Concentration in vanishing adiabatic exponent limit of solutions to the Aw–Rascle traffic model

1. Introduction

The celebrated Aw–Rascle (AR) model of traffic flow reads (cf. [1]): (1.1) ρ t + ( ρ u ) x = 0 , ( ρ ( u + p ( ρ ) ) ) t + ( ρ u ( u + p ( ρ ) ) ) x = 0 , where ρ and u represent the traffic density and velocity of the cars located at position x at time t, respectively; p is the velocity offset and called as the “pressure” inspired from gas dynamics. The model (1.1) is now widely used to study the formation and dynamics of traffic jams. It was proposed by Aw and Rascle [1] to remedy the deficiencies of second order models of car traffic pointed out by Daganzo [8] and had also been independently derived by Zhang [46]. Since its introduction, it had received extensive attention (see [19,23–25,27,31,36,37,42,44], etc.).

In this paper, we are concerned with the “pressure” function (1.2) p ( ρ ) = ρ γ , 0 < γ < 1 .

The Riemann solutions of (1.1) with classical pressure p ( ρ ) = ρ γ ( γ > 0 ) were obtained at low densities by Aw and Rascle [1]. Lebacque, Mammar, and Salem1 [17] also solved the Riemann problem of (1.1) with classical pressure p ( ρ ) = ρ γ ( γ > 0 ) with an extended fundamental diagram for all possible initial data. Sun [36,37] studied the interactions of elementary waves to system (1.1).

We are interested in the Riemann problem for (1.1)–(1.2) with initial data (1.3) ( ρ , u ) ( 0 , x ) = ( ρ − , u − ) , x < 0 , ( ρ + , u + ) , x > 0 , where ρ ± > 0 and u ± are given constant states. We assume that u + < u − .

System (1.1)–(1.2) is just like a hyperbolic system for conservation laws of the form (1.4) ∂ t U + ∂ x F ( U ) = 0 , with U = ρ ρ u + ρ γ + 1 , F ( U ) = ρ u ρ u 2 + u ρ γ + 1 .

When γ → 0 , the limiting system of (1.1)–(1.2) formally becomes the zero pressure gas dynamics, (1.5) ρ t + ( ρ u ) x = 0 , ( ρ u ) t + ( ρ u 2 ) x = 0 , which can be used to describe the process of the motion of free particles sticking under collision [3] and depict the formation of large-scale structure in the universe [9,28]. The solutions to the zero pressure gas dynamics were widely studied by many scholars (see [2,3,9,12,13,21,22,34], etc.). In particular, the existence of measure solutions of the Riemann problem was first proved by Bouchut [2] and the existence of the global weak solution was obtained by Brenier and Grenier [3] and E, Rykov and Sinai [9]. Sheng and Zhang [34] discovered that the δ-shocks and vacuum states do occur in the Riemann solutions to the zero pressure gas dynamics (1.5) by the vanishing viscosity method. Huang and Wang [13] proved the uniqueness of the weak solution for the case when the initial data is a Radon measure.

A distinctive feature for (1.5) is just that the δ-shocks and vacuum states do occur in the Riemann solutions. In paper [31], Shen and Sun studied the limits of Riemann solutions of (1.1) with classical pressure p ( ρ ) = ε ρ γ ( γ > 0 ) as ε → 0 + . They identified a special δ-shock in the limit of solutions, whose the propagation speed and the strength are different from those of the zero pressure gas dynamics (1.5). Then, they analyzed a perturbed Aw–Rascle model and proved that the limit of Riemann solutions to the perturbed Aw–Rascle model are those of (1.5) when ε → 0 + . The idea of vanishing pressure limits dates back to early works of Li [20], Chen and Liu [5,6], and the vanishing pressure limit method was also applied to other systems [4,7,10,11,18,19,23,26,27,29–33,38–43,45,47–51].

