Nonlinear Dyn DOI 10.1007/s11071-017-3477-2

ORIGINAL PAPER

A new continuum model based on full velocity difference model considering traffic jerk effect Rongjun Cheng · Fangxun Liu · Hongxia Ge

Received: 12 October 2016 / Accepted: 13 March 2017 © Springer Science+Business Media Dordrecht 2017

Abstract In this paper, a new continuum model is developed based on full velocity difference carfollowing model, which takes the traffic jerk effect into account. The critical condition for traffic flow is derived, and density waves occur in traffic flow because of the small disturbance. Near the neutral stability line, nonlinear analysis is taken to derive the KdV–Burgers equation for describing the density wave, and one of the solutions is given. Numerical simulation is carried out to show the local traffic described by the model.

phenomena [1–8] and explain the causes of the some traffic jams. Lighthill and Whitham [9,10] devoted themselves to continuum models and later Richards [11] drew the similar conclusion independently (for short, the LWR model). In this model, the relationship of the three basic parameters of the fluid is built by

Keywords Traffic flow · Continuum model · KdV– Burgers equation · Traffic jerk

where ρ, q, t, x, represent the density, flow, time and space, respectively. Only supplemented by the two equations of q = ρv and the equilibrium condition v = ve (ρ), Eq. (1) is a self-consistent model. Although the model above can be used to explain the majority of traffic flow phenomena, it still has some defects. Liu et al. [12] considered that LWR model is a simple and convenient model to study traffic flow, and some typical traffic phenomena, which come down to the non-equilibrium traffic flow dynamics, such as traffic hysteresis and phantom traffic jams, can hardly be explained. Payne [13] established a new high-order continuum traffic flow model, which can describe the real traffic effectively. What is more, this model can show the nonlinear wave propagation characteristics and be used to analyze the traffic phenomena, such as the small disturbance instability and the stop-and-go traffic. The dynamic equation is:

1 Introduction Now, the problem of traffic congestion has become the focus of attention. Many models have been put forward to simulate different kinds of continuum traffic flow R. Cheng · F. Liu · H. Ge (B) Faculty of Maritime and Transportation, Ningbo University, Ningbo 315211, China e-mail: [email protected] R. Cheng · F. Liu · H. Ge National Traffic Management Engineering and Technology Research Centre, Ningbo University Sub-centre, Ningbo 315211, China R. Cheng · F. Liu · H. Ge Jiangsu Province Collaborative Innovation Center for Modern Urban Traffic Technologies, Nanjing 210096, China

∂q ∂ρ + =0 ∂t ∂x

(1)

123

R. Cheng et al.

∂v ∂v α ∂ρ ve − v +v =− + ∂t ∂x ρT ∂ x T

(2)

where T is relaxation time, and α is the anticipation coefficient. Daganzo [14] pointed out that there are many advantages in Payne’s model. Nevertheless, one of the characteristic speeds in Eq. (2) is greater than the macroscopic flow speed which exists a gas-like behavior. And the phenomenon that the speed of following vehicle is faster than that of the front one would never exist in single-lane traffic. Based on the actual traffic conditions, it is widely acknowledged that vehicles respond only to frontal ones, which we call it anisotropic characteristics. According to the anisotropic characteristics of the high-order continuum models, scholars made more extensive and in-depth discussions. Zhang [15] put forward a new continuum traffic flow theory which overcomes vehicles driving backward problem happened in the high-order continuum models. Later Zhang [16] further discussed whether anisotropic property exists in multi-lane traffic. At the same time, Bando et al. [17] proposed the optimal velocity model (for short, OVM), which can be used to explain the qualitative characteristics of the actual traffic flow, such as the stop-and-go phenomenon, traffic instability and the congestion evolution and so on. The model is: dvn (t) = a [V (xn (t)) − vn (t)] dt

(3)

in which the item V (xn (t)) means an optimal velocity depended on space headway xn (t). But the traffic phenomena described by Treiber et al. [18] cannot be explained by OVM. Namely, if the preceding cars are much faster than the following ones, the vehicle would not brake, even if its headway is smaller than the safe distance. In order to solve this problem, Jiang et al. [5] proposed a full velocity difference model (for short, FVDM). The formula of the FVDM is dvn (t) = a [V (xn (t)) − vn (t)] + λv dt

(4)

