Riemann Problem for the Aw–Rascle Model of Traffic Flow with General Pressure

Abstract

Riemann problem for the Aw–Rascle model of traffic flow with general pressure law is solved analytically. The Riemann solutions with three or four kinds of different structures involving the delta shock wave in three cases are obtained. For the delta shock wave, the generalized Rankine–Hugoniot relation and entropy condition are clarified. The theoretical analysis is confirmed by the numerical results finally.

1 Introduction

The Aw–Rascle (AR) model of traffic flow in the conservative form reads

$$\begin{aligned} \left\{ \begin{array}{l} \rho _t+(\rho u)_x=0,\\ (\rho (u+p(\rho )))_t+(\rho u(u+p(\rho )))_x=0, \end{array}\right. \end{aligned}$$

(1.1)

which was proposed in 2000 by Aw and Rascle [1] to study the formation and dynamics of traffic jams. Here $\rho \ge 0$ denotes the traffic density, and $u\ge 0$ the traffic velocity. Inspired by gas dynamics, the velocity offset $p(\rho )$ is called as the “pressure”. In particular, the model (1.1) can be used to explain instabilities near the vacuum, i.e., for light traffic with few slow drivers.

The model (1.1) has attracted extensive attention since 2000. For example, the Riemann problem for the system (1.1) with the state equation of pressure

$$\begin{aligned} p(\rho )=\rho ^\gamma ,\quad \gamma >0 \end{aligned}$$

(1.2)

was solved in [1]. The interactions of elementary waves were studied by Sun [26]. Shen and Sun [24] analyzed the formation of delta-shocks and vacuum states in vanishing pressure limit of Riemann solutions to the perturbed system of (1.1), in which they replaced $\rho p(\rho )$ by $p(\rho )$.

Interestingly enough, the modified Chaplygin gas pressure law

$$\begin{aligned} p(\rho )=A\rho -\displaystyle \frac{B}{\rho ^\alpha },\quad 0<\alpha \le 1,\quad A,B>0 \end{aligned}$$

(1.3)

proposed by Benaoum [2] can be used to describe the current accelerated expansion of the universe. In 2011, Saul Perlmutter, Brian P. Schmidt and Adam G. Riess shared the Nobel Prize in Physics for their discovery that the expansion of the universe is accelerating through observations of distant supernovae. It is believed that the source of this acceleration may be due to an exotic type of fluid with negative pressure called commonly dark energy. As the dark energy was introduced, the different candidates that can effectively play the role of dark energy have been sought. So far, to interpret the behavior of dark energy, people proposed a variety of theoretical models, in which the Chaplygin gas and modified Chaplygin gas are plausible.

It is noted that, as $A=0$, (1.3) becomes the generalized Chaplygin gas [23]

$$\begin{aligned} p(\rho )=-\displaystyle \frac{B}{\rho ^\alpha }. \end{aligned}$$

(1.4)

Besides, if $\alpha =1$, it is just the Chaplygin gas

$$\begin{aligned} p(\rho )=-\displaystyle \frac{B}{\rho }, \end{aligned}$$

(1.5)

which was introduced by Chaplygin [6], Tsien [28] and von Karman [13] as a suitable mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics. Recently, Pan and Han [21] solved the Riemann problem for (1.1) and (1.5). Cheng and Yang [8] studied the formation of delta-shock in solutions to the system (1.1) as the pressure function (1.3) satisfying $\alpha =1$ tends to the pressure (1.5). Moreover, Wang [31] considered the Riemann problem for the system (1.1) with (1.4). Liu and Sun [18] studied the wave interactions and stability of Riemann solutions of (1.1) and (1.4).

As to the delta-shock, it is a kind of nonclassical waves on which at least one of the state variables may develop an extreme concentration. A delta-shock is more compressive than an ordinary shock in the sense that more characteristics enter the discontinuity. It can be used to interpret the process of concentration of mass or formation of the galaxies in the universe. The study of the delta-shock has been made a great many progresses. For instance, Korchinski [15] considered a Riemann problem for a $2\times 2$ system of conservation laws possessing no classical weak solution. Keyfitz and Kranzer [14] studied a strictly hyperbolic system of conservation laws admitting singular shock. By introducing measure value solution, Le Floch [19] established an existence and uniqueness of weak solutions for two nonstrictly hyperbolic systems. Tan and Zhang [29] independently discovered the delta shock wave in the study of Riemann problem for two-dimensional Burgers-type equations. Tan et al. [30] established the existence, uniqueness, and stability of delta-shocks to some viscous perturbations in the reduced one-dimensional system. Sheng and Zhang [25] studied the 1-D and 2-D Riemann problems for the well-known zero-pressure flow, and obtained the solutions involving delta-shock and vacuum with vanishing viscosity method. Yang [33] studied the Riemann problem for a class of coupled systems including the zero-pressure flow as a prototype, and proved the existence, uniqueness and stability of delta-shocks with vanishing viscosity method. For more details of delta-shock, please refer to the papers by Joseph [12], Bouchut [5], Weinan et al. [32], Brenier [3, 4], Li et al. [17], Li and Yang [16], Ding and Wang [10], Huang [11], Chen and Liu [7], Danilov and Shelkovich [9], Panov and Shelkovich [22], Yang and Liu [34], Yang and Zhang [35, 36], etc.