Let us turn to the Euler system of power law in Eulerian coordinates, (1.6) ρ t + ( ρ u ) x = 0 , ( ρ u ) t + ( ρ u 2 + p ( ρ ) ) x = 0 , When the pressure tends to zero or a constant, the Euler system (1.6) formally tends to the zero pressure gas dynamics. In earlier seminal papers, Chen and Liu [6] first showed the formation of δ-shocks and vacuum states of the Riemann solutions to the Euler system (1.6) for polytropic gas by taking limit ε → 0 + in the model p ( ρ ) = ε ρ γ / γ ( γ > 1 ), which describe the phenomenon of concentration and cavitation rigorously in mathematics. Further, they also obtained the same results for the Euler equations for nonisentropic fuids in [5]. The same problem for the Euler Eqs (1.6) for isothermal case ( γ = 1 ) was studied by Li [20]. Recently, Muhammad Ibrahim, Fujun Liu and Song Liu [14] showed the same phenomenon of concentration also exists in the mode p ( ρ ) = ρ γ ( 0 < γ < 1 ) as γ → 0 , which is the case that the pressure goes to a constant. Namely, they showed rigorously the formation of delta wave with the limiting behavior of Riemann solutions to the Euler Eqs (1.6).

Motivated by [14], for the Aw–Rascle model (1.1) with classical pressure (1.2), we show the same phenomenon of concentration also exists in the case 0 < γ < 1 and u + < u − as γ → 0 . We can see that, as γ → 0 , the Riemann solution converges to a special delta shock solution, whose the propagation speed and the strength are different from those of the PGD model (1.5), which means the Riemann solution of (1.1)–(1.2) don’t converge to the delta shock solution of (1.5).

In order to solve this problem, we motivated by [31], adding a suitable perturbation in the pressure term in the Aw–Rascle model (1.1)–(1.2). That is we consider the perturbed Aw–Rascle (PAR) model as follows: (1.7) ρ t + ( ρ u ) x = 0 , ( ρ u + 1 γ ρ γ ) t + ( ρ u 2 + u ρ γ ) x = 0 , where 1 < γ < 3 . For convenience and conciseness, we replace ρ p ( ρ ) with p ( ρ ) in (1.1) and take p ( ρ ) = ρ γ for γ ∈ ( 1 , 3 ) . In the system (1.7), p ( ρ ) = ρ γ can be regarded as the traffic pressure term and 1 < γ < 3 is analogous with the adiabatic exponent 0 < γ < 2 in the Aw–Rascle model (1.1)–(1.2). It is proved that when γ → 1 , the limit of the Riemann solutions containing two shock waves of the perturbed Aw–Rascle model is exactly a delta shock solution of the zero pressure gas dynamics (1.5).

Finally, by using the fifth-order weighted essentially non-oscillatory scheme and third-order Runge-Kutta method [16,35], some representative numerical simulations are exhibited, which are completely consistent with theoretical analysis.

The rest of the paper is organized as follows. For the sake of completeness, in Section 2, we briefly review the delta shock wave and vacuum state in the Riemann solutions of the zero pressure gas dynamics (1.5). In Section 3, we display some results on the Riemann solutions of (1.1)–(1.2) when 0 < γ < 1 . In Section 4, we discuss the limits of Riemann solutions of (1.1)–(1.2) as the adiabatic exponent vanishes. In Section 5, we display some results on the Riemann solutions of (1.7) when 1 < γ < 3 . In Section 6, we show rigorously the formation of delta shock wave with the limiting behavior of Riemann solutions of (1.7) as γ → 1 . In Section 7, we present some representative numerical simulations to demonstrate the validity of the theoretical analysis. Finally, a conclusion is carried out in Section 8.

2. Preliminaries

For the sake of completeness, in this section we briefly recall the delta shock wave and vacuum state in the Riemann solutions of the zero pressure gas dynamics (1.5). More details can be found in [15,22,32,34].

The system (1.5) has a double eigenvalue λ = u and only one right eigenvector r → = ( 1 , 0 ) T . The system is obviously nonstrictly hyperbolic, and λ is linearly degenerate by ∇ λ · r → ≡ 0 , in which ▽ denotes the gradient with respect to ( ρ , u ) . Therefore, in classical sense, the associated elementary waves involve only contact discontinuities. It can be seen from previous works [15,22,32,34] that the Riemann problem for (1.5) with initial data (1.3) can be solved by contact discontinuities, vacuum or delta shock wave connecting two constant states ( ρ ± , u ± ) .

