1 Introduction

Traffic flow can be considered as a one-dimensional (1D) compressible flow of particles in the macroscopic theory^[1-2]. The traffic state at the position x and time t can be described in terms of the spatial vehicle density ρ(x, t) and the average velocity v(x, t). Higher-order traffic flow models have been proposed with the consideration of not only mass conservation but also acceleration^{[1, 3-10]}. Therefore, the kinetic effects in the acceleration equation could enable most of these models to describe very complex nonlinear phenomena in traffic flow, e.g., phase transitions, metastable states, and stop-and-go waves.

The stop-and-go wave embodies a wide moving jam, where a stable non-trivial solution is constructed to be distinguished from the constant equilibrium traffic state. The characteristics of wide moving jams to higher-order traffic flow models have been investigated widely^{[2, 5, 11-17]}. Recently, some scholars have studied wide moving jams in very complex situations, e.g., the peculiarities of traffic flow on a loop road with ramp effects in a visco-elastic traffic model^[18], the effects of bi-directional information^[19] and bi-directional driving strategy^[20] in the macroscopic traffic models. The occurrence of the oscillations has been well explained by empirical studies on the growth patterns of traffic oscillations^[21] and the traffic transitions before and after the stop-and-go wave^[22]. A wide moving jam is a traveling wave solution, which can be uniquely determined by the model equations. Therefore, it is important to find the equations for the characteristic parameters. The characteristic parameters mainly include the propagation speed of the moving jam and the maximal and minimal densities and velocities.

Kerner et al.^[2] discussed the structure of the wide moving jams and narrow clusters to the Kerner-Konhäuser (KK) model^[5], and derived several simple equations to describe the formation of the traffic jams with large amplitude by using the asymptotic theory. Zhang and Wong^[13] obtained the parametric equations of the wide moving jam solutions to the Payne-Whitham (PW) model^{[1, 3]} with two conservation forms by using the weak solution theory, and indicated the essence of the conservation forms. The wide moving jam solutions of the Jiang-Wu-Zhu (JWZ) model^[8] and the Aw-Rascle (AR) model^[6] are investigated by the similar method used in Refs. [12] and [14], respectively. The investigations of the AR model showed that the lower-density equilibrium state of the wide moving jam can exist in the instable region, while the higher-density equilibrium state should exist in the stable region. Wu et al.^[15] derived the parameter equations of the wide moving jam solution to the Kühne model^[4] by using the boundary-layer method. The solution was "asymptotic" to that of the PW model, where v was a conservative variable. The asymptotic relation of the wide moving jams between the KK model and the PW model, where ρv is a conservative variable, can be derived from Refs. [2] and [13]. Accordingly, the essence of the conservation forms is indicated by using the asymptotic theory, and the internal relation between the asymptotic theory and the weak solution theory is obtained. However, there are still some problems to be studied, such as whether the above asymptotic relationship is suitable for other models and how to determine the existence of a wide moving jam solution to a traffic flow model.

In this paper, we will use the boundary-layer method with the asymptotic theory to obtain the asymptotic solution to a class of higher-order viscous traffic flow models. The equations for the characteristic parameters, which can be used to determine the wide moving jam, are obtained. For a very small viscosity coefficient, the asymptotic solutions to many higher-order viscous models are consistent with the wide moving jam solutions to the corresponding inviscid models derived from the weak solution theory. Therefore, the internal relation between the asymptotic theory and the weak solution theory is true not only for the Kühne and KK models but also for many other traffic flow models. The sufficient conditions and essential conditions for the formation of a wide moving jam are discussed in detail, respectively. These conditions are used to judge whether wide moving jams can be formed in many familiar higher-order traffic models. The numerical results and analytical conclusions are in good agreement. The remain content of this paper is organized as follows. In Section 2, the asymptotic analysis and equations for the determination of the characteristic parameters for the wide moving jams to higher-order viscous traffic flow models are presented. In Section 3, the numerical results are presented to show whether a wide moving jam can be formed. In Section 4, the conclusions of this paper are summarized.

