J Shanghai Univ (Engl Ed), 2009, 13(1): 6–11 Digital Object Identifier(DOI): 10.1007/s 11741-009-0102-3
Application of singular perturbation method in analyzing traffic density waves SHEN Fei-ying ()1 , GE Hong-xia ()1 , LEI Li (
)2
1. Faculty of Science, Ningbo University, Ningbo 315211, Zhejiang, P. R. China 2. School of Energy and Power Engineering, Shandong University, Jinan 250061, P. R. China (Communicated by NI Ming-kang) Abstract Car following model is one of microscopic models for describing traffic flow. Through linear stability analysis, the neutral stability lines and the critical points are obtained for the different types of car following models and two modified models. The singular perturbation method has been used to derive various nonlinear wave equations, such as the Kortewegde-Vries (KdV) equation and the modified Korteweg-de-Vries (mKdV) equation, which could describe different density waves occurring in traffic flows under certain conditions. These density waves are mainly employed to depict the formation of traffic jams in the congested traffic flow. The general soliton solutions are given for the different types of car following models, and the results have been used to the modified models efficiently. Keywords traffic flow, singular perturbation method, car following models, Korteweg-de-Vries (KdV) equations, modified Korteweg-de-Vries (mKdV) equations 2000 Mathematics Subject Classification 34E20
Introduction It is hardly necessary to emphasize the importance of transportation in our daily life. Traffic jams, as characteristics of the complex behavior of vehicular traffic, have been treated frequently as the limiting case of propagating density waves induced by the interaction between vehicles[1] . The singular perturbation method has been used to derive the different nonlinear wave equations such as the Burgers, Korteweg-de-Vries (KdV), and modified Korteweg-de-Vries (mKdV) equations, to describe the corresponding density waves respectively. Recently, the singular perturbation method is mainly applied in the microscopic car following model. In the microscopic models of vehicular traffic, attention is paid explicitly to each individual vehicle, each of which is represented by a particle, and the nature of the interactions among these particles is determined by the way the vehicles influence each other[2] . Therefore, vehicular traffic, modeled as a system of interacting particles driven far from equilibrium, is of current interest of researchers. The mKdV equation from the optimal velocity model was deduced by Komatsu, et al.[3] , and later Nagatani[4−6]. More recently, the author and her
co-workers have made many efforts in this aspect. In this paper, we will pay attention to the application of singular perturbation method in the microscopic car following models. We review the results in our previous study, and hope that the conclusion would be useful for mathematics researchers in this field.
1 Simplest car following models 1.1
Models
Firstly we outline the three simplest versions of car following models[7] . The vehicle motion in Model A proposed by Newell[8] and Whitham[9] is described by the following differential equation: dxj (t + τ ) = V (Δxj (t)), dt
(1)
where xj (t + τ ) is the position of car j at time t + τ ; Δxj (t) ≡ xj+1 (t) − xj (t) is the headway between car j and car j + 1 at time t; τ is introduced to denote the delay time within which the car velocity reaches the dx (t+τ ) is adjusted optimal velocity. The car velocity j dt according to the headway Δxj (t). Making the Taylor expansion of (1) and omitting the
Received Sept.10, 2008; Revised Nov.3, 2008 Project supported by the National Basic Research Program of China (Grant No.2006CB705500), the National Natural Science Foundation of China (Grant Nos.10532060, 10602025, 10802042), and the Natural Science Foundation of Ningbo (Grant No.2007A610050) Corresponding author GE Hong-xia, PhD, Assoc. Prof., E-mail:
[email protected]
J Shanghai Univ (Engl Ed), 2009, 13(1): 6–11
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higher order term of τ , we obtain the Model B[10] : d2 xj (t) dxj (t) = a V (Δx (t)) − , (2) j dt2 dt where a is the sensitivity of a driver, and its relation with τ is a = 1/τ . By transforming time derivatives to asymmetric forward differences, (1) could be rewritten as the difference equation, called Model C[11] :
3.0
(3)
The optimal velocity V is selected in a way similar to that proposed by Bando, et al.[10] : V (Δxj (t)) =
vmax [tanh (Δxj (t) − hc ) + tanh (hc )], (4) 2
2.5 2.0
Models A and B
1.5
where hc is the safety distance and vmax is the maximum velocity (for simplicity, vmax = 2).
1.2
Stable Model C
Sensitivity
xj (t + 2τ ) − xj (t + τ ) = τ V (Δxj (t)).
headway is always linearly stable for a > ac , while the uniform state in the neighborhood of hc is unstable for a < ac . The apex of each line corresponds to the critical point. The traffic flow is stable above the neutral stability line and traffic jams will not appear, while below the line, traffic flow is unstable and the density waves emerge.