When u − < u + , there is no characteristic passing through the region u − t < x < u + t and the vacuum appears in this region. The solution can be expressed as (2.1) ( ρ , u ) ( t , x ) = ( ρ − , u − ) , − ∞ < x < u − t , ( 0 , x t ) , u − t ⩽ x ⩽ u + t , ( ρ + , u + ) , u + t < x < + ∞ .

When u − = u + , the constant states ( ρ ± , u ± ) can be connected by a contact discontinuity. The solution can be expressed as (2.2) ( ρ , u ) ( t , x ) = ( ρ − , u − ) , − ∞ < x < u − t , ( ρ + , u + ) , u − t < x < + ∞ .

When u − > u + , the characteristic lines from initial data will overlap, so the Riemann solution cannot be constructed by using the classical waves, we seek a solution containing a weighted Dirac delta function with the support on a line.

To do so, a two-dimensional weighted delta function w ( s ) δ S supported on a smooth curve S = { ( t ( s ) , x ( s ) ) : a < s < b } is defined by (2.3) ⟨ w ( t ) δ S , φ ( t , x ) ⟩ = ∫ a b w ( t ( s ) ) φ ( t ( s ) , x ( s ) ) d s , for all test functions φ ( t , x ) ∈ C 0 ∞ ( [ 0 , + ∞ ) × ( − ∞ , + ∞ ) ) .

For the Riemann problem with u + < u − , we can construct a dirac-measured solution with parameter σ as follows, (2.4) ρ ( t , x ) = ρ 0 ( t , x ) + w ( t ) δ S , u ( t , x ) = u 0 ( t , x ) , where S = { ( t , σ t ) : 0 ⩽ t < + ∞ } , (2.5) ρ 0 ( t , x ) = ρ − , x < σ t , ρ + , x > σ t , (2.6) u 0 ( t , x ) = u − , x < σ t , σ , x = σ t , u + , x > σ t , and (2.7) w ( t ) = t ( σ [ ρ ] − [ ρ u ] ) , in which [ q ] = q + − q − denotes the jump of function q across the discontinuity. The dirac-measured solution ( ρ , u ) constructed above is called a delta shock solution of (1.5) in the sense of distributions if (2.8) ⟨ ρ , φ t ⟩ + ⟨ ρ u , φ x ⟩ = 0 , (2.9) ⟨ ρ u , φ t ⟩ + ⟨ ρ u 2 , φ x ⟩ = 0 , hold for any test function φ ( t , x ) ∈ C 0 ∞ ( [ 0 , + ∞ ) × ( − ∞ , + ∞ ) ) , where ⟨ ρ , φ ⟩ = ∫ 0 + ∞ ∫ − ∞ + ∞ ρ 0 ( t , x ) φ ( t , x ) d x d t + ⟨ w ( t ) δ S , φ ( t , x ) ⟩ , ⟨ ρ u , φ ⟩ = ∫ 0 + ∞ ∫ − ∞ + ∞ ρ 0 ( t , x ) u 0 ( t , x ) φ ( t , x ) d x d t + ⟨ σ w ( t ) δ S , φ ( t , x ) ⟩ . Then the following generalized Rankine–Hugoniot relation (2.10) d x d t = σ , d w ( t ) d t = σ [ ρ ] − [ ρ u ] , d ( w ( t ) σ ) d t = σ [ ρ u ] − [ ρ u 2 ] , holds, where [ ρ ] = ρ + − ρ − , with initial data (2.11) ( x , w ) ( 0 ) = ( 0 , 0 ) .

To guarantee uniqueness, the delta shock should satisfy the entropy condition: (2.12) u + < σ < u − , which means that all the characteristic lines on both sides of the discontinuity are incoming. So it is a overcompressive condition.