Based on FVDM, Jiang et al. [19] derived an anisotropic macroscopic continuum model—the velocity gradient model, which allows the vehicles’ speed deviate from speed–density relationship and can be used to analyze

123

the stop-and-go and the small disturbance instability traffic phenomena. The model is: Ve (ρ (x, t)) − v (x, t)  ∂v dv (x, t) = + dt T η ∂x

(5)

where c0 =  η means disturbance propagation speed. Recently, the anticipation effect is considered in lattice hydrodynamic models by Peng et al. [20,21]. For instance, the jerk effect [22] has been considered in continuum model now, but it is based on the optimal velocity model. And we consider the effect of jerk on the basis of the full velocity difference model. Due to the OVM will produce excessive acceleration and unrealistic deceleration, the FVDM can overcome this problem, and the simulation results are more in agreement with the measured results. So we will investigate the effect of jerk in the new continuum model. In this paper, the effect of jerk is introduced into the FVDM. In Sect. 2, the new model is derived and then the stability analysis is used. Through nonlinear analysis, the KdV–Burgers equation is derived in Sect. 3 and the simulation is carried out in Sect. 4. Finally, the conclusion is given in Sect. 5.

2 Model and stability analysis Based on the FVDM, the jerk term is added to the dynamic equation, which is: dvn (t) = a [V (xn (t)) − vn (t)]+K vn (t)−λJn (t) . dt (6) where vn (t) = vn+1 (t) − vn (t), Jn (t) = dvdnt(t) − dvn (t−1) , and a is the sensitivity which corresponds dt to the inverse of the delay time T , K (0 ≤ K ≤ 1) is the weight coefficient, and λ (0 ≤ λ ≤ 1) is the jerk parameter. Jiang et al. [19] give the headway-density formula x ≈

ρx ρx x 1 − 3 − 4. ρ 2ρ 6ρ

(7)

of which the first term on the right side of Eq. (7) means the relationship between the headway and density. The second term stands for pressure term that weakens the stability in the pneumodynamics, and the last is similar

A new continuum model based on full velocity difference model

to the viscosity item which can smooth the variation of the density wave. Where vn = v (x + , t)−v (x, t). Then, we use the following relation to rewrite the above micro variables into macro ones: vn (t) → v (x, t) , V (h) → Ve (ρ) , K = η1 ,

vn+1 (t) → v (x + , t) V  (xn (t)) → V  (h) a = T1

apply an infinitesimally perturbation to the homogeneous flow. 

where     vρ ρ   , A= U= 0 v − c0 v  0  E = ρx a [Ve (ρ) − v] − λvvxt + aVe (ρ) 2ρ +

ρ0 v0

 +

 ρˆk  vˆk

exp (ikx + σk t) .

Combining Eqs. (9)–(10) with (13) and then neglecting the nonlinear higher-order terms, we have ⎧ (σk + ikv0 ) ρˆk + ikρ0 vˆk =0 ⎪ ⎪  ⎨ σk vˆk + [v0 − c0 ] vˆk ik = a ρˆk Ve (ρ0 ) − vˆk − λv0 vˆk ikσ k ⎪ Ve (ρ0 ) Ve (ρ0 ) 2 ⎪ . +a ρˆk ⎩ 2ρ0 ik+ 6ρ 2 (ik) 0

(14)

(9)

Taking ρˆk and vˆk as the unknown quantities of the equations, we can obtain that σk satisfies the following quadratic equation (σk + ikv0 )2 + (λv0 ikσk + a − c0 ik)  ik (ik)2  + = 0. (σk + ikv0 ) + ikaρ0 Ve (ρ0 ) 1+ 2ρ0 6ρ02

(10)

(15)

(11)

ρx x 6ρ 2



and the eigenvalues of A are λ1 = v, λ2 = v − c0 .