1.0 2.5
Unstable 3.0
3.5
Linear stability theory
Linear stability analysis can be conducted for the car following models. It is obvious that the vehicles move with the uniform headway h and the optimal velocity V (h) is the steady state solution for (1)∼(3), which is given as x0j (t) = hj + V (h)t with h =
L , N
(5)
where N is the total number of vehicles, and L is the road length. Suppose yj (t) to be a small deviation from the steady-state solution x0j (t) : xj (t) = x0j (t) + yj (t). Substituting it into (1)∼(3), linearizing the resultant equations and expanding yj in the Fourier-modes give yj (t) = Cexp (ikj + zt), where z = z1 (ik)+ z2 (ik)2 + · · · . Finally we have the first- and second-order terms of coefficients in the expression of z respectively as follows: V (h) − τ z12 2 for Models A and B;
z1 = V (h) and z2 =
z1 = V (h) and z2 =
(6)
V (h) 3 2 − τ z1 2 2
for Model C.
(7)
4.0 4.5 Headway
5.0
Fig.1
Phase diagram in the headway-sensitivity space (vmax = 2.0, hc = 4.0)
1.3
KdV equation
For later convenience, we rewrite (1)∼(3) in their difference forms: dΔxj (t + τ ) = V (Δxj+1 (t)) − V (Δxj (t)), dt d2 Δxj (t) = a V (Δxj+1 (t)) − V (Δxj (t)) dt2 dΔxj (t) − , dt Δxj (t + 2τ ) − Δxj (t + τ ) = τ [V (Δxj+1 (t)) − V (Δxj (t))].
X = ε(j + bt), and T = ε3 t,
0 < ε 1,
where b is a constant to be determined. Let
1 2V (h) 1 τ> 3V (h)
Δxj (t) = h + ε2 R(X, T ).
for Models A and B, for Model C.
(8)
From Fig.1, we see that there exist the critical points (hc , ac ) of the neutral stability lines for the three models, such that the uniform state irrespective of vehicle
(9)
(10)
(11)
We apply the reductive perturbation method to (9)∼(11) and focus on the system behavior near the neutral stability lines. With such treatment, the nature of soliton solutions can be described by the KdV equation. We introduce slow scales for space variable j and time variable t[12−13] , and define the slow variables X and T as
For small disturbances with long wavelengths, the uniform traffic flow is unstable in the conditions that τ>
5.5
(12)
(13)
Substituting (12) and (13) into (9)∼(11) and making the Taylor expansions to the sixth order of ε lead to the following unified expression: 2 3 ε3 f1 ∂X R + ε4 f2 ∂X R + ε5 (∂T R + f3 ∂X R − f4 ∂X R2 ) 4 2 R − f7 ∂X R2 ) = 0, + ε6 (f5 ∂X ∂T R + f6 ∂X
(14)
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J Shanghai Univ (Engl Ed), 2009, 13(1): 6–11
where
(14) result in the simplified equation:
V (h) =
dV (Δxj ) d2 V (Δxj ) |Δxj =h , and V (h) = |Δxj =h . dΔxj dΔx2j
Model
f1
f2
A
b−V
b2 τ − V /2
B
b−
V
b2 τ
b−
V
3b2 τ /2
C
Table 2
−V
−V
T =
√ m1 T ,
√ X = − m1 X ,
R=
1 R. m2
f3
f4
f5
f6
f7
b3 τ 2 /2 − V /6
V /2
2bτ
b4 τ 3 /6 − V /24
V /4
−V 7b3 τ 2 /6
/6
−V
Coefficients mi of the three models and corresponding amplitudes A
Model
m1
m2
m3
m4
m5
A
V /24
V
V /2
V /48
V /4
3V
B
V /6
V
V /2
V /8
V /4
14V /3
C
V /27
V
V /2
V /54
V /4
3V
A
Thus (15) turns into 1 3 2 − m3 ∂X ∂T R + ∂X R + R ∂X R + ε √ R m1 m4 4 m5 2 2 + ∂ R + ∂ R = 0. m1 X m2 X
V /2 /6
V /2
2bτ
−V
3bτ
5b4 τ 3 /8
/24
−V
V /4 /24
V /4
In Table 2, the results of Model B calculated by the general solution (19) is in agreement with those given in [5]. The general soliton solution of the headway is τ τ A A 2 1− sech 1− Δxj (t) = h + m2 τs 12m1 τs
τ A t . · j + V (h) + 1− (20) 3 τs The soliton wave is shown in Fig.2.