Solving (2.10) with initial data (2.11) under the entropy condition (2.12), we have (2.13) w ( t ) = ρ − ρ + ( u − − u + ) t , σ = ρ + u + + ρ − u − ρ + + ρ − .

Therefore, a delta shock solution defined by (2.4) with (2.5), (2.6) and (2.13) is obtained.

3. Riemann solutions of the AR model (1.1)–(1.2)

In this section, we review the Riemann solutions of (1.1)–(1.2) with initial data (1.3), for which the detailed investigations can be found in Sun [36].

The system (1.1)–(1.2) has two eigenvalues (3.1) λ 1 = u − γ ρ γ , λ 2 = u , with the corresponding right eigenvectors r → 1 = ( 1 , − γ ρ γ − 1 ) T , r → 2 = ( 1 , 0 ) T satisfying ∇ λ 1 · r → 1 = − γ ( γ + 1 ) ρ γ − 1 < 0 , and ∇ λ 2 · r → 2 ≡ 0 . Therefore, system (1.1)–(1.2) is strictly hyperbolic for ρ > 0 , and λ 1 is genuinely nonlinear for ρ > 0 and the associated wave is either shock wave or rarefaction wave, while λ 2 is always linearly degenerate and the associated wave is the contact discontinuity.

Since (1.1), (1.2) and the Riemann data (1.3) are invariant under stretching of coordinates: ( t , x ) → ( τ t , τ x ) (τ is constant), we seek the self-similar solution ( ρ , u ) ( t , x ) = ( ρ , u ) ( ξ ) , ξ = x t . Then the Riemann problem (1.1), (1.2) and (1.3) is reduced to the following boundary value problem of the ordinary differential equations: (3.2) − ξ ρ ξ + ( ρ u ) ξ = 0 , − ξ ( ρ u + ρ γ + 1 ) ξ + ( ρ u 2 + u ρ γ + 1 ) ξ = 0 , with ( ρ , u ) ( ± ∞ ) = ( ρ ± , u ± ) .

For any smooth solution, system (3.2) can be written as (3.3) u − ξ ρ ( u − ξ ) ( u + ( γ + 1 ) ρ γ ) − ξ ρ + 2 ρ u + ρ γ + 1 ρ ξ u ξ = 0 0 . Besides the constant solution ( ρ , u ) ( ξ ) = constant ( ρ > 0 ) , it provides a rarefaction wave which is a continuous solution of (3.3) in the form ( ρ , u ) ( ξ ) . Then, for a given left state ( ρ − , u − ) , the rarefaction wave curves in the phase plane, which are the sets of states that can be connected on the right by a 1-rarefaction wave, are as follows: (3.4) R ( ρ − , u − ) : ξ = λ 1 = u − γ ρ γ , u − u − = − ( ρ γ − ρ − γ ) , ρ < ρ − , u > u − . Differentiating the second equation of (3.4) with respect to ρ yields u ρ = − γ ρ γ − 1 < 0 , and u ρ ρ = − γ ( γ − 1 ) ρ γ − 2 > 0 , which mean that for 0 < γ < 1 , the rarefaction wave curve R ( ρ − , u − ) is monotonic decreasing and convex in the ( ρ , u ) phase plane ( ρ > 0 ) . Moreover, it can be concluded from (3.4) that lim ρ → 0 + u = u − + ρ − γ for the rarefaction wave curve R ( ρ − , u − ) , which implies that R ( ρ − , u − ) intersects the u-axis at the point ( 0 , u ˜ ∗ ) , where u ˜ ∗ is determined by u ˜ ∗ = u − + ρ − γ .

For a bounded discontinuity at ξ = σ , the Rankine–Hugoniot relation (3.5) − σ [ ρ ] + [ ρ u ] = 0 , − σ [ ρ u + ρ γ + 1 ] + [ ρ u 2 + u ρ γ + 1 ] = 0 , holds, where [ ρ ] = ρ − ρ − , etc. Eliminating σ from (3.5), we obtain (3.6) [ ρ ] [ ρ u 2 ] − ( [ ρ u ] ) 2 = − [ ρ ] [ u ρ γ + 1 ] + [ ρ u ] [ ρ γ + 1 ] . Simplifying (3.6) yields ( u − u − ) 2 = − ( u − u − ) ( ρ γ − ρ − γ ) . If u − u − ≠ 0 , we have (3.7) u − u − = − ( ρ γ − ρ − γ ) and σ = u − ρ − ( ρ γ − ρ − γ ) ρ − ρ − , where σ, ( ρ − , u − ) and ( ρ , u ) are the shock speed, the left state and the right state, respectively.

