Analyses of Lattice Traffic Flow Model on a Gradient Highway

Transcription

1 Commun. Theor. Phys. 6 (014) Vol. 6, No. 3, September 1, 014 Analyses of Lattice Traffic Flow Model on a Gradient Highway Arvind Kumar Gupta, 1, Sapna Sharma, and Poonam Redhu 1 1 Department of Mathematics, Indian Institute of Technology Ropar, India School of Mathematical Sciences, University of Science and Technology of China, Hefei 3006, China (Received February 17, 014; revised manuscript received April 17, 014) Abstract The optimal current difference lattice hydrodynamic model is extended to investigate the traffic flow dynamics on a unidirectional single lane gradient highway. The effect of slope on uphill/downhill highway is examined through linear stability analysis and shown that the slope significantly affects the stability region on the phase diagram. Using nonlinear stability analysis, the Burgers, Korteweg-deVries (KdV) and modified Korteweg-deVries (mkdv) equations are derived in stable, metastable and unstable region, respectively. The effect of reaction coefficient is examined and concluded that it plays an important role in suppressing the traffic jams on a gradient highway. The theoretical findings have been verified through numerical simulation which confirm that the slope on a gradient highway significantly influence the traffic dynamics and traffic jam can be suppressed efficiently by considering the optimal current difference effect in the new lattice model. PACS numbers: 1.65.Mn, 1.45.Ff, 1.30.Fe, 1.65.Ef Key words: traffic flow, gradient highway, stability analysis, simulation 1 Introduction In recent years, due to rapid increase of automobiles on the roads, the problem of traffic jam has attracted considerable attention of scientists and researchers. To disclose the nature of traffic jams and investigate its properties and also to reduce it, various traffic flow models including car following models, cellular automaton models, hydrodynamic models and so on have been proposed by many scientists and engineers with different backgrounds. [1 8] Recently, lattice hydrodynamic model, firstly, proposed by Nagatani [9] has been given much attention. This model incorporates the idea that drivers adjust their velocity according to the observed headway. Later, many extended [10 5] lattice models have been developed by considering different factors like backward effect, [10] lateral effect of the lane width, [11] honk effect, [1] and anticipation effect of potential lane changing [13] etc. Recently, Peng [14] proposed a new lattice model by incorporating the effect of optimal current difference which suppressed the traffic jam effectively. Kang and Sun [15] introduced a lattice hydrodynamic model by taking into account driver s delay effect in sensing relative flux (DDSRF) and found that this effect has an important influence on the traffic jams. Most of the aforementioned models mainly focused on describing some traffic phenomena only on single or two-lane traffic system without any gradient. None of these models can be used to investigate the effect of road conditions on traffic flow as road conditions has not been considered. As it is well known that in real world all the highways are not uniform or straight, there are a large number of gradient highways or a portion of roadways with some slopes. These slopes being a special case of road conditions play an important role on traffic dynamics as the traffic behavior on slope is significantly different from that on the normal road. Moreover, both the angle and characteristics of the slope have a substantial affect on the traffic dynamics. In this direction, Lan et al. [6] proposed a cellular automata model to study the effect of slope on traffic flow. Li et al. [7] and He et al. [8] presented an extended OV model and examined the effect of slope with and without gravity, respectively. Komada et al. [9] analyzed the effect of gravitational force on a highway with slope by using force analysis. Zhu and Yu [30] further modified the extended OV model by taking the brake distance as a function of slope and investigated its effects. Recently, Chen et al. [31] proposed a macroscopic model by using extended car following model on a highway with slope. But, upto our knowledge no effort has not been made to develop a lattice hydrodynamic model on a gradient highway with different slopes. The paper is organized as follows: In the following section, a realistic lattice model considering the effect of slope is proposed. In Sec. 3, the linear stability analysis is performed for the proposed model. Section 4 is devoted to the nonlinear analysis in which Burger s, mkdv and KdV equations are derived. Numerical simulations are carried out in Sec. 5 and finally, conclusions are given in Sec. 6. Proposed Model The first lattice hydrodynamic model proposed by Nagatani [9] incorporates both the ideas of car-following models and macroscopic models to analyze the density The third author acknowledges Council of Scientific and Industrial Research (CSIR), India for providing financial assistance. This work was also supported by Chinese Universities Scientific Fund under Grant No. WK akgupta@iitrpr.ac.in c 014 Chinese Physical Society and IOP Publishing Ltd