2 Asymptotic analysis of a wide moving jam 2.1 Basic equations for a traveling wave solution

Most higher-order models can be written as follows^[23]:

(1)

(2)

where τ is the relaxation time. The velocity-density relationship v_e(ρ) is a monotonically decreasing function of the density ρ, i.e.,

The non-negative functions c₀(ρ) and c(ρ) are related to the sonic speed of perturbation, whereas

which is associated with the viscosity. The parameter β=0 or 1 represents the conservation variable v or ρv in the acceleration equation. The parameters c(ρ), c₀(ρ), and ν(ρ) of several higher-order models are shown in Table 1^{[1, 3-10]}.

The initial conditions are

(3)

Considering a traveling wave solution with the propagation speed a < 0 and substituting X=x-at into Eqs. (1)-(3), we have the higher-order model and initial conditions as follows:

(4)

(5)

(6)

The following equation can be deduced from Eq. (4):

(7)

where q^* > 0. This equation shows a simple relationship between the density ρ and the average velocity v. Therefore, Eq. (5) turns to be

(8)

where

2.2 Equations for the outer solution and correction term

Set the viscosity coefficient μ 1. The differential equation, in which an arbitrary small parameter μ multiplies the highest order derivation term, belongs to the singularly perturbation, and the boundary-layer normally occurs in the solution.

We propose the first assumption as follows: the functions G(v, q^*, a) and F(v, q^*, a) are smooth enough. According to the boundary-layer correction method^{[2, 24]}, a solution to Eq. (8) can be written as the combination of an outer solution , which is so-called because it is valid outside the boundary-layer, and a boundary-layer correction term v^(m)(ζ, μ), which is so-called because it has the property of the boundary-layer function and can correct the boundary-layer, as follows:

where

and the inner point X=X₀ is the boundary-layer point.

In the boundary-layer, the upstream front of the moving jam is contained. Moreover, the point X=X₀ divides the solution of Eq. (8) into two parts, i.e., v⁽¹⁾(X) for X < X₀ and v⁽²⁾(X) for X> X₀. In the outer part, the correction term vanishes with

(9)

At the point X=X₀, the functions v⁽¹⁾ and v⁽²⁾ satisfy the following boundary conditions:

(10)

An equilibrium flow-density function q_e(ρ)= ρv_e(ρ), which is concave for some lower density region but convex for the other, is adopted. Figure 1(a) can be derived from Eq. (7), where the phase plot (ρ, q) denotes a segment, and the traveling wave speed a equals the slope of the segment, where q(ρ)=ρv=q^*-ρa. The solution v=v(X) that describes the propagation of a wide moving jam is shown in Fig. 1(b).

The functions , v^(m)(ζ, μ) and the two sonic speeds of perturbation can be expanded as follows:

(11)

(12)

where

Substitute Eqs. (11) and (12) into Eq. (8). When μ→ 0, the coefficients of the zeroth-and first-order terms of μ on both sides of Eq. (8) are equated, and the higher order terms of μ are ignored. By separating the terms of X' and ζ, respectively, we yield the following two ordinary differential equations. One is about the outer solution as follows:

(13)

where

(14)

(15)

The other is about the correction term v₀^(m)(ζ) as follows:

(16)

Substituting Eqs. (11) and (12) into Eqs. (6), (9), and (10), we can obtain the boundary conditions and combined conditions as follows:

(17)

(18)

(19)

2.3 Correction term

We introduce a new variable v^*(ζ) defined by

Integrating Eq. (16) with respect to ζ, we have

(20)

where

(21)

The continuities of v^*(ζ) and d v^*(ζ)/dζ at ζ=0 in Eq. (19) make the integration constant B of Eq. (21) same for both ζ≤0 and ζ>0. Therefore, Eq. (18) can be rewritten as follows:

(22)

Equations (20)-(22) show that the equation f(v^*, q^*, a)=0 has two real positive roots as follows:

(23)

Therefore, the integration constant B can be written as follows:

(24)

We propose the second assumption as follows:

According to Eq. (20), we have

This means that the solution about the upstream front of the wide moving jam describes a decrease in the average velocity of vehicles (see Fig. 1(b)).