From the above analysis, one can observe that the researches of (1.1) mainly focus on the specific pressure laws (1.2)–(1.5). The different pressure laws highly effect the solutions of the model (1.1). In addition, the important feature of (1.1) with (1.4), (1.5) is that the delta-shock develops in solutions, which is different from (1.1) with the pressure laws (1.2), (1.3). Then what is the difference of the system (1.1) with the pressure functions (1.2)–(1.5) and general ones? Specially, under what conditions does the pressure result in the delta-shock appearing in solutions? These motivate us to consider the system (1.1) with general pressure satisfying the following four hypotheses

Here it is observed that $(H_3)$ and $(H_4)$ imply $p'(\rho )>0$, i.e., the pressure is increasing with respect to the traffic density. Consequently, the limit $\lim \limits _{\rho \rightarrow 0}p(\rho )$ possesses the three cases:

Case 1 $\lim \limits _{\rho \rightarrow 0}p(\rho )=-\infty$, Case 2 $\lim \limits _{\rho \rightarrow 0}p(\rho )=k\ (p(0)\ \text{ undefined })$, Case 3 $p(0)=k$,

where k is a negative constant.

In particular, if inequalities $(H_2)$ and $(H_4)$ are equal to 0, then taking into account $(H_1)$ and $(H_3)$, one can get $p(\rho )=-\frac{C}{\rho }$ (C is a positive constant), which is exactly the Chaplygin gas (1.5). Therefore, $(H_1)$–$(H_4)$ cover a lot of functions, such as the generalized Chaplygin gas (1.4), $p(\rho )=-\ln (1+\frac{1}{\rho })$, $p(\rho )=-\arctan \frac{1}{\rho }$, $p(\rho )=-\frac{1}{1+\rho ^\alpha } (0<\alpha \le 1)$, $p(\rho )=\arctan \rho -\frac{\pi }{2}$, and $p(\rho )=-\arctan \frac{1-\rho }{1+\rho }-\frac{\pi }{4}$, etc.

Here it is noticed that the pressure $p(\rho )$ above is only a function of the density $\rho$. If the pressure p is also related to the state variable u, the delta shock wave may be also involved in the Riemann solution. See Sun [27] for a macroscopic production model. However, this is a complex problem for the Chaplygin gas pressure law which will be studied in the future.

In the present paper, under the hypotheses $(H_1)$–$(H_4)$, we consider the Riemann problem for (1.1) with initial data

$$\begin{aligned} (\rho ,u)(0,x)=\left\{ \begin{array}{l} (\rho _-,u_-),\quad x<0,\\ (\rho _+,u_+),\quad x>0. \end{array}\right. \end{aligned}$$

(1.6)

Since our focus is on the delta shock wave, we assume $u_-+p(\rho _-)>0$ throughout this paper. Firstly, the classical elementary waves including rarefaction wave, shock wave and contact discontinuity are obtained. With the classical wave curves, the phase plane is divided into three domains ${\mathrm{I}},\ {\mathrm{II}}, \ {\mathrm{III}}(\rho _-,u_-)$ for Case 1 and four domains ${\mathrm{I}},\ {\mathrm{II}}, \ {\mathrm{III}},\ {\mathrm{IV}}(\rho _-,u_-)$ for Case 2 and Case 3, respectively. Using the method of phase plane analysis, the solutions with three or four kinds of different configurations are obtained. More precisely, for Case 1 and Case 2, the Riemann solutions are $R+J$ when $(\rho _+,u_+)\in$$I(\rho _-,u_-)$, and $S+J$ when $(\rho _+,u_+)\in {\mathrm{II}}(\rho _-,u_-)$. Nevertheless, there is no solution when $(\rho _+,u_+)\in$${\mathrm{IV}}(\rho _-,u_-)$ in Case 2. For Case 3, the Riemann solutions are $R+J$ when $(\rho _+,u_+)\in {\mathrm{I}}(\rho _-,u_-)$, $S+J$ when $(\rho _+,u_+)\in$${\mathrm{II}}(\rho _-,u_-)$, and $R+\text{ Vacuum }+J$ when $(\rho _+,u_+)\in$${\mathrm{IV}}(\rho _-,u_-)$. However, for $(\rho _+,u_+)\in {\mathrm{III}}(\rho _-,u_-)$ in the three cases, the delta shock wave occurs in solutions. For the delta-shock, the generalized Rankine–Hugoniot relation is clarified, and entropy condition is proposed. By solving the generalized Rankine–Hugoniot relation under entropy condition, the existence and uniqueness of the delta-shock solution is established.

This paper is organized as follows. Section 2 solves classical elementary waves and then constructs the classical Riemann solutions. In Sect. 3, we construct the delta-shock solution. Section 4 presents some representative numerical results.

2 Classical Riemann Solutions

The characteristic roots of the system (1.1) are

$$\begin{aligned} \lambda _1=u-\rho p'(\rho ),\quad \lambda _2=u, \end{aligned}$$

(2.1)

which means that it is strictly hyperbolic. The corresponding right characteristic vectors are

$$\begin{aligned} \overrightarrow{r}_1=(1,-p'(\rho ))^T, \quad \overrightarrow{r}_2=(1,0)^T. \end{aligned}$$

(2.2)

Moreover, one can check that (2.1) and (2.2) satisfy

$$\begin{aligned} \nabla \lambda _1\cdot \overrightarrow{r}_1=-(\rho p''(\rho )+2p'(\rho ))<0,\quad \nabla \lambda _2\cdot \overrightarrow{r}_2=0, \end{aligned}$$

which show that the first characteristic field is genuinely nonlinear and the associated wave is either rarefaction wave or shock wave and the second characteristic field is linearly degenerate and the associated wave is contact discontinuity.