2 394 Communications in Theoretical Physics Vol. 6 wave of traffic flow on a unidirectional simple roadway and is given by t ρ j + ρ 0 (ρ j v j ρ j 1 v j 1 ) = 0, (1) t (ρ j v j ) = a[ρ 0 V (ρ j+1 ) ρ j v j ], () where j indicates site-j on the one-dimensional lattice; ρ j and v j, respectively, represent the local density and velocity at site-j at time t; ρ 0 is the average density; a(= 1/τ) is the sensitivity of drivers; V ( ) is called optimal velocity function and it is taken as V (ρ) = V max [ tanh( 1 ρ h c ) ] + tanh(h c ), (3) here V max and h c, respectively, denote the maximal velocity and the safety distance on a highway without any slope. This optimal velocity function is monotonically decreasing, has an upper bound and an inflection point at ρ = 1/h c = ρ 0. The above model is further modified to take the effect of optimal current difference into account by Peng et al. [14] The continuity equation remains preserved while the evolution equation is modified by looking at the difference of optimal traffic currents on site-j + and j + 1. This effect plays an important role in stabilizing the traffic flow and suppressing effectively the traffic jams. Then, the modified evolution equation by considering optimal current difference effect is given by t (ρ j v j ) = a[ρ 0 V (ρ j+1 ) ρ j v j ] + aλ[ρ 0 V (ρ j+ ) ρ 0 V (ρ j+1 )], (4) where λ is the reaction coefficient of optimal current difference. Undoubtedly, all the improvements in lattice hydrodynamic theory are noteworthy but they have the same characteristics that lattices are distributed along a straight line representing a normal highway without slope. As the traffic behavior on a gradient highway is different from that of on a normal highway without any gradient, we consider a situation in which vehicles move along a unidirectional single channel gradient lattice under periodic boundary conditions. Figure 1 shows a schematic of a gradient highway illustrating the gravitational force acting on a lattice in an uphill and downhill situations. The slope of the gradient is represented by θ, the gravitational acceleration is g and the total mass of the vehicles on the lattice is m. A horizontal force of amplitude mg sin θ acts upon the lattice in both the situations at any instant if drivers do not operate the brake otherwise the gravitational force would be replaced by the brake control. Based on the lattice optimal current difference model, we propose a novel evolution equation for an extended lattice model to describe the motion of vehicles on a single lane gradient highway as follows: where t (ρ j v j ) = a[ρ 0 V (ρ j+1, θ) ρ j v j ] + aλ[ρ 0 V (ρ j+, θ) ρ 0 V (ρ j+1, θ)], (5) V (ρ, θ) = V max V g,u,max [ 1 ) tanh( ρ h c,u(θ) for an uphill gradient highway, and V (ρ, θ) = V max + V g,d,max [ 1 ) tanh( ρ h c,d(θ) ] + tanh(h c,u (θ)), (6) ] + tanh(h c,d (θ)), (7) for a downhill gradient highway. V (ρ, θ) is the optimal velocity function varying with slope θ, h c,u (θ) and h d,u (θ), respectively, represent the brake distances for the vehicles on the uphill and downhill gradient. V g,u,max and V g,d,max are the maximal reduced and enhanced velocity on uphill and downhill gradient highway, respectively and formulated as V g,u,max = V g,d,max = mg sin θ, (8) γ where γ is a longitudinal friction coefficient. The basic idea is that the maximal velocity under the influence of gravitational force on a gradient highway is proportional to the slope of the gradient. For θ = 0, the model reduces to that of Ref. [14]. With an increase in the gradient slope θ, the maximal velocities, respectively, V g,u,max and V g,d,max for uphill and downhill highway increase. For the sake of simplicity, we choose γ = mg. As it is well known that the brake distance of the vehicles varies with the slope of gradient. The idea is that in an uphill (downhill) situation, with the increase in slope the brake distance decreases (increases) as h c,u (θ) = h c (1 α sin θ), h c (θ) = h c,d (1 + β sin θ), (9) where α and β are constant and for simplicity, we take α = β = 1. Assume ( 1 ) ( ) V 0 (ρ, θ) = tanh ρ h(θ) + tanh h(θ), (10) where h( ) denotes h c,u and h d,u for uphill and downhill gradient, respectively. Then, the optimal velocity function is rewritten as follows ( Vmax sinθ ) V (ρ, θ) = V 0 (ρ, θ), (11) here correspond to an uphill and a downhill situation, respectively. Fig. 1 An illustration of gravitational force on a single lane gradient lattice: uphill and downhill.