2.4 Outer solution

We consider a trivial outer solution of Eq. (13), which satisfies for X'≥0. Then, Eqs. (14) and (7) can be rewritten as follows:

(25)

where .

The downstream front of the wide moving jam is a transition-layer between two homogeneous traffic flows with the average velocity of vehicles changing from the lower one to the higher one. Therefore, this front can only start at the point and end at the point (see Fig. 1(b)), where v_B < v_A. This means that

(26)

where

The locality of the boundary-layer can be determined by the Prandtl matched principle in the boundary-layer method^[24]. Some assumptions should be proposed to ensure that the boundary-layer point is an inner point. According to Eqs. (8), (17), and (26), we have the following results. If

the locality of the boundary-layer is in the neighborhood of the right endpoint. If

the locality is in the neighborhood of the left endpoint, where

Although the outer solution is discontinuous on the point X=X₀, the solution is continuous even on the boundary-layer point by the correction term.

The translation-layer is in the outer of the boundary-layer. According to Eq. (26) and the sign of the function G, we have the following results. The smooth function is negative when X₀→ +∞ and and positive when and X₀ < X < l. Therefore, there must be one point X'=X'₁>l such that

(27)

where . Taking Eq. (13) into consideration, we have F(v_C, q^*, a)=0.

Consequently, the roots ρ=ρ_i (i=A, B, C) of Eq. (25) can be regarded as the three intersection points at which the fundamental diagram q_e(ρ)=ρv_e(ρ) is intersected by a line q(ρ)=q^*-ρa. The average velocities are

(28)

where v_B < v_C<v_A. Therefore, the equilibrium flow function should be non-concave (see Fig. 1(a)). According to Eqs. (15), (27), and (28), we have

(29)

From Fig. 1(a), we know that, the function is positive when and negative when . Combining with Eq. (13) and the sign of the function , we can obtain

This means that the solution about the downstream front of the wide moving jam describes an increase in the average velocity of vehicles (see Fig. 1(b)).

Therefore, we propose the third assumption as follows: G(v₀, q^*, a) is negative when v₀⁽¹⁾>v_C and positive when v₀⁽²⁾ < v_C, where v_C satisfies G(v_C, q^*, a)=0.

At the left side of the boundary-layer, the outer solution must be constant, i.e., v_A. In fact, we have by combining with Eqs. (17) and (26). If we assume that the solution corresponds to a decrease in the average velocity of vehicles at X'>-∞, i.e.,

Then,

Moreover, for all near v_A and . Therefore, we have

This contradicts the initial assumption. The solution corresponds to an increase in the average velocity at X'>-∞. Therefore, we have

At the point X'=0, the outer solution .

Substituting Eqs. (15), (26), and (28) into Eqs. (24) and (25), we have

(30)

(31)

Equations (29)-(31) can determine the five parameters ρ_A, ρ_B, ρ_C, a, and q^*, which are the characteristic parameters of the wide moving jam to the higher-order viscous traffic flow model for the viscosity coefficient μ1.

2.5 Sufficient conditions for the formation of a wide moving jam

We consider the three assumptions for the formation of a wide moving jam. The first one is that the functions G(v, q^*, a) and F(v, q^*, a) are smooth enough. The second one is that the function f(v^*, q^*, a) is negative when v₂^* < v^* < v₁^*. The third one is that the function is negative when and positive when . Substituting the variables v₂^* and into Eqs. (15) and (21), we have

(32)

(33)

where β=0, 1. Then, we have

Based on Eqs. (24), (26), (32), and (33), we have

and

Therefore, the sufficient conditions for the formation of a wide moving jam are as follows:

(ⅰ) The functions G(v, q^*, a) and F(v, q^*, a) are smooth enough.

(ⅱ) G(v, q^*, a) is negative when v>v_e(ρ_C) and positive when v < v_e(ρ_C), where v=v_e(ρ_C) satisfies G(v, q^*, a)=0.

If a higher-order traffic flow model satisfies the sufficient conditions, a wide moving jam can be formed, and its parameters are determined by Eqs. (29)-(31). The boundary-layer and transition-layer of the wide moving jam are shown as Fig. 1(b).