Since both (1.1) and (1.6) remain invariant under the transformation $x\rightarrow \beta x$, $t\rightarrow \beta t$, $\beta >0$, we look for the self-similar solution $(\rho ,u)(t,x)=(\rho ,u)(\xi )(\xi =x/t)$. Then the Riemann problem (1.1) and (1.6) is reduced to a boundary value problem for ordinary differential equations

$$\begin{aligned} \left\{ \begin{array}{l} -\xi \rho _\xi +(\rho u)_\xi =0,\\ -\xi (\rho ( u+p(\rho )))_\xi +(\rho u(u+p(\rho )))_\xi =0 \end{array}\right. \end{aligned}$$

(2.3)

with

$$\begin{aligned} (\rho ,u)(\pm \infty )=(\rho _\pm ,u_\pm ). \end{aligned}$$

(2.4)

Any smooth solution of (2.3) satisfies

$$\begin{aligned} \left( \begin{array}{cc} u-\xi &{}\quad \rho \\ 0 &{}\quad \rho (u-\rho p'(\rho )-\xi ) \end{array}\right) \left( \begin{array}{c} {\mathrm{d}}\rho \\ {\mathrm{d}}u \end{array}\right) =0, \end{aligned}$$

which provides either the constant state $(\rho ,u)(\xi )=\text{ constant },$ or singular solutions

$$\begin{aligned} R:\ \ \left\{ \begin{array}{l} \xi =\lambda _1=u-\rho p'(\rho ),\\ {\mathrm{d}}u+p'(\rho ){\mathrm{d}}\rho =0 \end{array}\right. \end{aligned}$$

(2.5)

and

$$\begin{aligned} \left\{ \begin{array}{l} \xi =u,\\ \rho =0. \end{array}\right. \end{aligned}$$

(2.6)

(2.6) is called as vacuum state, which develops only in Case 3.

From (2.5), it follows that

$$\begin{aligned} \frac{{\mathrm{d}}\lambda _1}{{\mathrm{d}}\rho }=\frac{\partial \lambda _1}{\partial u}\frac{{\mathrm{d}}u}{{\mathrm{d}}\rho }+\frac{\partial \lambda _1}{\partial \rho }=-(\rho p''(\rho )+2p'(\rho ))<0. \end{aligned}$$

Integrating (2.5) and considering the requirement $\lambda _1(\rho ,u)>\lambda _1(\rho _-,u_-)$, we obtain the rarefaction wave

$$\begin{aligned} R:\ \ \left\{ \begin{array}{l} \xi =\lambda _1=u-\rho p'(\rho ),\\ u=u_-+p(\rho _-)-p(\rho ),\ \ \ \ \rho <\rho _-. \end{array}\right. \end{aligned}$$

(2.7)

A discontinuity at $\xi =\sigma$ should satisfy the Rankine–Hugoniot relation

$$\begin{aligned} \left\{ \begin{array}{l} -\sigma [\rho ]+[\rho u]=0,\\ -\sigma [\rho (u+p(\rho ))]+[\rho u(u+p(\rho ))]=0, \end{array}\right. \end{aligned}$$

(2.8)

where $[g]=g_--g$ is the jump of g across the discontinuity.

Solving (2.8), we obtain the shock wave

$$\begin{aligned} S:\left\{ \begin{array}{l} \sigma =u_--\displaystyle \frac{\rho p(\rho _-)-\rho p(\rho )}{\rho _--\rho },\\ u=u_-+p(\rho _-)-p(\rho ) \end{array}\right. \end{aligned}$$

(2.9)

and the contact discontinuity

$$\begin{aligned} J:\left\{ \begin{array}{l} \sigma =u,\\ u=u_-. \end{array}\right. \end{aligned}$$

(2.10)

By the Lax entropy inequalities, the shock wave satisfies

$$\begin{aligned} \sigma<\lambda _1(\rho _-,u_-)<\lambda _2(\rho _-,u_-),\quad \lambda _1(\rho ,u)<\sigma <\lambda _2(\rho ,u), \end{aligned}$$

(2.11)

which are equivalent to $\rho >\rho _-$. So the admissible shock wave can be shown as

$$\begin{aligned} S:\left\{ \begin{array}{l} \sigma =u_--\displaystyle \frac{\rho p(\rho _-)-\rho p(\rho )}{\rho _--\rho },\\ u=u_-+p(\rho _-)-p(\rho ),\quad \rho >\rho _-. \end{array}\right. \end{aligned}$$

(2.12)

The curves in the $(\rho ,u)$-plane expressed by the second equations of (2.7), (2.10) and (2.12) are called as the rarefaction wave curve, contact discontinuity curve and shock wave curve, respectively. Obviously, the contact discontinuity line $u=u_-$ is parallel to $\rho$-axis. The shock wave and rarefaction wave curves possess the following properties.

Lemma 2.1

For the shock wave curve,

$$\begin{aligned} S(\rho _-,u_-): u=u_-+p(\rho _-)-p(\rho ),\quad \rho >\rho _-, \end{aligned}$$

we have $u_\rho <0$, $u_{\rho \rho }>0$ and $\lim \limits _{\rho \rightarrow +\infty }u=u_-+p(\rho _-)$.

Proof

Differentiating the second equation of (2.12) with respect to $\rho$, we get $u_\rho =-p'(\rho )<0, \ u_{\rho \rho }=-p''(\rho )>0$, which mean that it is monotonic decreasing and convex. By virtue of $(H_1)$, one can calculate

$$\begin{aligned} \lim \limits _{\rho \rightarrow +\infty }u=u_-+p(\rho _-) -\lim \limits _{\rho \rightarrow +\infty }p(\rho )=u_-+p(\rho _-), \end{aligned}$$

which shows that the shock wave curve has the line $u=u_-+p(\rho _-)$ as its asymptote. The proof is completed. $\square$

Lemma 2.2

For the rarefaction wave curve,

$$\begin{aligned} R(\rho _-,u_-): u=u_-+p(\rho _-)-p(\rho ),\quad \rho <\rho _-, \end{aligned}$$

we have $u_\rho <0$, $u_{\rho \rho }>0$. For Case 1, it has u-axis as its asymptote. For Case 2, it tends to the point $u_-+p(\rho _-)-k$ as $\rho \rightarrow 0$. For Case 3, it intersects u-axis at the point $(u_-+p(\rho _-)-k,0)$.

