Advances in Mathematical Physics

Volume 2018, Article ID 7104527, 7 pages

https://doi.org/10.1155/2018/7104527

## The Perturbed Riemann Problem for the Aw-Rascle Model with Modified Chaplygin Gas Pressure

School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255000, China

Correspondence should be addressed to Yujin Liu; moc.621@89uiljy

Received 12 January 2018; Revised 19 May 2018; Accepted 22 May 2018; Published 2 July 2018

Academic Editor: Antonio Scarfone

Copyright © 2018 Yujin Liu and Wenhua Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.