We choose several higher-order traffic flow models in Table 1 for verification. Substituting c(ρ)=0 and c₀(ρ)=c₀ into Eq. (33), we have the function G of the Kühne and KK models as follows:

When G(v, q^*, a)=0, we have v=v_e(ρ_C)=c₀+a. It is easy to see that, the function G(v, q^*, a) is negative when v>v_e(ρ_C) and positive when v < v_e(ρ_C). Therefore, the models satisfy the sufficient conditions, and can form wide moving jams. Equations (29)-(31) determine an asymptotic solution, which is smooth even at X=X₀. Since the correction term almost vanishes with a very small viscosity coefficient μ, the solution of the Kühne or KK models is "asymptotic" to that of the PW model with v or ρv being a conservative variable (see Refs. [13] and [15] for the detailed discussion).

Substituting c(ρ)=c and c₀(ρ)=0 into Eq. (33), we can obtain the function G of the JWZ viscous model as follows:

When G(v, q^*, a)=0, we have v=v_e(ρ_C)=c+a. It is easy to see that, G(v, q^*, a) is negative when v>v_e(ρ_C) and positive when v < v_e(ρ_C). Therefore, the model satisfies the sufficient conditions, and can form the wide moving jam. The solution to Eqs. (29)-(31) is "asymptotic" to that of Ref. [12] which was derived by using the weak solution theory.

Substituting c(ρ)=ρp'(ρ), c₀(ρ)=0, and β=1 into Eq. (33), we have the function G of the AR viscous model as follows:

Combining with Eq. (7), we have

where p(ρ)=αρ^γ, α>0, and γ>0. Its derivative is

Therefore, the function G increases with the density ρ, while decreases with the velocity v. When G(v, q^*, a)=0, we have v=v_e(ρ_C)=ρp'(ρ)+a. It is easy to see that the function G satisfies the condition (ⅱ). Thus, the model satisfies the sufficient conditions, and can form the wide moving jam. The solution to Eqs. (29)-(31) is "asymptotic" to that of Ref. [14], which was derived by using the weak solution theory.

Therefore, the relation between the asymptotic theory and the weak solution theory is established by the wide moving jam solutions of these higher-order traffic flow models.

2.6 Essential conditions for the formation of a wide moving jam

Lemma 1 If the derivable function f(x) is strictly convex, f(x₂)> f(x₁)+f'(x₁)(x₂-x₁) for all x₁, x₂ ∈ I and x₁≠ x₂; if the derivable function f(x) is strictly concave, f(x₂) < f(x₁)+f'(x₁)(x₂-x₁) for all x₁, x₂ ∈ I and x₁≠ x₂.

We consider the non-concave equilibrium flow function q_e(ρ), which satisfies

Equation (31) indicates that the three points (ρ_A, q_A), (ρ_B, q_B), and (ρ_C, q_C) are on one line (see Fig. 1(a)). According to the lemma and two locations of the point ρ_C, we have

(34)

(35)

The linear stability of an equilibrium solution requires that the kinetic wave speed must lie between the two characteristic velocities^{[1, 23]}, which is equivalent to

(36)

According to Eqs. (29) and (31), we yield the traveling speed

(37)

Therefore, Eqs. (34)-(36) show that the density ρ=ρ_C must be located in an instable region.

For μ1, Eq. (8) and Fig. 1(a) indicate that the function G(v, q^*, a) is positive when v_B < v < v_C. Because the functions c₀(ρ) and c(ρ) usually do not exist at the same time in the higher-order traffic flow model, we can derive a>v-c(ρ) when c₀=0 and a>v-c₀(ρ) when c=0 from the inequality G(v, q^*, a)>0. Combining with Eqs. (34) and (35), we have

and

Based on Eq. (36), we can see that the density ρ=ρ_B must be always located in the stable region.

Therefore, the essential conditions for the formation of a wide moving jam are as follows:

(ⅰ) The density ρ=ρ_C must be located in the instable region.

(ⅱ) The maximal density of the wide moving jam ρ=ρ_B is always located in the stable region when c₀=0 or c=0.

Because the Zhang model is always stable and the XD model is always unstable, the essential conditions cannot be satisfied, and thus such models cannot form wide moving jams.