3 No. 3 Communications in Theoretical Physics 395 By taking the difference form of Eqs. (1) and (5) and eliminating speed v j, the evolution equation of density is obtained as ρ j (t + τ) = ρ j (t + τ) τρ 0 A(θ)[V 0(ρ j+1, θ) V 0 (ρ j, θ)] τρ 0 λa(θ)[v 0(ρ j+, θ) V 0 (ρ j+1, θ) + V 0 (ρ j, θ)], (1) where A(θ) = (V f,max sin θ)/. 3 Linear Stability Analysis To investigate the effect of uphill/downhill gradient highway on traffic flow, we conducted linear stability analysis in this section. The traffic density and optimal velocity under uniform traffic condition is taken as ρ 0 and V 0 (ρ 0, θ), respectively, where ρ 0 is a constant. Hence, the steady-state solution of the homogeneous traffic flow is given by ρ j (t) = ρ 0, V 0 (ρ j (t), θ) = V 0 (ρ 0, θ). (13) For simplicity, we avoid the subscript of V. Let y j (t) be a small perturbation to the steady-state density on site-j. Then, ρ j (t) = ρ 0 + y j (t). (14) Putting this perturbed density profile into Eq. (1) and linearizing it, we obtain where y j (t + τ) y j (t + τ) + τρ 0 A(θ)V (ρ 0, θ) y j (t) + τρ 0 λa(θ)v (ρ 0, θ) ( y j (t)) = 0, (15) y j (t) = y j+1 (t) y j (t). Putting y j (t) = exp(ikj + zt) in Eq. (15), we get e τz e τz + τρ 0 A(θ)V (ρ 0, θ)(e ik 1) Inserting +τρ 0 λa(θ)v (ρ 0, θ)(e ik e ik + 1) = 0. (16) z = z 1 (ik) + z (ik) into Eq. (16), we obtained the first and second-order terms of the coefficient ik and (ik), respectively, as z 1 = ρ 0A(θ)V (ρ 0, θ), (17) z = 3τ(A(θ)) z1 ρ 0 A(θ)V (ρ 0, θ) λρ 0A(θ)V (ρ 0, θ). (18) When z < 0, the uniform steady-state flow becomes unstable for long-wavelength waves. For z > 0 the uniform flow will remain stable. Thus, the neutral stability curve is given by 1 + λ τ = 3ρ 0 A(θ)V (ρ 0, θ). (19) The instability condition for the homogeneous traffic flow can be described as 1 + λ τ > 3ρ 0 A(θ)V (ρ 0, θ). (0) Equation (0) clearly shows that the slope θ on a gradient highway plays an important role on the stability of traffic flow on a single lane. When θ = 0, the result of stability condition is same as that of Ref. [14]. Solid (dotted) curves in Figs. (a) and (b) are the neutral stability (coexisting) curves through linear (nonlinear) analysis in the phase space corresponding to uphill and downhill situations, respectively, at λ = 0 for different values of θ. Figures 3(a) and 3(b) explain the phase diagram under the same situation as that in Fig. except for λ = 0.1. The apex of each curve indicates the critical point (ρ c, a c ). The phase plane is divided into three regions: stable, metastable, and unstable. It can be easily depicted from Figs. (a) and 3(a) that in an uphill situation for any value of λ the amplitude of these curves decreases with an increase in θ which means that larger value of θ leads to enlargement of stability region and hence, the traffic jam is suppressed efficiently. But in the downhill situation, stable region reduces with increase in slope for any value of λ. On comparing Figs. (a) and 3(a), it can also be concluded that increase in the value of λ results a further enlarges the stable region which means that reaction coefficient reduces traffic jams significantly on a gradient highway. Moreover, the critical point (ρ c, a c ) remains preserved for a change in reaction coefficient while varies with respect to θ. Fig. Phase diagram in parameter space (ρ,a) for λ = 0 (a) uphill, and (b) downhill situation.

