a highway with only onelane. Let,
ρ (x , t) be the density of the cars (in vehicles per kilometre),x denote any point along the highway,t denote the time.v (x , t) denote the velocity of cars.Q (x , t) = ρ (x , t) v (x , t) denote the number of cars passthrough x at time t .( flux of the vehicles given by vehiclesper unit time).
The number of cars which are the interval (x1, x2) at time t is∫ x2
x1
ρ (x , t) dx . (1)
Assumption: ρ and Q are continuous functions.
/centre for analysis, scientific computing and applications
Mathematical model derivationConsider the traffic flow of the cars on a highway with only onelane. Let,
ρ (x , t) be the density of the cars (in vehicles per kilometre),x denote any point along the highway,t denote the time.v (x , t) denote the velocity of cars.Q (x , t) = ρ (x , t) v (x , t) denote the number of cars passthrough x at time t .( flux of the vehicles given by vehiclesper unit time).
The number of cars which are the interval (x1, x2) at time t is∫ x2
x1
ρ (x , t) dx . (1)
Assumption: ρ and Q are continuous functions.
/centre for analysis, scientific computing and applications
Mathematical model derivationConsider the traffic flow of the cars on a highway with only onelane. Let,
ρ (x , t) be the density of the cars (in vehicles per kilometre),x denote any point along the highway,t denote the time.v (x , t) denote the velocity of cars.Q (x , t) = ρ (x , t) v (x , t) denote the number of cars passthrough x at time t .( flux of the vehicles given by vehiclesper unit time).
The number of cars which are the interval (x1, x2) at time t is∫ x2
x1
ρ (x , t) dx . (1)
Assumption: ρ and Q are continuous functions.
/centre for analysis, scientific computing and applications
Mathematical model derivationConsider the traffic flow of the cars on a highway with only onelane. Let,
ρ (x , t) be the density of the cars (in vehicles per kilometre),x denote any point along the highway,t denote the time.v (x , t) denote the velocity of cars.Q (x , t) = ρ (x , t) v (x , t) denote the number of cars passthrough x at time t .( flux of the vehicles given by vehiclesper unit time).
The number of cars which are the interval (x1, x2) at time t is∫ x2
x1
ρ (x , t) dx . (1)
Assumption: ρ and Q are continuous functions.
/centre for analysis, scientific computing and applications
Mathematical model derivationConsider the traffic flow of the cars on a highway with only onelane. Let,
ρ (x , t) be the density of the cars (in vehicles per kilometre),x denote any point along the highway,t denote the time.v (x , t) denote the velocity of cars.Q (x , t) = ρ (x , t) v (x , t) denote the number of cars passthrough x at time t .( flux of the vehicles given by vehiclesper unit time).
The number of cars which are the interval (x1, x2) at time t is∫ x2
x1
ρ (x , t) dx . (1)
Assumption: ρ and Q are continuous functions.
/centre for analysis, scientific computing and applications
Mathematical model derivationConsider the traffic flow of the cars on a highway with only onelane. Let,
ρ (x , t) be the density of the cars (in vehicles per kilometre),x denote any point along the highway,t denote the time.v (x , t) denote the velocity of cars.Q (x , t) = ρ (x , t) v (x , t) denote the number of cars passthrough x at time t .( flux of the vehicles given by vehiclesper unit time).
The number of cars which are the interval (x1, x2) at time t is∫ x2
x1
ρ (x , t) dx . (1)
Assumption: ρ and Q are continuous functions.
/centre for analysis, scientific computing and applications
Mathematical model derivationConsider the traffic flow of the cars on a highway with only onelane. Let,
ρ (x , t) be the density of the cars (in vehicles per kilometre),x denote any point along the highway,t denote the time.v (x , t) denote the velocity of cars.Q (x , t) = ρ (x , t) v (x , t) denote the number of cars passthrough x at time t .( flux of the vehicles given by vehiclesper unit time).
The number of cars which are the interval (x1, x2) at time t is∫ x2
x1
ρ (x , t) dx . (1)
Assumption: ρ and Q are continuous functions.
/centre for analysis, scientific computing and applications
Mathematical model derivation
Figure: Derivation of the conservative law,where a = Q(x1, t),b = Q(x2, t)
/centre for analysis, scientific computing and applications
Mathematical model derivation
ddt
∫ x2
x1
ρ (x , t) dx = Q (x1, t)−Q (x2, t) .
Then we have, ∫ x2
x1
[∂ρ
∂t+
∂
∂xQ(x , t)
]dx = 0 (2)
∂ρ
∂t+
∂
∂x(ρv) = 0. (3)
/centre for analysis, scientific computing and applications
Mathematical model derivation
ddt
∫ x2
x1
ρ (x , t) dx = Q (x1, t)−Q (x2, t) .