(16)
The perturbed term in (16) gives the condition of selecting a unique member from the continuous family of KdV solitons. Next, supposing that R (X , T ) = R0 (X , T ) + εR1 (X , T ), we take into account the O(ε) correction. To determine the selected value of the amplitude A for the soliton solution (17), it is necessary to consider the solvability condition[14] : +∞ (R0 , M [R0 ]) ≡ dX R0 M [R0 ] = 0, (18) −∞
M [R0 ]
where is the perturbation terms in (16). Through integration, we obtain the selected value A: (19)
Headway
Equation (16) is the KdV equation with an O(ε) correction term. First, we ignore the O(ε) term in (16), and get the KdV equation with the soliton solution: A A 2 X − T . R0 (X , T ) = A sech (17) 12 3
21m1 m2 m3 . A= −5m2 m4 + 24m1 m5
(15)
Coefficients fi of the three models
/2 /2
4 2 R + m5 ∂X R2 ) = 0. + m4 ∂X
The coefficients are given in Table 2. In order to obtain the standard KdV equation with higher order correction, we make the following transformations for (15):
For simplicity, in Tables 1∼2 we denote V (h) = V , and V (h) = V . Near the neutral stability line τs = 1/(2V (h)) for Models A and B; τs = 1/(3V (h)) for Model C; τ = (1 − ε2 )τs . Taking b = V (h) and eliminating the third- and fourth-order terms of ε from Table 1
3 2 R − m2 R∂X R) + ε6 (−m3 ∂X R ε5 (∂T R − m1 ∂X
6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 0
t
5
10
15 20 25 Car number
Fig.2
1.4
30
35
40
Soliton wave
mKdV equation
We use the slow variables X and T defined as before. From the parabolic form of the neutral stability line near the critical point, it is expected that the amplitude of the density wave scales a ∝ ε[3] . The headway is set as Δxj (t) = hc + εR(X, T ).
(21)
Using the analogous method, we obtain 2 3 ε2 k1 ∂X R + ε3 k2 ∂X R + ε4 (∂T R + k3 ∂X R − k4 ∂X R3 ) 4 2 R − k7 ∂X R3 ) = 0, + ε5 (k5 ∂X ∂T R + k6 ∂X
(22)
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In Tables 3∼4 we denote V (hc ) = V , and V (hc ) = V . Near the critical point (hc , ac ), τ = (1 + ε2 )τc , taking b = V (hc ), and eliminating the second- and thirdorder terms of ε from (22) lead to the simplified equation:
where V (hc ) =
dV (Δxj ) , dΔxj Δxj =hc
and V (hc ) =
d3 V (Δxj ) . dΔx3j Δxj =hc
3 2 4 ε4 [∂T R − g1 ∂X R + g2 ∂X R3 ] + ε5 [g3 ∂X R + g4 ∂X R 2 R3 ] = 0. + g5 ∂X
Coefficients ki of the three models
Table 3 k1
Model
k2
b2 τ
b3 τ 2 /2
−V
k4 /6
k5
V /6
2bτ
k6 b4 τ 3 /6
b−V
b−V
b2 τ − V /2
−V /6
V /6
2bτ
−V /24
V /12
C
b−V
3b2 τ /2 − V /2
7b3 τ 2 /6 − V /6
V /6
3bτ
5b4 τ 3 /8 − V /24
V /12
T = g1 T, R =
g1 R, g2
we have the standard mKdV equation with an O(ε) correction term: 1 3 3 2 4 g3 ∂X ∂T R − ∂X R + ∂X R + ε R + g4 ∂X R g1 g1 g5 2 3 = 0. (24) + ∂ R g2 X We ignore the O(ε) term in (24), and get the mKdV equation with the kink-antikink soliton solution: √ c R0 (X, T ) = c tanh (X − cT ). (25) 2 The next step is the same as the process of deriving the amplitude A for the KdV equation. We obtain the selected velocity c: c=
5g2 g3 . 2g2 g4 − 3g1 g5
(26)
In Table 4, the results of Model C calculated by the general equation (26) are consistent with those given by Nagatani for the case of γ = 0 in [4]. Table 4
Coefficients gi of the three models and the corresponding values of c
Model
g1
g2
g3
g4
A
V /24
−V /6
V /2
V
B
V /6
−V /6
V /2
V /8
V /12
5
C
V /27
−V /6
V /2
V /54
V /12
27
/48
g5 V
/12
c 24
The general kink-antikink soliton solution of the
V
/12
B
Making the following transformations for (23):
−V
k7 /24
A
headway (V = 1, V = −2) is g1 c τ c τ Δxj (t) = hc + − 1 tanh −1 g2 τc 2 τc τ
−1 t . · j + 1 − cg1 τc
(27)
The kink-antikink soliton wave is shown in Fig.3.