4 396 Communications in Theoretical Physics Vol. 6 Fig. 3 Phase diagram in parameter space (ρ,a) for λ = 0.1 (a) uphill, and (b) downhill situation. 4 Nonlinear Stability Analysis Using reduction perturbation method, now, we investigate the evolution characteristics describing the collective motion on coarse-grained scales. Long-wavelength expansion method is used to understand the slowly varying behavior of traffic dynamics. The slow variables X and T for a small positive scaling parameter ǫ (0 < ǫ 1) are defined as X = ǫ(j + bt), T = ǫ m t, (1) where b is a constant to be determined. Let ρ j satisfy the following equation: ρ j (t) = ρ c + ǫ l R(X, T). () Here, different values of index m and l represent different phases of the traffic flow. The values of the index m =, l = 1; m = 3, l = ; m = 3, l = 1 correspond to stable, metastable and unstable traffic flow regions, respectively. By expanding Eq. (1) to fifth order of ǫ with the help of Eqs. (1) and (), we obtain the following nonlinear equation: where ǫ l+1 b X R + ǫ l+ 3b τ X R + ǫl+3 7b3 τ X 3 6 R + ǫl+4 5b4 τ 3 X 4 8 R + ǫl+m T R + ǫ l+m+1 3bτ T X R [ + ρ c A(θ) V ( ǫ l+1 X R + ǫ l+ 1 X R + 1 ǫl X R + 1 ) ǫl X R + ǫ l+3( 1 3 R 3 XR + X R XR + ǫ l+4 7 ) 1 4 X R + V V = + V ( ǫ l+1 X R + ǫ l+ 1 X R [ )) + V ( ǫ 3l+1 X R 3 + ǫ 3l+ 1 6 XR 3)] + λρ ca(θ) V ( ǫ l+ XR + ǫ l+3 XR 3 (ǫl+ X R + ǫ l+3 (R X 3 R + 6 XR X V ] R)) + 6 (ǫ3l+ X R3 ) = 0, (3) dv (ρ, θ) dρ ρ=ρc, V = d V (ρ, θ) dρ ρ=ρc, V = d3 V (ρ, θ) dρ 3 ρ=ρc. Case 1 m =, l = 1 We now discuss the profile of triangular shock waves of the traffic flow under stable condition. Putting the specific values of index m and l into nonlinear partial differential Eq. (3), we get ǫ [b + A(θ)ρ cv ] X R + ǫ 3[ ( 3b T R + A(θ)ρ cv τ R X R + + ρ ca(θ)v + λρ ca(θ)v ) ] XR = 0. (4) By taking b = A(θ)ρ cv and eliminating the second order terms of ǫ, we obtain [ 3 T R + A(θ)ρ cv R X R = b τ + ρ ca(θ) V + λρ ca(θ)v ] XR. (5) Since the coefficient (3/)b τ + ρ c A(θ)(V /) + λρ c A(θ)V has a negative value in the stable region satisfying the stability criterion (Eq. (0)). Therefore, Eq. (5) is a Burger s equation whose asymptotic solution for T 1 is a train of N-triangular shock waves and given as below: 1 R(X, T) = ρ c A(θ)V T [ tanh [ X 1 (η n + η n+1 ) D 4 ρ ca(θ)v T (η n+1 η n )(X ξ n ) ] 1 ρ c A(θ)V T (η n+1 η n ) ], (6)