Then we have, ∫ x2
x1
[∂ρ
∂t+
∂
∂xQ(x , t)
]dx = 0 (2)
∂ρ
∂t+
∂
∂x(ρv) = 0. (3)
/centre for analysis, scientific computing and applications
Mathematical model derivation
Since x1, x2 ∈ R, t1, t2 > 0 are arbitrary, we conclude that
∂ρ
∂t+
∂
∂x(ρv ) = 0, (4)
with Initialρ (x ,0) = ρ0 (x),∀x ∈ R
Let flux Q = Q(ρ) , then Q(ρ) = ρV (ρ).Thus equation ( 4) can be written as
∂ρ
∂t+ c(ρ)
∂ρ
∂x= 0, where c(ρ) = Q′(ρ).
/centre for analysis, scientific computing and applications
Mathematical model derivation
Figure: Flow density curve in the traffic flow
Traffic modelsLighthill-Whitham-Richards model
v(ρ) = vmax
(1− ρ
ρmax
), 0 ≤ ρ ≤ ρmax . (5)
/centre for analysis, scientific computing and applications
Mathematical model derivation
Greenberg modelv(ρ) = a log ρmax
ρ , 0 < ρ ≤ ρmax .
where a is in kilometres per hour.we are going to solve this model numerically using Godunovscheme .
/centre for analysis, scientific computing and applications
Godunov Scheme for the Greenberg model
Godunov Scheme for Nonlinear Conservation lawsconsider the initial value problem
∂u∂t
+∂f (u)
∂x= 0, x ∈ R, t > 0, (6a)
u (x ,0) = u0 (x),∀x ∈ R. (6b)
Let us introduce a control volumes or cells Vj as follows:
Vj =[xj− 1
2, xj+ 1
2
), xj+ 1
2=(xj + xj+1
), j = 0,±1,±2, . . . ,
(7)
/centre for analysis, scientific computing and applications
Godunov Scheme for the Greenberg model
Associated with unj is the function u(x , t), defined as the
solution of the following initial value problem with piecewiseconstant initial at t = tn:
∂u∂t
+∂f (u)
∂x= 0, x ∈ R, t > tn, (8a)
u(x , tn) = un
j , x ∈ Vj (j = 0,±1,±2, . . . ). (8b)
To compute the numerical flux;
u(x , tn) =
unj , if x < xj+ 1
2,
unj+1, if x > xj+ 1
2.
/centre for analysis, scientific computing and applications
Godunov Scheme for the Greenberg model
The solution of this Riemann problem is a similarity of the form
u(x , t) = uR(η; unj ,u
nj+1), η =
x − xj+ 12
t − tn . (9)
Since η = 0 for x = xj+ 12, the computation of the numerical flux
we simply find
F (unj ,u
nj+1) = f (uR(0; un
j ,unj+1)). (10)
Thus the Godunov scheme is given by
un+1j = un
j −4t4x
(F (un
j ,unj+1)− F (un
j−1,unj )), (11)
with the numerical flux as defined in (10).
/centre for analysis, scientific computing and applications
Godunov Scheme for the Greenberg model
And the stability condition of the method is given by
4t4x
max |f ′(unj )| ≤ 1, (j = 0,±1,±2, . . . ).
Implementation of the scheme to the Greenberg model
x∗ =xL, t∗ =
tL/a
=atL, q(x∗, t∗) =
ρ(x∗, t∗)ρmax
, ρ > 0. (12)
/centre for analysis, scientific computing and applications
Godunov Scheme for the Greenberg model
Where L the characteristic distance along the highway. Thus qsatisfies the conservation law
∂q∂t∗
+∂f (q)
∂x∗= 0 (13)
with the flux function f (q) = −q log q, q > 0.The corresponding Riemann problem has the followingpiecewise constant initial condition
q(x ,0) =
{ql , if x < 0,qr , if x > 0.
/centre for analysis, scientific computing and applications
Godunov Scheme for the Greenberg model
The Riemann Problem has two kinds of solutions;b(q) = f ′(q) = − log q − 1,
Case 1: b(ql) > b(qr ) =⇒ ql < qr .
q(x , t) =
{ql if x/t < s,qr if x/t > s.
where s = f (ql )−f (qr )ql−qr
.
Case 2: b(ql) < b(qr ) =⇒ ql > qr .
q(x , t) =
ql if x/t < b(ql),
e−(1+ xt ) ifb(ql) < x/t < b(qr ),
qr if x/t > b(qr ).
/centre for analysis, scientific computing and applications
Godunov Scheme for the Greenberg model
In order to implement the Godunov scheme lets us consider theRiemann problem given below
∂q∂t
+∂
∂x(−q log q) = 0, x ∈ R, t > tn, (14)
q(x , tn) =
qnj if x < xj+ 1
2,
qnj+1 if x > xj+ 1
2.
For the similarity solution of this problem we also distinguishtwo cases.