 =

(13)

Because c0 =  η > 0 represents the propagating velocity of the small disturbance, so we call it traffic sound speed. In order to analyze more conveniently, we rewrite the system (9)–(10) as follows: ∂ U ∂ U + A = E ∂t ∂x



(8)

where  represents the distance between two adjacent vehicles. Through the density ρ and the mean headway h = 1/ρ, we define the equilibrium speed Ve (ρ) and have V  (h) = −ρ 2 Ve (ρ). Considering the continuous conservation equation, we get ∂ (ρv) ∂ρ + =0 ∂t ∂x ∂v ∂v + [v − c0 ] = a [Ve (ρ) − v] − λvvxt ∂t ∂ x  ρx ρx x + 2 . + aVe (ρ) 2ρ 6ρ

ρ (x, t) v (x, t)

(12)

From the above two eigenvalues, we can easily find that the macroscopic traffic flow speed v is bigger than the characteristic speed because the equilibrium speed Ve (ρ) declines along with the propagation speed c0 , i.e., c0 > 0, which means our model possesses the anisotropic property of traffic flow. To analyze the qualitative properties of the model, the linear stability method is used below. Considering that the steady state is a uniform flow, we initially



Under the condition that both roots of σk have negative real parts, we will obtain the stable traffic flow. So the neutral stability condition for this steady state is given by  2   k2 k2 2  as = −2ρ0 Ve (ρ0 ) 1 − 2 − 2c0 ρ0 1 − 2 6ρ0 6ρ0      k2  3 + k 1− 2 (16) 6ρ0 Considering the Taylor expansion, we can get     Im (σk ) ≈ −k v0 + ρ0 Ve (ρ0 ) + O k 3 .

(17)

Based on Eq. (17), we deduce the c (ρ0 ) = v0 + ρ0 Ve (ρ0 ) .

(18)

which is similar to those mentioned in Refs. [23–28].

3 Nonlinear analysis We are interested in the system behavior near the neutral stability condition determined by Eq. (18). Then

123

R. Cheng et al.

a coordinate is introduced to transform the reference system [24].

Combining Eqs. (20) with (23) and turning the ρˆ to ρ, we can get

z = x − ct.

  −cρz + (ρVe )ρ + (ρVe )ρρ ρ ρz +b1 ρzz +b2 ρzzz = 0.

(19)

(27) Then we get −cρz + qz = 0

(20)

−cvz + [v − c0 ] vz = a [Ve (ρ) − v] − λv (−cvzz )   ρx ρx x  (21) + 2 + aVe (ρ) 2ρ 6ρ where q is defined as the product of density and speed. From Eq. (20), we get vz =

qρz cρz − 2. ρ ρ

(22)

Among Eqs. (23)–(24), b1 and b2 are determined by making coefficients of the terms ρz and ρzz equal to zero. So we obtain: b1 = b2 =

Ve (ρ) 2 Ve (ρ) 6ρ

+ +

(Ve (ρ)−c)2 +c0 (c−Ve (ρ)) a λcVe (ρ) − Ve (ρ)) . (c a

(25)

Near the neutral stability condition, using the Taylor expansions to rewrite Eq. (23), we have   ρˆ ρVe (ρ) ≈ ρh Ve (ρh ) + (ρVe )ρ  ρ=ρh  1  + (ρVe )ρρ  ρˆ 2 . (26) ρ=ρh 2

123

UT + UU X − mb1 U X X − m 2 b2 U X X X = 0.

(28)

One of the solutions is 3 (−mb1 )2   25 −m 2 b2   ⎤ ⎡ 2 1) 1 1 + 2 tanh ± −mb X + 256(−mb 2 2 b ) T + ζ0 10m −m ( 2    ⎦. ×⎣ 6(−mb1 )2 1 T + ζ X + + tanh2 ± −mb 0 2 2 10m 25(−m b2 )

(23)

two parameters b1 and b2 can get easily solved because the flow q is homogeneous and stable. Substituting Eqs. (22)–(23) into (21), we have  2    c + cc0 ρz a ρz 2 (c0 + 2c) ρz q= + − 3q + ρ ρ2 ρ ρ q aVe (ρ) − λ (−c) ρ   cρz2 cρz2 2qρz2 cρzz qρzz − 2 − 2 − 2 + 3 × ρ ρ ρ ρ ρ   ρz ρzz (24) + aVe (ρ) + 2 . 2ρ 6ρ



Then considering Eq. (27), we can obtain the KdV– Burgers equation [33–35]:

U =−

The flow q can be expanded as q = ρVe (ρ) + b1 ρz + b2 ρzz .

Performing the following transformations [29–32]:   U = − (ρVe )ρ + (ρVe )ρρ ρ , X =mx, T = − mt.

(29) in which ζ0 is an arbitrary constant.