5 No. 3 Communications in Theoretical Physics 397 where D = [(3/)b τ +ρ c A(θ)(V /)+λρ c A(θ)V ], the coordinates of shock fronts are given by ξ n, (n = 1,, 3,..., N) and η n denotes the coordinates of the intersection of slope with the x-axis. As T, R(X, T) 0, which means that any density wave expressed by Eq. (6) in the stable traffic flow region evolves into a uniform flow in due course of time. Case m = 3, l = 1 Here, mkdv equation is derived near the critical point and the kink-antikink soliton wave of traffic flow is discussed under the unstable condition. Putting the specific values of index m and l into nonlinear partial differential Eq. (3), we get ǫ [b + A(θ)ρ cv ] X R + ǫ 3[( 3b τ + ρ ca(θ) V + λρ ca(θ)v ) ] XR + ǫ 4[ ( 7b 3 τ T R + + ρ ca(θ)v (1 + 6λ) ) X R + ρ ca(θ)v X R 3] 6 + ǫ 5[ ( 5b 4 τ 3 3bτ T X R + + ρ c A(θ)V (1 + 14λ) ) XR + ρ c A(θ)V (1 + λ) 1 XR 3] = 0. (7) Near the critical point, τ = τ c (1 + ǫ ). By taking b = ρ ca(θ)v and eliminating second and third order term of ǫ in Eq. (7), we get ǫ 4 ( T R g 1 X 3 R + g X R 3 ) + ǫ 5 (g 3 X R + g 4 X 4 R + g 5 X R3 ) = 0, (8) where the coefficients g, are given in Table 1. Table 1 The coefficients g i of the model. g 1 g g 3 g 4 g 5 7b3 τ c + b(1 + 6λ) 6 ρ ca(θ)v 6 3b τ c 5b 4 τ 3 c 8 + ρ c A(θ)V (1 + 14λ) 4 + 3bτ cg 1 ρ c A(θ)V (1 + λ) 1 ba(θ)τcρ c V To convert Eq. (8) into a standard mkdv equation, the following transformations are used: T = g 1 T, R = g1 g R. Hence, Eq. (8) becomes where T R 3 X R + X R 3 + ǫm[r ] = 0, (9) M[R ] = 1 ( g 3 X g R + g 1g 5 X 1 g R 3 + g 4 X 4 R ). After ignoring the correction term in Eq. (9), we get the mkdv equation with kink-antikink solution as follows: R 0 (X, T ) = [ c ] c tanh (X ct ). (30) In order to determine the value of propagation velocity for the kink-antikink solution, it is necessary to satisfy the following condition: (R 0, M[R 0 ]) dxr 0 M[R 0 ] = 0, (31) with M[R 0 ] = M[R ]. By solving Eq. (31), the selected value of c is: 5g g 3 c =. (3) g g 4 3g 1 g 5 Hence, the kink-antikink solution is given by g1 c ( c ) ρ j = ρ c + ǫ tanh g (X cg 1T), (33) with ǫ = τ/τ c 1 and the amplitude D of the solution is g1 D = ǫ g c. (34) Case 3 m = 3, l = Now, KdV equation is derived and the soliton wave of traffic flow is discussed under the metastable condition. Putting the specific values of index m and l into nonlinear partial differential Eq. (3), we get ǫ 3 [b + A(θ)ρ c V ] X R + ǫ 4[( 3b τ + ρ c A(θ)V + λρ c A(θ)V ) ] X R + ǫ 5[ ( 7b 3 τ T R + + ρ c A(θ)V (1 + 6λ) ) XR + ρ ca(θ)v X R ] + ǫ 6[ ( 5b 4 τ 3 3bτ T X R + + ρ ca(θ)v (1 + 14λ) ) X R + ρ ca(θ)v (1 + λ) X 4 R] = 0. (35) Near the critical point, τ = τ c (1 + ǫ ). By taking b = ρ c A(θ)V and eliminating second and third order term of ǫ in Eq. (35), we get ǫ 5 ( T R h 1 3 XR + h X R ) + ǫ 6 (h 3 XR + h 4 4 XR + h 5 XR ) = 0, (36) where, the coefficients h i are given in the Table.