/centre for analysis, scientific computing and applications
Godunov Scheme for the Greenberg model
If qnj < qn
j+1,
qR(η; qnj ,q
nj+1) =
{qn
j if η < snj ,
qnj+1 if η > sn
j .
where snj =
qj+1 log qj+1−qj log qjqj−qj+1
.if qn
j > qnj+1,
qR(η; qnj ,q
nj+1) =
qn
j if η < b(qnj ),
η if b(qnj ) < η < b(qn
j+1),
qnj+1 if η > b(qn
j+1).
/centre for analysis, scientific computing and applications
Godunov Scheme for the Greenberg model
For the numerical fluxF (un
j ,unj+1) = −qR(0; qn
j ,qnj+1) log qR(0; qn
j ,qnj+1);
If qnj < qn
j+1, then
F (qnj ,q
nj+1) =
{−qn
j log qnj if sn
j > 0,−qn
j+1 log qnj+1 if sn
j < 0.
If qnj > qn
j+1, then
F (qnj ,q
nj+1) =
−qn
j log qnj if qn
j < e−1,
e−1 if qnj+1 < e−1 < qn
j ,
−qnj+1 log qn
j+1 if qnj+1 < e−1.
/centre for analysis, scientific computing and applications
Numerical experiments
In this case the initial condition for q is given by
q(x ,0) =
{0.1 if x < 0.2,1 if x > 0.2.
/centre for analysis, scientific computing and applications
Numerical experiments
In this case the initial condition for q is given by
q(x ,0) =
{0.5 if x < 0.8,0.1 if x > 0.8.
/centre for analysis, scientific computing and applications
Higher Order Effects
q = q(ρx , ρ).Assumptions:q = Q(ρ)− νρx , v = V (ρ)− ν
ρρx .There are two additional effects one may wish to include thetheory: Diffusion of waves, and Response Time. To incorporatethese effects,
ρt + c(ρ)ρx = νρxx .
DvDt
= vt + vvx = −1τ
[v − V (ρ) +
ν
ρρx
]. (15)
where τ - measure of the response time. The equation (15)is to be solved together with the conservation equation.
/centre for analysis, scientific computing and applications
Higher Order Effects
If equation (15) and the conservation equation are linearized forsmall perturbations about ρ = ρ0, v = v0 = V (ρ0), bysubstituting
ρ = ρ0 + r , v = v0 + w ,
and retaining only the first powers of r and w , we have
τ (wt + v0wx ) = −[w − V ′(ρ0)r +
ν
ρ0rx
],
rt + v0rx + ρ0wx = 0.
/centre for analysis, scientific computing and applications
Higher Order Effects
The kinematic wave speed is
c0 = ρ0V ′(ρ0) + V (ρ0);
hence V ′(ρ0) = −(v0 − c0)/ρ0. Introducing this expression andthen eliminating w , we have
∂r∂t
+ c0∂r∂x
= ν∂2r∂x2 − τ
(∂
∂t+ v0
∂
∂x
)2
r . (16)
/centre for analysis, scientific computing and applications
Higher Order Effects
The effect of the finite response time τ is more complicated butcan be approximated as follows.
∂
∂t≈ −c0
∂
∂x. (17)
If this approximation is used in the right hand side of (16) , theequation reduces to
∂r∂t
+ c0∂r∂x
=[ν − (v0 − c0)2τ
] ∂2r∂x2 . (18)
/centre for analysis, scientific computing and applications
Shock structure
We need a steady profile solution of (15) and the conservationequation with
ρ = ρ(X ), v = v(X ), X = x − Ut ,
where U is the constant translational velocity.Then,
−Uρx + (vρ)x = 0 (19)
and may be integrated to
ρ(U − v) = A, where A is a constant. (20)
/centre for analysis, scientific computing and applications
Shock structureEquation (15) becomes
τρ(v − U)vx + νρx + ρv −Q(ρ) = 0. (21)
Since v = U − A/ρ,(ν − A2
ρ2 τ
)ρx = Q(ρ)− ρU + A. (22)
For τ 6= 0, the possibility that ν − A2τ/ρ2 may vanish introducesthe new effects.We are interested in the solution curves between ρ1 at X = +∞and ρ2 at X = −∞. For traffic flow c′(ρ) = Q′′(ρ) < 0, soρ2 < ρ1 and the right hand side of (22) is positive forρ2 < ρ < ρ1.
/centre for analysis, scientific computing and applications
Shock structureIf ν − A2τ/ρ2 remains positive in this range, then ρx > 0 and wehave a smooth profile as in the figure below. In view of (20), thecondition for ν − A2τ/ρ2 to remain positive may be written
ν > (v − U)2τ, that is v −√ν/τ < U < v −
√ν/τ . (23)
Figure: Continuous wave structure
/centre for analysis, scientific computing and applications
Conclusions
The traffic model is based on first order approximation, andhence the original assumptions are not goodapproximation.Include higher order effects and shock structure in order tohave a better solutions.
/centre for analysis, scientific computing and applications
Thank you