Otherwise, for case u = u − (i.e., [ u ] = 0 ), we have σ = u = u − .

The classical Lax entropy conditions imply that the propagation speed σ for the 1-shock wave has to be satisfied with σ < λ 1 ( ρ − , u − ) , λ 1 ( ρ , u ) < σ < λ 2 ( ρ , u ) . From the first equation of (3.5), we obtain σ = ρ u − ρ − u − ρ − ρ − = u − + ρ ρ − ρ − ( u − u − ) . If u > u − , then from (3.7), we have ρ < ρ − , and σ − u − = ρ ρ − ρ − ( u − u − ) = − ρ ( ρ γ − ρ − γ ) ρ − ρ − = − ρ γ ρ ‾ γ − 1 , for some ρ ¯ ∈ ( ρ , ρ − ) . By direct calculation, we have γ ρ − γ − ρ γ ρ ‾ γ − 1 > γ ( ρ − γ − ρ γ ) > 0 , which implies that σ − u − > − γ ρ − γ . This contradicts with σ < λ 1 ( ρ − , u − ) . Then, given a left state ( ρ − , u − ) , the possible states that can be connected to ( ρ − , u − ) on the right by shock wave in the 1-family are as follows: (3.8) S ( ρ − , u − ) : σ = u − ρ − ( ρ γ − ρ − γ ) ρ − ρ − , u − u − = − ( ρ γ − ρ − γ ) , ρ > ρ − , u < u − . Differentiating u with respect to ρ in the second equation of (3.8) gives that for ρ > ρ − , u ρ = − γ ρ γ − 1 < 0 and u ρ ρ = − γ ( γ − 1 ) ρ γ − 2 > 0 , which means that the shock wave curve S ( ρ − , u − ) is monotonic decreasing and convex in the ( ρ , u ) phase plane ( ρ > ρ − ). It can also be derived from (3.8) that lim ρ → + ∞ u = − ∞ for the shock wave curve S ( ρ − , u − ) , which indicates that the shock wave curve intersects with the ρ-axis at a point.

Since λ 2 is linearly degenerate, the set of states ( ρ , u ) can be connected to a given left state ( ρ − , u − ) by a contact discontinuity on the right if and only if (3.9) J : ξ = u = u − .

In the ( ρ , u ) phase plane ( ρ , u ⩾ 0 ) , through a given point ( ρ − , u − ) , we draw the elementary wave curves. We find that the elementary wave curves divide the quarter phase plane ( ρ , u ⩾ 0 ) into three regions, I = { ( ρ , u ) | u < u − } , II = { ( ρ , u ) | u − < u < u ˜ ∗ } , and III = { ( ρ , u ) | u > u ˜ ∗ } , where u ˜ ∗ = u − + ρ − γ , see Fig. 1. According to the right state ( ρ + , u + ) in the different regions, one can construct the unique global Riemann solution connecting two constant states ( ρ ± , u ± ) as follows: (1)  ( ρ + , u + ) ∈ I ( ρ − , u − ) : S + J , (2)  ( ρ + , u + ) ∈ II ( ρ − , u − ) : R + J , (3)  ( ρ + , u + ) ∈ III ( ρ − , u − ) : R + Vac + J (see Fig. 1), where “+” means “followed by”.

OPEN IN VIEWER

Fig. 1.

( ρ , u ) -plane.

In this section, we study the limiting behavior of the Riemann solutions of (1.1)–(1.2) with the assumption u + < u − as γ tends to zero, that is, the formation of delta shock as γ → 0 in the case u + < u − .