6 398 Communications in Theoretical Physics Vol. 6 Table The coefficients h i of the model. h 1 h h 3 h 4 h 5 7b3 τ c + b(1 + 6λ) 6 ρ ca(θ)v 3b τ c 5b 4 τ 3 c 8 + ρ c A(θ)V (1 + 14λ) 4 + 3bτ ch 1 ρ c A(θ)V (1 + λ) 4 ba(θ)τcρ c V To convert Eq. (36) into a standard KdV equation, the following transformations are used: T = h 1 T, R = h1 /h R. Hence, Eq. (36) becomes where T R 3 X R + X R + ǫm[r ] = 0, (37) M[R ] = 1 ( h 3 X h R + h 1h 5 X 1 h R + h 4 X 4 R ). After ignoring the correction term in Eq. (37), we get the KdV equation with soliton solution as follows: R 0 (X, T ) = csech [ c (X ct )]. (38) 1 3 In order to determine the value of propagation velocity for the soliton solution, it is necessary to satisfy the following condition: (R 0, M[R 0]) dxr 0M[R 0] = 0, (39) with M[R 0 ] = M[R ]. By solving Eq. (39), the selected value of c is: 1h 1 h h 3 c =. (40) 4h 1 h 5 5h h 4 Hence, the soliton solution is given by ρ j = ρ c + cǫ sech ( c ( X ch )) T, (41) with ǫ = τ/τ c 1. 5 Numerical Simulation To check whether the proposed model is capable of describing the effect of slope on traffic flow dynamics and validate linear as well as nonlinear stability analysis, numerical simulation is carried out for the proposed model under periodic boundary conditions. To study the triangular shock and kink-antikink waves in the proposed lattice model, we use two different nonrandom initial conditions. Firstly, the triangular shock waves are simulated in the stable region with the initial condition chosen as: ρ 1, 0 j < L, ρ j (1) = ρ j (0) = ρ, L j < L, where L is the total number of sites taken as 00 and other parameters are chosen as: ρ 1 = 0.5, ρ = 0.4, ρ c = 0.5. Fig. 4 Density profile of the triangular shock waves at time t = s, when λ = 0 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively for uphill situation. Figures 4 and 5 describe the density profiles in a stable region at time t = 000 s with a = 1.5 on an uphill and a downhill highway with different slopes, respectively, for λ = 0. It is clear from Figs. 4 and 5 that initial disturbance leads to a triangular shock wave. The amplitude of the triangular shock wave varies with θ and increases in the uphill case while decreases in the downhill case with the increase of the slope. Figures 6 and 7 show the spa-

7 No. 3 Communications in Theoretical Physics 399 tiotemporal evolution of triangular shock waves after sufficiently long time, namely 10 4 steps corresponding to Figs. 4 and 5, respectively. These numerical results are consistent with the theoretical findings in Sec. 4. Moreover, we also examine the effect of λ on the triangular shock wave in stable region for uphill/downhill case and find that the density profiles and spatiotemporal patters in both the cases are similar for λ = 0 and λ = 0.1. This can be understood as the increase in the value of λ will only reduce the instability region while keeping critical point intact. Therefore, it is reasonable to conclude that slope of highway affects significantly the traffic dynamics even in stable region. Now, we examine the kink-antikink soliton waves in the unstable region described by mkdv equation. The initial conditions are adopted as follows: ρ 0, j L, L + 1, ρ j (0) = ρ 0 σ, j = L, ρ 0 + σ, j = L + 1, where σ is the initial disturbance and the value of other parameters are chosen as : σ = 0.1, L = 100, and ρ 0 = ρ c. Fig. 5 Density profile of the triangular shock waves at time t = s, when λ = 0 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively for downhill situation. Fig. 6 Spatiotemporal evolutions of density for λ = 0 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively for uphill situation.