3 Numerical simulation

We use the fifth-order accurately weighted essentially nonoscillatory (WENO) scheme (see Ref. [25] for the detailed discussion for the WENO scheme) to perform the numerical simulation. The conservation form of the higher-order traffic flow model is

The semi-discrete scheme of this equation reads

where these numerical fluxes are acquired by proceeding the WENO reconstruction, and v_xx in R(u) is approximated by v_xx=(v_i+1-2v_i+v_i-1)/(Δx)². The third-order total-variation-diminishing (TVD) Runge-Kutta method is chosen for the time discretization. The Courant-Friedrichs-Lewy (CFL) condition, which is necessary for the numerical stability used here, is Δt≤ 0.6/(λ/Δx+2μ/(Δx)²), where .

We use the periodic boundary condition as follows:

the initial condition as follows:

and the velocity-density function^[12] as follows:

where v_f and ρ_m are the free flow velocity and the maximal density, respectively.

Let the road length L=10 000 m, the maximal velocity v_f=30 m/s, and the mesh number N=1 000. Then, we can obtain the density evolution under the initial equilibrium flow ρ=ρ₀ with a small perturbation. Figure 2 shows the evolutions of the Kühne^[4], KK^[5], JWZ^[8], and AR^[6] models, and Fig. 3 shows the evolutions of the Zhang model^[10] and the XD model^[9], where all relative parameters are dimensionless.

According to the stability conditions Eq. (36), we know that the Kühne model is unstable for the density ρ∈ (0.206 527, 0.393 874) with the parameter c₀=0.6, the KK model is unstable for the density ρ∈ (0.186 528, 0.420 098) with the parameter c₀=0.5, the JWZ model is unstable for the density ρ∈ (0.198 453, 0.404 273) with the parameter c=0.56, and the AR model is unstable for the density ρ∈ (0.150 555, 0.440 170) with the parameters α=2.1, γ=0.25, and the function p(ρ)=αρ^γ. Figure 2 shows that wide moving jams can be formed in the Kühne, KK, JWZ, and AR models from an initial unstable equilibrium flow. It is easy to see that the maximal densities of the wide moving jams are all in the stable origins. These results are consistent with the analytic ones.

Figure 3 shows that, the small perturbation decays in the Zhang model whereas amplifies in the XD model with the increase of time. A wide moving jam cannot be formed in these two models. These results are also consistent with the analytic ones.

At the end of this section, we point out that the grids should be sufficiently refined to obtain the discussed wide moving jam solution although the fifth-order accurately WENO scheme is applied in the numerical simulation. The comparison of the evolutions of the density at different mesh sizes in the AR model is shown in Fig. 4, where the other parameters are the same as those in Fig. 2(d). By comparing Figs. 2(d) and 4(a), we know that a longer evolution time is required to form a wide moving jam when the mesh number is smaller. Choosing three mesh sizes, wide moving jams will be formed (see Fig. 4(b)). We obtain the minimum and maximum densities, i.e., ρ_A=0.144 118 and ρ_B=0.906 202 for the mesh number N=400; ρ_A=0.143 505 and ρ_B=0.943 899 for the mesh number N=800; ρ_A=0.143 377 and ρ_B=0.952 308 for the mesh number N=1 000. By comparing with the analytical result ρ_A=0.142 860 and ρ_B=0.968 573^[14], we know that the numerical solution is closer to the analytical one when the mesh number is larger.

4 Conclusions

We derive an asymptotic solution to a wide moving jam to a class of higher-order viscous traffic flow models by using the boundary-layer method with the asymptotic theory. The equations for determining the characteristic parameters of the solution are obtained. The solution is "asymptotic" to that of the corresponding inviscid model with a very small viscosity coefficient. An intrinsic link between the asymptotic theory and the weak solution theory is established. We discuss the sufficient conditions and essential conditions for the formation of a wide moving jam, respectively. As examples, these conditions are used to judge whether a wide moving jam can be formed in several higher-order viscous traffic flow models. We use the fifth-order accurately finite difference WENO scheme to perform numerical simulation, and the simulation results are in good agreement with the analytical results.