Abstract
This paper is concerned with the perturbed Riemann problem for the Aw-Rascle model with the modified Chaplygin gas pressure. We obtain constructively the solutions when the initial values are three piecewise constant states. The global structure and the large-time asymptotic behaviors of the solutions are discussed case by case. Further, we obtain the stability of the corresponding Riemann solutions as the initial perturbed parameter tends to zero.
1. Introduction
In the present paper we study the Aw-Rascle (AR) macroscopic model of traffic flow which is expressed bywhere , are the density and the velocity, respectively, and is the velocity offset which is called the “pressure" inspired from gas dynamics. The AR model was established to remedy the deficiencies of second order models of car traffic presented by Daganzo [1] and was independently obtained by Zhang [2]. For the related work, we refer to [3–10] and the references cited therein.
In [10], Wang obtained the Riemann solutions to (1) withwhich is called the generalized Chaplygin gas. Wang observed that the delta shock wave occurs in the solutions. Let and , we call it the Chaplygin gas which was introduced by Chaplygin [11]. Chaplygin gas (see [12–15]) is a good candidate for dark energy.
In [16], we constructed the solutions of the elementary wave interactions for the generalized AR model (1) with (2) by resolving two initial value problems case by case and found that the Riemann solutions are globally stable.
In [17], Wang, Liu, and Yang constructed the Riemann solutions for the AR model (1) withwhich is called the modified Chaplygin gas. The authors analyzed the limit behavior when the pressure tends to zero.
In [18], Cheng and Yang studied the Riemann problem for (1) withand they investigated the limits of the solutions when the pressure tends to the Chaplygin gas pressure.
In our present paper, we study the wave interactions for (1) with the state equation (3) which can be regarded as a combination of the perfect fluid and Chaplygin gas. In order to construct the solutions of the all possible wave interactions, we consider (1) and (3) with the following three piecewise constants:where the perturbation parameter is sufficiently small. The initial data (5) is a small perturbation on the corresponding Riemann initial valueswhere . We will face the interesting question that whether the solutions of the perturbed Riemann problem (1), (3), and (5) converge to the corresponding Riemann solutions of (1), (3), and (6) as tends to zero.
The paper is organized as follows. In Section 2, for readers’ convenience we restate the results of the Riemann problem (1), (3), and (6). In Section 3, the elementary wave interactions are investigated case by case. And we obtain that the perturbed Riemann solutions of the modified AR model (1) and (3) converge to the corresponding Riemann solutions when tends to zero which shows that the corresponding Riemann solutions of (1), (3), and (6) are globally stable.
2. Preliminaries
In this section we first sketch some results on the Riemann solutions to the modified AR model (1) and (3) and the detailed study can be found in [17].
The AR model (1) has two eigenvalues: , , and the corresponding right eigenvectors are given byBy a direct calculation, we haveTherefore, is genuinely nonlinear, and is always linearly degenerate. And we know that the Riemann solutions of (1), (3), and (6) can be constructed by shock wave, rarefaction wave, or the contact discontinuity connecting the two constant states and .
For the reason that system (1) and the initial values (6) are invariant under stretching of coordinates, i.e., ( is constant), we look for the self-similar solution , . Thus we get the following boundary value problem of the ordinary differential equations:and . By solving the above problem, we obtain the constant state solution , and the rarefaction wave solution
For a bounded discontinuity at , the Rankine-Hugoniot conditions are as follows:where , , , etc.
By solving (11) and using the Lax entropy inequalities, we get the shock wave solution
Because is always linearly degenerate, from (11) we get the contact discontinuity
, , and are called the elementary waves of (1). Notice that system (1) can be written by [19]; i.e., it is the Temple type; we can see that the shock curves coincide with the rarefaction curves in the phase plane (Figure 1). This property simplifies the discussion of the wave interactions.
Since and , the above two wave curves are monotonic decreasing and convex. And we know that the shock wave curve interacts with -axis, and the rarefaction wave curve has the -axis as its asymptote.
Based on the above analysis, we have the following result.
Theorem 1. There exists a unique Riemann solution for (1) and (4) with the initial data (6). When I or IV, the unique Riemann solution is ; when II or III, the unique Riemann solution is .
3. Wave Interactions and Large-Time Behaviors
In this section, we consider the perturbed problem (1) and (3) with initial data (5). The data (5) is a small perturbation to the corresponding Riemann initial values (6). We want to determine whether the solutions of the perturbed Riemann problem (1), (3), and (5) converge to the corresponding Riemann solutions of (1), (3), and (6) as tends to zero, where are the solutions of (1), (3), and (5).