8 400 Communications in Theoretical Physics Vol. 6 Figures 8 and 9 depict the spatiotemporal evolution of density for different values of θ after sufficiently long time at λ = 0 in an uphill and downhill situation, respectively. It is clear from the figures that initial disturbance leads to the kink-antikink soliton, which propagates in the backward direction in all cases. The initial small amplitude perturbation evolves into congested flow as the stability condition is not satisfied. In both the situations, the number of stop-and-go waves decreases with the increase in slope. Patterns 8(a) and 9(a) are corresponding to no slope (θ = 0). Figures 10 and 11 describe the density profile at time t = 0 00 s corresponding to panel of Figs. 8 and 9, respectively. The amplitude of kinkantikink soliton wave in an uphill (downhill) situation increases (decreases) with the increase in slope (Table 3). This is in accordance with the theoretical results that the critical safe density increases (decreases) with the increase of slope in an uphill (downhill) situation. For higher slope, the traffic congestion in the form of stop-and-go waves will disappear. Fig. 7 Spatiotemporal evolutions of density for λ = 0 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively for downhill situation. Fig. 8 Spatiotemporal evolutions of density when λ = 0 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively in an uphill situation.

9 No. 3 Communications in Theoretical Physics 401 Fig. 9 Spatiotemporal evolutions of density when λ = 0 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively in an downhill situation. Table 3 Amplitude of the kink-antikink waves varying with the slope. λ = 0 λ = 0.1 Situation Slope Maximum Minimum Maximum Minimum Uphill No slope Downhill Fig. 10 Density profile of the kink-antikink at time t = s, when λ = 0 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively for uphill situation.

10 40 Communications in Theoretical Physics Vol. 6 Fig. 11 Density profile of the kink-antikink at time t = s, when λ = 0 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively for downhill situation. Fig. 1 Spatiotemporal evolutions of density when λ = 0.1 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively in an uphill situation. Now, we investigate the effect of λ in the unstable region for uphill/downhill case. Figures 1 and 13 show the spatiotemporal evolution of density for different values of θ after sufficiently long time at λ = 0.1 in an uphill and downhill situation, respectively. The results corresponding to λ = 0.1 are qualitatively similar to those obtained for λ = 0. The kink-antikink soliton occurs and propagates in the backward direction. Figures 14 and 15 describe the density profile at time t = 0 00 s corresponding to panel of Figs. 1 and 13, respectively. It can be seen from Figs. 14 and 15 that the amplitude of the stopand-go waves varies with slope. In both the situations, the amplitude of the density waves weakened for λ = 0.1 as compared to the case of without optimal current difference effect. This also satisfies theoretical predictions that the stability of the lattice model is better than that of Nagatani s lattice model by taking the optimal current difference effect into account even in the gradient highway situation. Therefore, it is reasonable to conclude that the optimal current difference effect plays a significant role in one-dimensional lattice hydrodynamic model on gradient highways with slopes and enhances the stability of traffic flow for all possible values of the slope.

11 No. 3 Communications in Theoretical Physics 403 Fig. 13 Spatiotemporal evolutions of density when λ = 0.1 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively in a downhill situation. Fig. 14 Density profile of the kink-antikink at time t = s, when λ = 0.1 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively for uphill situation. Fig. 15 Density profile of the kink-antikink at time t = s, when λ = 0.1 (a) θ = 0, (b) θ =, (c) θ = 4, and (d) θ = 6, respectively for downhill situation.