4 Numerical simulation Firstly, Eqs. (9)–(10) need to be discretized as follows: j+1

ρi

j

= ρi +

 t   t j  j j j j j ρi vi − vi+1 + vi ρi−1 − ρi . x x (30) j

(a) If the traffic is heavy (i.e., vi < c0 )   j+1 j j = vi + at Ve − vi vi   t  j j j j 1 − vi − c0 vi+1 − vi − λvi x     x j+1 j j+1 j × vi+1 − vi+1 − vi − vi  j j ρi+1 − ρi  + at Ve j 2ρi x ⎤ j

j

j

ρ − 2ρ + ρi−1 ⎥ + i+1 2 i ⎦ j 6 ρi (x)2

(31)

A new continuum model based on full velocity difference model

Fig. 1 The time evolution of traffic on a range of 32.2 km circumference with a uniform initial traffic and a localized perturbation of amplitude ρ0 = 0.01 veh/m for: a ρ0 = 0.046 veh/m, b ρ0 = 0.058 veh/m, c ρ0 = 0.068 veh/m, d ρ0 = 0.0 74 veh/m

123

R. Cheng et al.

Fig. 2 Temporal evolution of traffic density (ρ0 = 0.058 veh/m) for: a K = 0, λ = 0.2, b K = 0, λ = 0.5, c K = 0, λ = 0.7, d K = 0, λ = 0.9 j

(b) If the traffic is light (i.e., vi ≥ c0 ) j+1

vi

 t  j j j vi − c0 = vi + at Ve − vi − x   j j j 1 × vi − vi−1 − λvi x     j+1 j j+1 j × vi+1 − vi+1 − vi − vi ⎡ 



j

j

j

j

⎤ j

− ρi ρi+1 − 2ρi + ρi−1 ⎥ ⎢ρ + at Ve ⎣ i+1 j + ⎦  2 j 2ρi x 6 ρ (x)2 i

(32)

123

where indices i and j represent space and time secj j tion, respectively; ρi and vi mean the density and speed on the condition (i, j). Then, x and t mean the spatial step and the time step, respectively. In order to observe the evolution of the microvariations more intuitively, the average density ρ0 given by Herrmann and Kerner [24,25] is used as the initial variation:

A new continuum model based on full velocity difference model

Fig. 3 Temporal evolution of traffic density (ρ0 = 0.058 veh/m) for: a K = 0.05, λ = 0.2, b K = 0.05, λ = 0.5, c K = 0.05, λ = 0.7, d K = 0.05, λ = 0.9

"    160 5L x− ρ (x, 0) = ρ0 + ρ0 cosh−2 L 16   # 40 11L 1 x− (33) − cosh−2 4 L 32

where we choose L = 32.2 km, ρ0 as the road length and density perturbation, respectively. We adopt the periodic boundary conditions as follows:

123

R. Cheng et al.

Fig. 4 Temporal evolution of traffic density (ρ0 = 0.058 veh/m) for: a K = 0.11, λ = 0.2, b K = 0.11, λ = 0.5, c K = 0.11, λ = 0.7, d K = 0.11, λ = 0.9

ρ (L , t) = ρ (0, t) ,

v (L , t) = v (0, t) .

(34)

Then, we introduce the equilibrium speed-density relationship derived by Zhou and Shi [26]:

123

 Ve (ρ) = vf

ρ/ρm − 0.25 1 + exp 0.06

−1

−3.72 × 10

−6

(35)

A new continuum model based on full velocity difference model

Fig. 5 (Density-Space) profile of traffic density (ρ0 = 0.068 veh/m) at t = 1800 s for: a K = 0, λ = 0.2, b K = 0.05, λ = 0.5, c K = 0.05, λ = 0.7, d K = 0.11, λ = 0.9

where vf is the free flow speed, and ρm is the maximum density. For the purpose of simulation, 100 m is selected as a unit length during the whole space domain, and 1s is chosen as the time cell.