Headway
−V
k3 /2
(23)
t
6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 10
20
Fig.3
Kink-antikink soliton wave
30 40 Car number
50
60
2 Extended models For describing the traffic flow with the navigation, we put forward two different models, which is a velocitydifference model (TVDM)[15] , and a forward looking anticipation optimal velocity model (FLAM)[16−17] . The governing equations are as follows:
dxj (t) d2 xj (t) = a V (Δx (t)) − j dt2 dt + λ(pΔvn + (1 − p)Δvn+1 ), (28) where λ is a sensitivity coefficient different from a, and p the weighting value. The dynamic equation of FLAM reads dxj (t + τ ) = V (Δxj (t), Δxj+1 (t), · · · , dt Δxj+n−1 (t)),
(29)
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J Shanghai Univ (Engl Ed), 2009, 13(1): 6–11
where n denotes the number of vehicles ahead considered. Making a linear stability analysis similar to those for the original models, we obtain the criteria for stability respectively: TVDM : τ =
FLAM : τ =
1 2V n l=1
(h)
− 2λ
Near the neutral stability lines a = (1 + ε2 )as , taking b = V (h) and eliminating the third- and fourth-order terms of ε from (14) lead to the simplified (16) with the final coefficients listed in Table 6 (V (h) = V , and V (h) = V ). Substituting the corresponding coefficients of (16) into (19), we get the amplitude A. So the general soliton solutions of the headway are obtained by substituting the values of A into (20). Next, we derive the mKdV equations for the two models. By inserting (12) and (21) into (28) and (29), and expanding it to the fifth order of ε, we have (22) with the coefficients listed in Table 7 (V (hc ) = V , and V (hc ) = V ).
, (30)
βl (2l − 1)
. 3V Inserting (12) and (13) into (28) and (29), and expanding it to the sixth order of ε, we have (14) with the coefficients given in Table 5 (V (h) = V , and V (h) = V ). Table 5
Coefficients fi of the TVDM and FLAM
Model
f1
f2
f3
f4
TVDM
b−V b−V
−V /6 − λbτ /2 − λ(1 − p)bτ n P 7b3 τ 2 /6 − V βl (3l2 − 3l + 1)/6
V /2
FLAM
b2 τ − V /2 − λbτ n P 3b2 τ /2 − V βl (2l − 1)/2
Model
f5
f6
f7
TVDM
(2b − λ)τ
FLAM
3bτ
−V /24 − λbτ /6 − λ(1 − p)bτ n P βl (4l3 − 6l2 + 4l − 1)/24 5b4 τ 3 /8 − V
l=1
V /4 V
l=1
Table 6
V /2
l=1
n P l=1
βl (2l − 1)/4
Coefficients mi of the TVDM and FLAM
Model
m1
m2
m3
TVDM
V
V /2
FLAM
V /6 + λV τs /2 + λ(1 − p)V τs n P βl (3l2 − 3l + 1)/6 −7b3 τs2 /6 + V
V
3b2 τs /2
Model
m4
m5
l=1
[(2V 2 − λV )(τs + 9λτs2 − 6pλτs2 )
TVDM
(2V V − λV )τs /2 − V /4
−(7λ − 6pλ)V τs ]/6 − V /24 n P −23b4 τs3 /8 + bτs V βl (3l2 − 3l + 1)/2
FLAM
−V
n P l=1
V [6bτs −
l=1
βl
(4l3
−
6l2
Table 7
+ 4l − 1)/24
n P l=1
βl (2l − 1)]/4
Coefficients ki of the TVDM and FLAM
Model
k1
k2
k3
TVDM
b−V
FLAM
b−V
b2 τ − V /2 − λbτ n P 3b2 τ /2 − V βl (2l − 1)/2
−V /6 − λbτ /2 − λ(1 − p)bτ n P 7b3 τ 2 /6 − V βl (3l2 − 3l + 1)/6
Model
k4
k5
k6
TVDM
V /6
(2b − λ)τ
FLAM
V /6
−V /24 − λbτ /6 − λ(1 − p)bτ n P 5b4 τ 3 /8 − V βl (4l3 − 6l2 + 4l − 1)/24
l=1
3bτ
l=1
l=1
k7 V /12 V
n P l=1
βl (2l − 1)/12
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Around the critical point (hc , ac ),a = (1 − ε2 )ac , taking b = V (hc ) and eliminating the second- and third-order terms of ε from (22) lead to the simplified equation (23) with the coefficients presented in Table 8 (V (hc ) = V , and V (hc ) = V ). Table 8
Analogously, substituting the above coefficients of (23) into (26), we get the density waves velocity c of the mKdV equations for the two models. Then the general kink-antikink soliton solutions of the headways are obtained by substituting the values of c into (27).