12 404 Communications in Theoretical Physics Vol. 6 6 Conclusion We have proposed a new lattice hydrodynamic model of traffic flow on a gradient highway by considering the optimal current difference effect. The traffic behavior has been analyzed through linear and nonlinear analysis. Through nonlinear stability analysis, we derived the Burgers, mkdv and KdV equations in stable, unstable and metastable regions, respectively. Phase diagrams in the density-sensitivity space with the neutral stability and coexisting curves are analyzed and found that slope plays an important role in influencing the critical density as well as stable and unstable regions while the reaction coefficient enlarges the stable region for both the situations for any value of slope. Moreover, a series of numerical simulations are carried out to reproduce the density waves and verify the theoretical findings on a gradient highway. It is concluded that the reaction coefficient does not have any effect on the density profiles in the stable region while have a significant effect in the unstable region for all possible values of slope. The simulation results are compared and found in good accordance with the theoretical findings, which verifies that our consideration is reasonable. Therefore, it is worth to conclude that the newly developed lattice hydrodynamic model for a gradient highway with the consideration of optimal current difference can efficiently explain the effect of slope on traffic dynamics. References [1] R. Jiang, Q.S. Wu, and Z.J. Zhu, Transp. Res. Part B 36 (00) 405. [] T.Q. Tang, C. Li, H. Huang, and H. Shang, Nonlinear Dynamics 67 (01) 55. [3] T.Q. Tang, Y.P. Wang, X.B. Yang, and Y.H. Wu, Nonlinear Dynamics 70 (01) [4] T.Q. Tang, Y.F. Shi, Y. Wang, and G. Yu, Nonlinear Dynamics 70 (01) 09. [5] T.Q. Tang, C. Li, H. Huang, and H. Shang, Nonlinear Dynamics 67 (01) 437. [6] A.K. Gupta and V.K. Katiyar, Physica A 38 (005) [7] A.K. Gupta, Int. J. Mod. Phys. C 5 (013) [8] A.K. Gupta and V.K. Katiyar, Transportmetrica 3 (007) 73. [9] T. Nagatani, Physica A 61 (1998) 599. [10] H.X. Ge and R.J. Cheng, Physica A 387 (008) 695. [11] G.H. Peng, X.H. Cai, B.F. Cao, and C.Q. Liu, Phys. Lett. A 375 (011) 83. [1] G.H. Peng, X.H. Cai, C.Q. Liu, and B.F. Cao, Int. J. Mod. Phys. C (011) 967. [13] G.H. Peng, X.H. Cai, C.Q. Liu, and M.X. Tuo, Phys. Lett. A 376 (01) 447. [14] G.H. Peng, Commun. Nonl. Sci. Numer. Simul. 18 (013) 559. [15] Y.R. Kang and D.H. Sun, Nonlinear Dynamics 71 (013) 531. [16] G.H. Peng, Commun. Theor. Phys. 60 (013) 485. [17] G.H. Peng, Int. J. Mod. Phys. C 4 (013) [18] G.H. Peng, Commun. Nonl. Sci. Numer. Simul. 18 (013) 801. [19] T. Nagatani, Physica A 65 (1999) 97. [0] A.K. Gupta and P. Redhu, Phys. Lett. A 377 (013) 07. [1] A.K. Gupta and P. Redhu, Commun. Nonl. Sci. Numer. Simul. 19 (013) [] G.H. Peng, Nonlinear Dynamics 73 (013) [3] A.K. Gupta and P. Redhu, Physica A 39 (013) 56. [4] T. Nagatani, Phys. Rev. E 60 (1999) [5] T. Nagatani, Phys. Rev. E 59 (1999) [6] S.Y. Lan, Y.G. Liu, B.B. Liu, P. Sheng, T. Wang, and X.S. Li, Int. J. Mod. Phys. C (011) 319. [7] X.I. Li, T. Song, H. Kuang, and S.Q. Dai, Chin. Phys. B 17 (008) [8] H.D. He, W.Z. Lu, Y. Xue, and L.Y. Dong, Chin. Phys. B 18 (009) 703. [9] K. Komada, S. Masukura, and T. Nagatani, Physica A 388 (009) 880. [30] W.X. Zhu and R.L. Yu, Physica A 391 (01) 954. [31] J. Chen, Z. Shi, Y. Hu, L. Yu, and Y. Fang, Int. J. Mod. Phys. C 4 (013)