What is more, these parameters are given as follows: vf = 30 m/s, ρm = 0.2 veh/m, x = 100 m, T = 10, c0 = 11 m/s

123

R. Cheng et al.

Considering the values above, here are the simulation results. In Fig. 1, the traffic flow density is below the critical density, and the perturbation dissipates with time and the traffic flows is stable. As the initial density increases to 0.046 (Fig. 1a), the density fluctuation appears. When the density goes to 0.058 (Fig. 1b), the stop-and-go traffic phenomenon appears. Furthermore, when it reaches to 0.068 (Fig. 1c), the traffic density is similar to the solution described by KdV–Burgers equation. Last, the density fluctuation appears at 0.074 (Fig. 1d). Figure 2 displays the traffic patterns with initial density ρ0 = 0.058 veh/m; K = 0 for different values of λ (0.2 for 2a, 0.5 for 2b, 0.7 for 2c, 0.9 for 2d). We can clearly find that the density wave becomes weaker and weaker as the parameter λ increases. So jerk plays a positive role in the stability of traffic flow. Figure 3 shows the traffic patterns with initial density ρ0 = 0.058 veh/m; K = 0.05 for different values of λ (0.2 for 3a, 0.5 for 3b, 0.7 for 3c, 0.9 for 3d). Compared to Fig. 2, in Fig. 3, with the variable K increased from 0 to 0.05, we can clearly see the density wave becomes very smooth, and the fluctuations are smaller. The conclusion is shown that traffic jams can be suppressed considering the influence of variable K . Figure 4 shows the traffic patterns with initial density ρ0 = 0.058 veh/m; K = 0.11 for different values of λ (0.2 for 4a, 0.5 for 4b, 0.7 for 4c, 0.9 for 4d). Compared to Figs. 2 and 3, in Fig. 4, with the variable K increased to 0.11, we can see that the density wave becomes remarkably smooth, and the fluctuation becomes the least. That means that with the increase in the variable K , it will make the traffic become more and more stable. In Fig. 5, with initial density ρ0 = 0.068 veh/m at t = 1800 s for: (5a) K = 0, λ = 0.2 (5b) K = 0.05, λ = 0.5 (5c) K = 0.05, λ = 0.7 (5d) K = 0.11, λ = 0.9. We can clearly see that with the increase of variables K and λ, traffic flow fluctuations will increasingly reaching stable, and it will eventually makes the traffic reaches a steady.

5 Conclusion In this paper, a new continuum model has been proposed based on full velocity difference model, in which the jerk effect is considered. We study the new con-

123

tinuum model by using the relationship between the micro and macro variables. The neutral stability has been obtained and the KdV–Burgers equation and its solution have been derived to describe the evolution of density wave happened in the traffic congestion. Numerical simulation is also given, and the simulation results show that when considering the effects of variables K and λ, the traffic jams can be suppressed efficiently. Some local cluster phenomena can be found by using our model with certain conditions. Acknowledgements Supported by the National Natural Science Foundation of China under Grant No. 71571107, the Scientific Research Fund of Zhejiang Provincial, China (Grant Nos. LY15A020007, LY15E080013, LY16G010003). The Natural Science Foundation of Ningbo under Grant Nos. 2015A610167, 2015A610168 and the K.C. Wong Magna Fund in Ningbo University, China.

References 1. Tang, T.Q., Shi, W.F., Shang, H.Y., Wang, Y.P.: A new carfollowing model with consideration of inter-vehicle communication. Nonlinear Dyn. 76, 2017–2023 (2014) 2. Liu, H.Q., Zheng, P.J., Zhu, K.Q., Ge, H.X.: KdV–Burgers equation in the modified continuum model considering anticipation effect. Phys. A 438, 26–31 (2015) 3. Treiber, M., Kesting, A., Helbing, D.: Delays, inaccuracies, and anticipation in microscopic traffic models. Phys. A 360, 71–88 (2006) 4. Tang, T.Q., Huang, H.J., Shang, H.Y.: Influences of the driver’s bounded rationality on micro driving behavior, fuel consumption and emissions. Trans. Res. Part D 41, 423–432 (2015) 5. Jiang, R., Wu, Q.S., Zhu, Z.J.: Full velocity difference model for a car-following theory. Phys. Rev. E 64, 017101–017104 (2001) 6. Li, Z.P., Gao, X.B., Liu, Y.C.: An improved car-following model for multiphase vehicular traffic flow and numerical tests. Commun. Theor. Phys. 46, 367 (2006) 7. Ngoduy, D.: Generalized macroscopic traffic model with time delay. Nonlinear Dyn. 77(1–2), 289–296 (2014) 8. Yu, S.W., Shi, Z.K.: An extended car-following model considering vehicular gap fluctuation. Measurement 70, 137– 147 (2015) 9. Lighthill, M.J., Whitham, G.B.: On kinematic waves. I. Flood movement in long rivers. Proc. R. Soc. Lond. Ser. A 229, 281–316 (1955) 10. Lighthill, M.J., Whitham, G.B.: On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A 229, 317–345 (1955) 11. Richards, P.I.: Shock waves on the highway. Open. Res. A 4, 42–51 (1956) 12. Liu, G.Q., Lyrintzis, A.S., Michalopoulos, P.G.: Improved high-order model for freeway traffic flow. Transp. Res. Rec. 1644, 37–46 (1998)