Proof

From the second equation of (2.7), it follows that $u_\rho =-p'(\rho )<0, \ u_{\rho \rho }=-p''(\rho )>0$. The facts show that it is monotone decreasing and convex. In addition, for Case 1, we have

$$\begin{aligned} \lim \limits _{\rho \rightarrow 0}u(\rho )=u_-+p(\rho _-) -\lim \limits _{\rho \rightarrow 0}p(\rho )=+\infty , \end{aligned}$$

while

$$\begin{aligned} \lim \limits _{\rho \rightarrow 0}u(\rho )=u_-+p(\rho _-)-k \end{aligned}$$

for Case 2. One can easily obtain $u(0)=u_-+p(\rho _-)-k$ for Case 3. The proof is finished.

Now we construct the Riemann solutions. Fixing a left state $(\rho _-,u_-)$ in the $(\rho ,u)$-plane, we draw the wave curves. As shown in Figs. 1, 2, 3, the phase plane is divided into three domains ${\mathrm{I}},\ {\mathrm{II}}, \ {\mathrm{III}}(\rho _-,u_-)$ for Case 1 and four domains ${\mathrm{I}},\ {\mathrm{II}}, \ {\mathrm{III}},\ {\mathrm{IV}}(\rho _-,u_-)$ for Case 2 and Case 3, respectively. $\square$

With the classical elementary waves R, S and J, the unique global Riemann solutions for Case 1 can be constructed as follows:

• (i) when $(\rho _+,u_+)\in {\mathrm{I}}(\rho _-,u_-)$, the solution is $R+J$;

• (ii) when $(\rho _+,u_+)\in {\mathrm{II}}(\rho _-,u_-)$, the solution is $S+J$.

For Case 2, two kinds of solutions are listed:

• (i) when $(\rho _+,u_+)\in {\mathrm{I}}(\rho _-,u_-)$, the solution is $R+J$;

• (ii) when $(\rho _+,u_+)\in {\mathrm{II}}(\rho _-,u_-)$, the solution is $S+J$.

One can see that, as $(\rho _+,u_+)\in {\mathrm{IV}}(\rho _-,u_-)$, i.e., $u_+\ge u_-+p(\rho _-)-k$, there is no solution.

For Case 3, the unique global Riemann solutions are expressed in the following:

• (i) when $(\rho _+,u_+)\in {\mathrm{I}}(\rho _-,u_-)$, the solution is $R+J$;

• (ii) when $(\rho _+,u_+)\in {\mathrm{II}}(\rho _-,u_-)$, the solution is $S+J$;

• (iii) when $(\rho _+,u_+)\in {\mathrm{IV}}(\rho _-,u_-)$, the solution is $R+\text{ Vacuum }+J$.

In these solutions, the nonvacuum intermediate state $(\rho _*,u_*)$ satisfies

$$\begin{aligned} u_*=u_+,\quad u_*=u_-+p(\rho _-)-p(\rho _*). \end{aligned}$$

(2.13)

Fig. 1

Elementary wave curves for Case 1

Fig. 2

Elementary wave curves for Case 2

Fig. 3

Elementary wave curves for Case 3

However, when $(\rho _+,u_+)\in {\mathrm{III}}(\rho _-,u_-)$ in the three cases, i.e., as $u_+\le u_-+p(\rho _-)$, the Riemann solutions cannot be constructed by the classical waves. Meanwhile, the delta-shock will appear in solutions.

3 Delta-Shock Solutions

This section gives a detailed discussion when $(\rho _+,u_+)\in {\mathrm{III}}(\rho _-,u_-)$ in Cases 1–3. In these situations, the characteristic lines from initial data will overlap in a domain $\Omega$ shown in Fig. 4, so the singularity of solution must happen in $\Omega$. Because the Rankine–Hugoniot relation is no longer valid on the bounded jump, the singularity is impossible to be a jump with finite amplitude. In other words, there is no piecewise smooth and bounded solution. Motivated by [25, 33, 35], the delta-shock solution will be sought. To this end, the definitions of two-dimensional weighted delta function and delta-shock solution are introduced as follows.

Fig. 4

Analysis of characteristics of the delta-shock

Definition 3.1

A two-dimensional weighted delta function $\omega (s)\delta _S$ supported on a smooth curve $S=\{(t(s),x(s)): c\le s\le d\}$ is defined by

$$\begin{aligned} \big <\omega (s)\delta _S, \phi (t,x)\big >=\displaystyle \int _c^{\mathrm{d}}\omega (s)\phi (t(s),x(s)){\mathrm{d}}s \end{aligned}$$

(3.1)

for all test functions $\phi \in C^\infty _0([0,+\infty )\times (-\infty ,+\infty ))$.