In order to cover all the cases completely, we divide our discussions into the following four cases: , , , and .
Case 1 (). In this case, we consider the interaction of emitted from and emitted from (Figure 2), where “+” means “followed by”. The occurrence of this case depends on the condition .
The propagation speed of and is and , respectively. Since , we know that will overtake at a point which is determined byIt follows thatAt the interaction point , the new Riemann problem with the initial data and is formed, which is resolved by a new and a new . For the reason that propagates with the same speed as that of , we obtain that the propagation direction of is unchanged. From we get that has the same propagation speed as that of which implies that is parallel to .
When , due to and , by a direct computation it follows that . Then we know will overtake at which satisfiesFrom (17) we get . When , a new shock wave will occur and its propagating speed is , which yields that cannot overtake forever.
When tends to zero, we get that and ; and will coincide with each other. Thus, as the time is large enough, the solution of the perturbed Riemann problem is . For this case, the limit of the perturbed Riemann solution to (1), (3), and (5) is no other than the corresponding Riemann solution to the initial problem (1), (3), and (6) which implies the global stability of the Riemann solution to (1), (3), and (6).
Case 2 (). In this case, we investigate the interaction of emitted from and emitted from (Figure 3).
Due to , we know that will overtake at the point which satisfiesA direct computation yields thatWhen , begins to penetrate , and during the penetration satisfiesDifferentiating with respect to in the second equation of (20), we know thatDifferentiating with respect to in the third equation of (20), we obtain thatSubstituting the first equation of (20) into the above representations yields thatandFrom the above discussions, we conclude that accelerates during the process of penetration.
From (23) and the condition , we conclude that the contact discontinuity can penetrate completely the rarefaction wave in a finite time.
After the penetration, a new and a new will occur. Due to and , the propagation direction of keeps unchanged during the whole penetration. Notice that is parallel to since .
When tends to zero, the points and tend to and coincides with . Since the propagation speed of the wave back in equals that of the wave front in , it follows that coincides with when tends to zero. Thus, we know for this case the result is for large enough time. And the perturbed Riemann solution converges to the corresponding Riemann solution.
Case 3 (). In this case, we discuss the interaction of emitted from and emitted from (Figure 4).
By similar discussions to that in Case 1, a new shock and a new will occur after the interaction of with . Notice that propagates with at the same speed and is parallel to due to . intersects with at which is determined by (14).
Because the propagation speed of the wave front in is larger than the propagation speed in , it follows that will penetrate at which satisfieswhere and are, respectively, the propagation speed of the wave front in and the propagation speed of .
By a direct computation, we get . When , begins to penetrate and we haveFrom the first and the second expression in (26) we know thatand from the third expression we obtain thatSubstituting (28) into (27), due to we haveFrom (28)From (26), (28), and (29), we havetelling us that decelerates during the above penetration. From (29) we conclude that for the case , will penetrate completely due to at the finite time. While for the other case , cannot penetrate forever because when and has as its asymptote.
(a) Wave interactions as
(b) Wave interactions as
For the large time, the solution is (Figure 4(a)) as , and the solution is (Figure 4(b)) as . For this case, we obtain the stability of the Riemann solution to the initial problem (1), (3), and (6).
Case 4 (). In this case, we study the interaction of emitted from and emitted from (Figure 5).
By similar discussions to that of Case 2, a new and a new will occur after the intersection of and . Notice that keeps the same propagation speed and is parallel to because .
Similarly to Case 3, as , we obtain that will penetrate completely in a finite time and a new shock will occur whose propagation speed is less than that of . It implies that for the large time the solution is (Figure 5(a)). As , cannot penetrate forever and it has as its asymptote, which shows that for the large time the solution is (Figure 5(b)).
For this case, we get the global stability of the corresponding Riemann solution.
(a) Wave interactions as
(b) Wave interactions as
Now we have discussed all possible wave interactions in the phase plane . Based on the above discussions, we obtain our main result.
Theorem 2. The limits of the perturbed Riemann solutions of (1), (3), and (5) are exactly the corresponding Riemann solutions of (1), (3), and (6). The asymptotic behavior of the perturbed Riemann solutions is governed completely by the states . The Riemann solutions of the initial value problem (1), (3), and (6) are stable under such small perturbation to the initial values.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study is partially supported by NSFC 11326156 and by the Foundation for Young Scholars of Shandong University of Technology (no. 115024).