Coefficients gi of the TVDM and FLAM
Model
g1
g2
g3
TVDM
−V /6
V /2
FLAM
V /6 + λV τc /2 + λ(1 − p)V τc n P βl (3l2 − 3l + 1)/6 − 7b3 τc2 /6 V
−V /6
3b2 τc /2
Model
g4
g5
TVDM
FLAM
l=1
[(2V 2 − λV )(τc + 9λτc2 − 6pλτc2 ) −(7λ − 6pλ)V τc ]/6 − V /24 n P −23b4 τc3 /8 + bτc V βl (3l2 − 3l + 1)/2 −V
n P l=1
l=1
βl
(4l3
−
6l2
+ 4l − 1)/24
3 Summary We have investigated three versions of the simplest car following models. Using the singular perturbation method, the traffic characteristics have been analytically analyzed. The KdV and mKdV equations have been derived to describe the traffic behavior near the neutral stability lines and around the critical points, respectively. We have concluded two general forms to present the KdV and mKdV equations and have given the general solutions which are verified further. Using the general solutions, the complex calculating process could be simplified. They are applicable to the new models—TVDM and FLAM, which can be derived as the same form as general (20) and (27).
References
(2V V − λV )τc /6 − V /12
−V
hP n l=1
i βl (2l − 1) − 6bτc /12
[7] Ge H X, Cheng R J, Dai S Q. KdV and kink-artikink solitons in car following models [J]. Physica A, 2005, 357(3-4): 466–476. [8] Newell G F. Nonlinear effects in the dynamics of car following [J]. Oper Res Phys, 1961, 9(2): 209–229. [9] Whitham G B. Exact solutions for a discrete system arising in traffic flow [J]. Proc R Soc London, Ser A, 1990, 428(1874): 49–69. [10] Bando M, Hasebe K, Nakayama A, et al. Dynamical model of traffic congestion and numerical simulation [J]. Phys Rev E, 1995, 51(2): 1035–1042. [11] Nagatani T, Nakanishi K, Emmerich H. Phase transition in a difference equation model of traffic flow [J]. J Phys A, 1998, 31: 5431–5438. [12] Nagatani T. Thermodynamic theory for the jamming transition in traffic flow [J]. Phys Rev E, 1998, 58(4): 4271–4276.
[1] Nagatani T. The physics of traffic jams [J]. Rep Prog Phys, 2002, 65(9): 1331–1386.
[13] Nayfeh A H. Introduction to perturbation technique [M]. New York: John Wiley & Sons, 1981.
[2] Chowdhury D, Santen L, Schadschneider A. Statistical physics of vehicular traffic and some related systems [J]. Phys Rep, 2000, 329(4): 199–329.
[14] Ge H X, Cheng R J, Li Z P. Two velocity difference model for a car following theory [J]. Physica A, 2008, 387(21): 5239–5245.
[3] Komatsu T, Sasa S. Kink solution charactering traffic congestion [J]. Phys Rev E, 1995, 52(5): 5574–5582. [4] Nagatani T. Stabilization and enhancement of traffic flow by the next-nearest-neighbor interaction [J]. Phys Rev E, 1999, 60(6): 6395–6401. [5] Masakuni M, Nagatani T. Soliton and kink jams in traffic flow with open boundaries [J]. Phys Rev E, 1999, 60(1): 180–187. [6] Nagatani T. Density waves in traffic flow [J]. Phys Rev E, 2000, 61(4): 3564–3570.
[15] Ge H X, Dai S Q, Dong L Y, et al. Stabilization effect of traffic flow in an extended car-following model based on an intelligent transportation system application [J]. Phys Rev E, 2004, 70(6): 066134. [16] Ge H X, Dai S Q, Dong L Y. An extended carfollowing model based on intelligent transportation system application [J]. Physica A, 2006, 365(2): 543–548. [17] Ge H X, Zhu H B, Dai S Q. Effect of looking backward on traffic flow in a cooperative driving car following model [J]. Eur Phys J B, 2006, 54(4): 503–507. (Editor PAN Chun-zhi)