A new continuum model based on full velocity difference model 13. Payne, H.J.: Mathematical models of public systems. Simul. Counc. Proc. Ser. 1, 51–61 (1971) 14. Daganzo, C.F.: Require for second-order fluid approximations of traffic flow. Transp. Res. B 29, 277 (1995) 15. Zhang, H.M.: A theory of non-equilibrium traffic flow. Transp. Res. B 32, 485–498 (1998) 16. Zhang, H.M.: Anisotropic property revisited-does it hold in multi-lane traffic? Transp. Res. B 37(6), 561–577 (2003) 17. Bando, M., Hasebe, K., Shibata, A., Sugiyama, Y.: Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51, 1035–1042 (1995) 18. Treiber, M., Hennecke, A., Helbing, D.: Derivation, properties, and simulation of a gas-kinetic-based, nonlocal traffic model. Phys. Rev. E 59, 239–253 (1999) 19. Jiang, R., Wu, Q.S., Zhu, Z.J.: A new continuum model for traffic flow and numerical tests. Transp. Res. B 36, 405–419 (2002) 20. Peng, G.H., Cheng, R.J.: A new car-following model with the consideration of anticipation optimal velocity. Phys. A 392(17), 3563–3569 (2013) 21. Ge, H.X., Lai, L.L., Zheng, P.J., Cheng, R.J.: The KdV– Burgers equation in a new continuum model with consideration of driver’s forecast effect and numerical tests. Phys. A 377, 3193–3198 (2013) 22. Liu, H.Q., Cheng, R.J., Zhu, K.Q., Ge, H.X.: The study for continuum model considering traffic jerk effect. Nonlinear Dyn. 83, 57–64 (2016) 23. Helbing, D., Tilch, B.: Generalized force model of traffic dynamics. Phys. Rev. E 58, 163–188 (1998) 24. Berg, P., Woods, A.: Traveling waves in an optimal velocity model of freeway traffic. Phys. Rev. E 64, 035602 (2001)

25. Herrmann, M., Kerner, B.S.: Local cluster effect in different traffic flow models. Phys. A 255, 163–188 (1998) 26. Zhou, J., Shi, Z.K.: Lattice hydrodynamic model for traffic flow on curved road. Nonlinear Dyn. 83(3), 1217–1236 (2016) 27. Zhou, J., Shi, Z.K., Cao, J.L.: Nonlinear analysis of the optimal velocity difference model with reaction-time delay. Phys. A 396, 77–87 (2014) 28. Zhou, J., Shi, Z.K., Wang, C.P.: Lattice hydrodynamic model for two-lane traffic flow on curved road. Nonlinear Dyn. 85(3), 1423–1443 (2016) 29. Nagatani, T.: Jamming transition in the lattice models of traffic. Phys. Rev. E 59, 4857–4864 (1999) 30. Nagatani, T.: Thermodynamic theory for the jamming transition in traffic flow. Phys. Rev. E 58, 4271–4276 (1998) 31. Nagatani, T.: TDGL and MKDV equation for jamming transition in the lattice models of traffic. Phys. A 264, 581–592 (1999) 32. Nagatani, T.: Jamming transitions and the modified Korteweg-de Vries equation in a two-lane traffic flow. Phys. A 265, 297–310 (1999) 33. Lewandowski, Jerome L.V.: A marker method for the solution of the damped Burgers’ equation. Numer Methods Partial Differ 22(1), 48–68 (2005) 34. Darvishi, M.T., Khnai, S.: A numerical solution of the Kdv–Burgers’ equation by spectral collocation method and Darvishi’s preconditionings. Int. J. Contemp. Math. Sci. 2(22), 1085–1095 (2007) 35. Liao, W.Y.: An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation. Appl. Math. Comput. 206(2), 755–764 (2008)

123