Definition 3.2

A pair $(\rho ,u)$ is called a delta-shock solution to the system (1.1) in the sense of distributions if there exist a smooth curve $S=\{(t,x(t)):0\le t\le \infty \}$ and a weight $\omega$ such that $\rho$ and u are represented in the following form

$$\begin{aligned} \rho (t,x)=\rho _0(t,x)+\omega (t)\delta _S,\quad u(t,x)=u_0(t,x),\quad u(t,x)|_S=u_\delta (t), \end{aligned}$$

(3.2)

where $\rho _0(t,x)=\rho _l(t,x)-[\rho ]H(x-x(t))$, $u_0(t,x)=u_l(t,x)-[u]H(x-x(t))$, in which $(\rho _l,u_l)(t,x)$ and $(\rho _r,u_r)(t,x)$ are piecewise smooth solutions to the system (1.1), H(x) is the Heaviside function

$$\begin{aligned} H(x)=\ \ \left\{ \begin{array}{l} 0 \quad x<0,\\ 1 \quad x>0, \end{array}\right. \end{aligned}$$

$\omega \in C^1(S)$, $u_\delta (t)$ is the tangential derivative of curve S, $p(\rho )$ is set as

$$\begin{aligned} p(\rho )=\left\{ \begin{array}{ll} p(\rho _l),&{}\quad x<x(t), \\ 0,&{}\quad x=x(t),\\ p(\rho _r),&{}\quad x>x(t) \end{array}\right. \end{aligned}$$

(3.3)

by the assumption $(H_1)$, and it satisfies

$$\begin{aligned}&\langle \rho ,\phi _t\rangle +\langle \rho u, \phi _x\rangle =0, \nonumber \\&\langle \rho (u+p(\rho )),\phi _t\rangle +\langle \rho u(u+p(\rho )),\phi _x\rangle =0 \end{aligned}$$

(3.4)

for all test function $\phi \in C_{0}^{\infty }([0,+\infty )\times (-\infty ,+\infty ))$, where

$$\begin{aligned} \langle \rho ,\phi \rangle= & {} \displaystyle \int _{0}^{+\infty }\int _{-\infty }^{+\infty }\rho _0\phi {\mathrm{d}}x {\mathrm{d}}t+\langle \omega \delta _S,\phi \rangle ,\\ \langle \rho (u+p(\rho )), \phi \rangle= & {} \displaystyle \int _{0}^{+\infty }\int _{-\infty }^{+\infty }\rho _0(u_0+p(\rho _0))\phi {\mathrm{d}}x {\mathrm{d}}t+\langle \omega u_\delta \delta _S,\phi \rangle . \end{aligned}$$

From these definitions, the delta-shock solution to the system (1.1) can be also written in the following form

$$\begin{aligned} (\rho ,u)(t,x)=\left\{ \begin{array}{ll} (\rho _l,u_l)(t,x),&{}\quad x<x(t),\\ \big ( \omega (t)\delta (x-x(t)),u_\delta (t)\big ),&{}\quad x=x(t),\\ (\rho _r,u_r)(t,x),&{}\quad x>x(t), \end{array}\right. \end{aligned}$$

(3.5)

where $\delta (\cdot )$ is the standard Dirac measure.

Theorem 3.1

A pair $(\rho ,u)$ of the form (3.5) is a solution to the system (1.1) in the sense of distributions if the relation

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{{\mathrm{d}}x(t)}{{\mathrm{d}}t}=u_\delta (t),\\ \displaystyle \frac{{\mathrm{d}}\omega (t)}{{\mathrm{d}}t}=-[\rho ]u_\delta (t)+[\rho u],\\ \displaystyle \frac{{\mathrm{d}}\omega (t)u_\delta (t)}{{\mathrm{d}}t}=-[\rho ( u+p(\rho ))]u_\delta (t)+[\rho u(u+p(\rho ))] \end{array}\right. \end{aligned}$$

(3.6)

holds.

Proof

For any test function $\phi \in C^{\infty }_{0}([0,+\infty )\times (-\infty ,+\infty ))$, with Green’s formula and integrating by parts, one can calculate

$$\begin{aligned}&\langle \rho (u+p(\rho )),\phi _t\rangle +\langle \rho u(u+p(\rho )),\phi _x\rangle \\&\quad =\displaystyle \int ^{+\infty }_{0}\int ^{x(t)}_{-\infty } (\rho _l(u_l+p(\rho _l))\phi _t+\rho _lu_l(u_l+p(\rho _l))\phi _x) {\mathrm{d}}x{\mathrm{d}}t\\&\qquad +\,\displaystyle \int ^{+\infty }_{0}\int _{x(t)}^{+\infty } (\rho _r(u_r+p(\rho _r))\phi _t +\rho _ru_r(u_r+p(\rho _r))\phi _x) {\mathrm{d}}x{\mathrm{d}}t \\&\qquad +\,\displaystyle \int ^{+\infty }_{0}\omega (t)u_\delta (t)(\phi _t+u_\delta (t)\phi _x){\mathrm{d}}t\\&\quad =\displaystyle \int ^{+\infty }_{0}\int ^{x(t)}_{-\infty }\{(\rho _l(u_l+p(\rho _l))\phi )_t+(\rho _lu_l(u_l+p(\rho _l))\phi )_x\}{\mathrm{d}}x{\mathrm{d}}t\\&\qquad +\,\displaystyle \int ^{+\infty }_{0}\int _{x(t)}^{+\infty }\{(\rho _r(u_r+p(\rho _r))\phi )_t+(\rho _ru_r(u_r+p(\rho _r))\phi )_x\}{\mathrm{d}}x{\mathrm{d}}t \\&\qquad +\,\displaystyle \int ^{+\infty }_{0}\omega (t)u_\delta (t)\frac{d\phi }{{\mathrm{d}}t}{\mathrm{d}}t\\&\quad =\displaystyle \oint _{\partial \Omega _-}-\rho _l(u_l+p(\rho _l))\phi {\mathrm{d}}x+\rho _lu_l(u_l+p(\rho _l))\phi {\mathrm{d}}t \\&\qquad +\displaystyle \oint _{\partial \Omega _+}\{-(\rho _r(u_r+p(\rho _r)))\phi {\mathrm{d}}x\\&\qquad +\,\rho _ru_r(u_r+p(\rho _r))\phi \}{\mathrm{d}}t-\displaystyle \int ^{+\infty }_{0}\frac{{\mathrm{d}}\omega (t)u_\delta (t)}{{\mathrm{d}}t}\phi {\mathrm{d}}t\\&\quad =\displaystyle \int ^{+\infty }_{0}\Big (-\rho _l(u_l+p(\rho _l))\frac{{\mathrm{d}}x}{{\mathrm{d}}t}+\rho _lu_l(u_l+p(\rho _l)) +\rho _r(u_r+p(\rho _r))\frac{{\mathrm{d}}x}{{\mathrm{d}}t}\\&\qquad -\,\rho _ru_r(u_r+p(\rho _r))\Big )\phi {\mathrm{d}}t -\displaystyle \int ^{+\infty }_{0}\frac{{\mathrm{d}}\omega (t)u_\delta (t)}{{\mathrm{d}}t}\phi {\mathrm{d}}t\\&\quad =\displaystyle \int ^{+\infty }_{0}\Big \{-[\rho (u+p(\rho )]u_\delta (t)+[\rho u(u+p(\rho ))]-\frac{{\mathrm{d}}\omega (t)u_\delta (t)}{{\mathrm{d}}t}\Big \}\phi {\mathrm{d}}t, \end{aligned}$$

where $\Omega _-={\{(t,x):t\in R^+,-\infty<x<x(t)\}},$$\Omega _+=\{(t,x):t\in R^+,x(t)<x$$<+\infty \},$$\partial \Omega _\pm$ is the boundaries of $\Omega _\pm$.

Analogously, one can get

$$\begin{aligned} \langle \rho ,\phi _t\rangle +\langle \rho u, \phi _x\rangle =\displaystyle \int ^{+\infty }_{0}\Big \{-[\rho ]u_\delta (t)+[\rho u]-\frac{{\mathrm{d}}\omega (t)}{{\mathrm{d}}t}\Big \}\phi {\mathrm{d}}t. \end{aligned}$$

The proof is finished. $\square$

Relation (3.6) in Theorem 3.1 is called as the generalized Rankine–Hugoniot relation, which reflects the exact relationship among the limit states on two sides of the discontinuity, the weight, propagation speed and the location of the discontinuity.

In addition, to guarantee uniqueness, the discontinuity satisfies the over-compressive entropy condition

$$\begin{aligned} \lambda _1(\rho _r,u_r)<\lambda _2(\rho _r,u_r)\le \displaystyle \frac{{\mathrm{d}}x(t)}{{\mathrm{d}}t}\le \lambda _1(\rho _l,u_l)<\lambda _2(\rho _l,u_l), \end{aligned}$$

(3.7)

which means that all characteristics on both sides of the discontinuity are incoming.

A discontinuity in Definition 3.2 satisfying (3.6), (3.7) is called as a delta shock wave of the system (1.1), symbolized by $\delta$.

In what follows, we proceed to solve the Riemann problem (1.1) and (1.6) when $(\rho _+,u_+)\in {\mathrm{III}}(\rho _-,u_-)$ in the three cases. At this moment, the Riemann solution is a delta-shock of the form, besides two constant states,

$$\begin{aligned} (\rho ,u)(t,x)=\left\{ \begin{array}{ll} (\rho _-,u_-),&{}\quad x<x(t),\\ \big ( \omega (t)\delta (x-x(t)),u_\delta (t)\big ),&{}\quad x=x(t),\\ (\rho _+,u_+),&{}\quad x>x(t). \end{array}\right. \end{aligned}$$

(3.8)

To determine $x(t),\omega (t)$ and $u_\delta (t)$ uniquely, we solve the generalized Rankine–Hugoniot relation (3.6) with initial data

$$\begin{aligned} t=0: \ x(0)=0, \quad \omega (0)=0 \end{aligned}$$

(3.9)

under the over-compressive entropy condition (3.7), i.e.,

$$\begin{aligned} \lambda _2(\rho _+,u_+)=u_+\le u_\delta (t)\le u_--\rho _-p'(\rho _-)=\lambda _1(\rho _-,u_-). \end{aligned}$$

(3.10)

Solving the initial value problem (3.6) and (3.9) provides

$$\begin{aligned} \left\{ \begin{array}{l} \omega (t)=-[\rho ]x(t)+[\rho u]t,\\ \omega (t)u_\delta (t)=-[\rho (u+p(\rho ))]x(t)+[\rho u(u+p(\rho ))]t. \end{array}\right. \end{aligned}$$

(3.11)

In consideration of knowledge concerning delta-shocks in [25, 30], one can find that $u_\delta (t)$ is a constant. Thus $x(t)=u_\delta t$. So (3.11) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \omega (t)=-[\rho ]u_\delta t+[\rho u]t,\\ \omega (t)u_\delta =-[\rho (u+p(\rho ))]u_\delta t+[\rho u(u+p(\rho ))]t. \end{array}\right. \end{aligned}$$

(3.12)

It follows that

$$\begin{aligned}{}[\rho ]u_\delta ^2-[2\rho u+\rho p(\rho )]u_\delta +[\rho u(u+ p(\rho ))]=0. \end{aligned}$$

(3.13)

Obviously, if $\rho _-=\rho _+$, (3.13) is a linear equation of $u_\delta$, we have

$$\begin{aligned} \left\{ \begin{array}{l} u_\delta =\displaystyle \frac{u_-+u_++p(\rho _-)}{2},\\ x(t)=\displaystyle \frac{u_-+u_++p(\rho _-)}{2}t,\\ \omega (t)=\rho _-(u_--u_+)t. \end{array}\right. \end{aligned}$$

(3.14)

In addition, one can calculate

$$\begin{aligned} u_\delta -\lambda _2(\rho _+,u_+)=\displaystyle \frac{u_-+p(\rho _-)-u_+}{2}\ge 0 \end{aligned}$$

and

$$\begin{aligned} u_\delta -\lambda _1(\rho _-,u_-)=\displaystyle \frac{u_+-u_- +p(\rho _-)+2\rho _-p'(\rho _-)}{2}\le 0. \end{aligned}$$

So the solution (3.14) satisfies the entropy condition (3.10).

If $\rho _-\ne \rho _+$, (3.13) is a quadratic equation with respect to $u_\delta$. By virtue of $u_+\le u_-+p(\rho _-)$, we have

$$\begin{aligned} \Delta= & {} [2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]\\= & {} 4\rho _-\rho _+(u_--u_+)(u_-+p(\rho _-)-u_ +-p(\rho _+))+(\rho _-p(\rho _-)-\rho _+p(\rho _+))^2{>}0. \end{aligned}$$

A routine calculation yields

$$\begin{aligned} \left\{ \begin{array}{l} u_\delta =\displaystyle \frac{[2\rho u+\rho p(\rho )]-\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2[\rho ]},\\ \omega (t)=\displaystyle \frac{-[\rho p(\rho )]+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2}t,\\ x(t)=\displaystyle \frac{[2\rho u+\rho p(\rho )]-\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2[\rho ]}t \end{array}\right. \end{aligned}$$

(3.15)

and

$$\begin{aligned} \left\{ \begin{array}{l} u_\delta =\displaystyle \frac{[2\rho u+\rho p(\rho )]+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2[\rho ]},\\ \omega (t)=\displaystyle \frac{-\big ([\rho p(\rho )]+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}\big )}{2}t,\\ x(t)=\displaystyle \frac{[2\rho u+\rho p(\rho )]+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2[\rho ]}t. \end{array}\right. \end{aligned}$$

(3.16)

With the entropy condition (3.10), let us choose the admissible solution from (3.15) and (3.16) for the Riemann problem (1.1) and (1.6). Noticing $(\rho _+,u_+)\in {\mathrm{III}}(\rho _-,u_-)$, we can obtain the following four estimates

$$\begin{aligned}&[\rho ](\lambda _2(\rho _+,u_+))^2-[2\rho u+\rho p(\rho )]\lambda _2(\rho _+,u_+)+[\rho u(u+ p(\rho ))]\\&\quad =\rho _-(u_--u_+)(u_-+p(\rho _-)-u_+) \ge 0,\\&\qquad -2[\rho ]\lambda _1(\rho _-,u_-)+[2\rho u+\rho p(\rho )]+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}\\&\quad =2\rho _+(u_--\rho _-p'(\rho _-)-u_+)+(\rho _-p(\rho _-)-\rho _+p(\rho _+))+2\rho _{-}^{2}p'(\rho _-)\\&\qquad +\sqrt{4\rho _-\rho _+(u_--u_+)(u_--u_++p(\rho _-)-p(\rho _+))+(\rho _-p(\rho _-)-\rho _+p(\rho _+))^2}\\&\quad \ge 2\rho _+(u_--\rho _-p'(\rho _-)-u_+)+2\rho _{-}^{2}p'(\rho _-)>0, \\&\qquad -2[\rho ]\lambda _2(\rho _+,u_+)+[2\rho u+\rho p(\rho )] =2\rho _-(u_--u_++p(\rho _-))\\&\qquad -(\rho _-p(\rho _-)+\rho _+p(\rho _+))\\&\quad \ge -(\rho _-p(\rho _-)+\rho _+p(\rho _+)) >0,\\&\qquad [\rho ](\lambda _1(\rho _-,u_-))^2-[2\rho u+\rho p(\rho )]\lambda _1(\rho _-,u_-)+[\rho u(u+p(\rho ))]\\&\quad =-\rho _+(u_--u_+-\rho _-p'(\rho _-))(u_--u_+-\rho _-p'(\rho _-)-p(\rho _+)) \\&\qquad +\rho _{-}^{2}p'(\rho _-)(p(\rho _-)+\rho _-p'(\rho _-))\\&\quad \le \rho _{-}^{2}p'(\rho _-)(p(\rho _-)+\rho _-p'(\rho _-)) \le 0. \end{aligned}$$

From the estimates above, for the solution (3.15), we have

$$\begin{aligned}&u_\delta -\lambda _2(\rho _+,u_+)\nonumber \\&\quad =\displaystyle \frac{[2\rho u+\rho p(\rho )]-\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2[\rho ]}-\lambda _2(\rho _+,u_+)\nonumber \\&\quad =\displaystyle \frac{2\big ([\rho ](\lambda _2(\rho _+,u_+))^2-[2\rho u+\rho p(\rho )]\lambda _2(\rho _+,u_+)+[\rho u(u+p(\rho ))]\big )}{(-2[\rho ]\lambda _2(\rho _+,u_+)+[2\rho u+\rho p(\rho )])+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}} \nonumber \\ \ge 0 \end{aligned}$$

(3.17)

when $\rho _-<\rho _+$, and

$$\begin{aligned}&u_\delta -\lambda _1(\rho _-,u_-)\nonumber \\&\quad =\displaystyle \frac{[2\rho u+\rho p(\rho )]-\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2[\rho ]}-\lambda _1(\rho _-,u_-)\nonumber \\&\quad =\displaystyle \frac{2\big ([\rho ](\lambda _1(\rho _-,u_-))^2-[2\rho u+\rho p(\rho )]\lambda _1(\rho _-,u_-)+[\rho u(u+p(\rho ))]\big )}{(-2[\rho ]\lambda _1(\rho _-,u_-)+[2\rho u+\rho p(\rho )])+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}} \nonumber \\&\quad \le 0 \end{aligned}$$

(3.18)

when $\rho _->\rho _+$. While for the solution (3.16), we have

$$\begin{aligned}&u_\delta -\lambda _2(\rho _+,u_+)\nonumber \\&\quad =\displaystyle \frac{[2\rho u+\rho p(\rho )]+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2[\rho ]}-\lambda _2(\rho _+,u_+)\nonumber \\&\quad =\displaystyle \frac{-2[\rho ]\lambda _2(\rho _+,u_+)+[2\rho u+\rho p(\rho )]+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2[\rho ]}\nonumber \\&\quad <0 \end{aligned}$$

(3.19)

when $\rho _-<\rho _+$, and

$$\begin{aligned}&u_\delta -\lambda _1(\rho _-,u_-)\nonumber \\&\quad =\displaystyle \frac{[2\rho u+\rho p(\rho )]+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2[\rho ]}-\lambda _1(\rho _-,u_-)\nonumber \\&\quad =\displaystyle \frac{-2[\rho ]\lambda _1(\rho _-,u_-)+[2\rho u+\rho p(\rho )]+\sqrt{[2\rho u+\rho p(\rho )]^2-4[\rho ][\rho u(u+p(\rho ))]}}{2[\rho ]}\nonumber \\&\quad >0 \end{aligned}$$

(3.20)

when $\rho _->\rho _+$. These indicate that the solution (3.16) does not satisfy the entropy condition (3.10). Therefore we arrive at the following conclusion.

Theorem 3.2

Let $u_+\le u_-+p(\rho _-)$. Under the hypotheses $(H_1)$–$(H_4)$, the Riemann problem (1.1) and (1.6) possesses one and only one entropy solution in the sense of distributions of the form

$$\begin{aligned} (\rho ,u)(t,x)=\left\{ \begin{array}{ll} (\rho _-,u_-),&{}\quad x<u_\delta t,\\ \big ( \omega (t)\delta (x-u_\delta t),u_\delta \big ),&{}\quad x=u_\delta t,\\ (\rho _+,u_+),&{}\quad x>u_\delta t, \end{array}\right. \end{aligned}$$

(3.21)

where $x(t), u_\delta$ and $\omega (t)$ are shown in (3.14) for $\rho _-=\rho _+$, and (3.15) for $\rho _-\ne \rho _+$.

4 Numerical Simulations

In this section, we employ the Nessyahu–Tadmor scheme [20] with 500 cells and $CFL=0.475$ to present some representative numerical results so that one can understand better the configurations of solutions for the Riemann problem (1.1) and (1.6). Here, we present lots of numerical tests to make sure what we show are not numerical artifacts.

For Case 1, we take the pressure $p(\rho )=-\ln (1+\frac{1}{\rho })$ and initial data $(\rho _-,u_-)=(1.0,1.2)$.

Configuration 1 $R\ +J\$

For this case, the initial data is taken as $(\rho _+,u_+)=(0.8,1.4)$. The numerical results at $t=0.6$ are shown in Fig. 5.

Fig. 5

Numerical results for $R\ +J\$

Configuration 2 $S\ +J\$

In this case, we take the initial data $(\rho _+,u_+)=(2.0,0.95)$ and provide the numerical results at $t=0.8$ in Fig. 6.

Fig. 6

Numerical results for $S\ +J\$

Configuration 3: Delta-shock solution

For this case, the initial data $(\rho _+,u_+)=(1.0,0.05)$ is chosen. The numerical results are shown at $t=1.4$ in Fig. 7.

Fig. 7

Numerical results for delta-shock

For Case 2, one can simulate the three kinds of configurations of solutions similar to the Case 1 by taking the pressure $p(\rho )=-\arctan \frac{1}{\rho }$ and suitable initial data.

For Case 3, we take the pressure $p(\rho )=\arctan \rho -\frac{\pi }{2}$, and $(\rho _-,u_-)=(1.0,1.0)$.

Configuration 1 $R\ +J\$

In this case, we take the initial data $(\rho _+,u_+)=(1.0,1.02)$ and present the numerical results at $t=0.7$ in Fig. 8.

Fig. 8

Numerical results for $R\ +J\$

Configuration 2 $S\ +J\$

For this case, the initial data $(\rho _+,u_+)=(2.0,0.5)$ is taken. The numerical results are given at $t=1.2$ in Fig. 9.

Fig. 9

Numerical results for $S\ +J\$

Configuration 3: Delta-shock solution

In this case, the initial data is chosen as $(\rho _+,u_+)=(2.0,0.05)$. The numerical results are shown at $t=1.5$ in Fig. 10.

Fig. 10

Numerical results for delta-shock

Configuration 4 $R+\text{ Vacuum }+J$

Fig. 11

Numerical results for $R+\text{ Vacuum }+J$

For this case, the initial data $(\rho _+,u_+)=(0.3,7.0)$ is taken. The numerical results are presented at $t=0.04$ in Fig. 11.

From the above analysis, one can observe that the numerical results are consistent with the theoretical analysis.