Instabilities in generic second-order traffic models with relaxation

Abstract

We prove the existence of weak solutions for a class of second-order traffic models with relaxation, without requiring the sub-characteristic stability condition to hold. With the help of numerical simulations, we show how, in this unstable setting, large but bounded oscillations may arise from small perturbations of equilibria, thus reproducing the formation of stop-and-go waves commonly observed in traffic dynamics. An analysis of the corresponding traveling waves completes the study.

1 Introduction

Macroscopic traffic flow models have been used since several decades [31, 35] to describe the spatiotemporal evolution of aggregated quantities, like vehicle density and average speed. In particular, second-order models [2, 33, 40, 41], which consist in the mass conservation equation coupled with a momentum balance equation accounting for the speed dynamics, have been developed to describe specific traffic characteristics that cannot be captured by scalar models, such as scattered fundamental diagrams and the appearance and persistence of stop-and-go waves. Indeed, traffic equilibria are generally unstable [4, 38], and small perturbances in drivers’ behavior can generate so-called “phantom” jams with no apparent cause. The same behavior can be observed at a macroscopic scale considering relaxation source terms that violate the usual stability conditions [18, 37].

In this paper, we study a generic second-order model (GSOM) [29] with relaxation, which extends the well-known Aw–Rascle–Zhang (ARZ) system [2, 41], allowing for more general speed formulation. In the literature, hyperbolic systems of balance laws with relaxation source terms are usually studied under a stability condition ensuring well-posedness and convergence results as the relaxation parameter tends to zero [8, 9]. We recall that the stability condition is related to the dissipativity of the evolution equation obtained by a first-order expansion around equilibria, as observed in [8]. Under this hypothesis, one can recover uniform compactness estimates, which ensure convergence to the equilibrium equation, see, e.g., [23, 26, 30, 36] for traffic flow applications.

Aiming at reproducing traffic flow instabilities that can be observed in reality, here we drop the stability assumption, and we study the model properties for a fixed positive relaxation parameter. In particular, we provide a global existence proof for weak solutions based on wave-front tracking approximations. Moreover, we perform a study of self-sustained oscillations through an analysis of periodic traveling waves and a Chapman–Enskog expansion around equilibria, which yields an advection–diffusion equation where the sign of the diffusion coefficient is related to the stability condition of the system. We observe that a similar study was conducted in [25] for the ARZ model and its mesoscopic BGK formulation, also showing the onset of stop-and-go waves if the sub-characteristic condition is violated in a suitable proper subset of the admissible density domain. Our results are general enough to hold for a wide class of traffic flow models, without too specific assumptions, showing that these models are able to capture the formation and persistence of stop-and-go waves. In particular, we prove that, even in unstable regimes, solutions remain in the invariant domain the initial conditions belong to, thus preventing uncontrolled blow-up. We also provide numerical experiments that confirm the theoretical results and give a deeper insight on the solution behavior.

The presentation is organized as follows. We introduce the GSOM system with relaxation in Sect. 2 and we describe the corresponding Riemann problem in Sect. 3. Section 4 provides the compactness estimates on the sequence of wave-front tracking approximate solutions, which are used to pass to the limit providing the existence of weak solutions in Sect. 5. Traveling waves are studied in Sects. 6 and 7, while numerical simulations are presented in Sect. 8. Finally, conclusions and perspectives are outlined in Sect. 9.

2 Model description

We consider the generic second-order traffic model [29], which consists in the following $2\times 2$ system of balance laws

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho + \partial _x (\rho v) =0, \\ \partial _t (\rho w) + \partial _x (\rho w v) =\rho \dfrac{V(\rho )-v}{\tau }, \end{array}\right. } \qquad x\in {\mathbb {R}},~t>0, \end{aligned}$$

(2.1)

defined on a domain of the form

$$\begin{aligned} \Omega := \left\{ U = (\rho ,w)\in {\mathbb {R}}^2 :\rho \in [0,R(w_{max})], w\in [w_{min},w_{max}]\right\} , \end{aligned}$$

(2.2)

for some $0<w_{min}\le w_{max}<+\infty$. Above, the average speed of vehicles v is a function of the density $\rho =\rho (t,x)$ and of a Lagrangian vehicle property $w=w(t,x)$, namely $v={{\mathcal {V}}}(\rho ,w)$ for some speed function ${{\mathcal {V}}}:\Omega \rightarrow {\mathbb {R}}_{\ge 0}$ satisfying the the following hypotheses [16]:

$$\begin{aligned}&{{\mathcal {V}}}(\rho ,w)\ge 0, \quad {{\mathcal {V}}}(0,w) = w, \end{aligned}$$

(2.3a)

$$\begin{aligned}&2\partial _\rho {{\mathcal {V}}}(\rho ,w)+\rho \, \partial _\rho ^2{{\mathcal {V}}}(\rho ,w) < 0\text { for }w > 0 , \end{aligned}$$

(2.3b)

$$\begin{aligned}&\partial _w {{\mathcal {V}}}(\rho ,w)>0, \end{aligned}$$

(2.3c)

$$\begin{aligned}&\forall w>0 \quad \exists \ R(w)>0:\quad {{\mathcal {V}}}(R(w),w)=0. \end{aligned}$$

(2.3d)

As in [16], we observe that (2.3b) implies that $Q(\rho ,w):=\rho {{\mathcal {V}}}(\rho ,w)$ is strictly concave and $\partial _\rho {{\mathcal {V}}}(\rho ,w)<0$ for $w>0$, if ${{\mathcal {V}}}$ is a $\mathbf {C^{2}}$ function in $\rho$. We also remark that in (2.3d) we can have $R(w)= {\bar{R}}$ for all $w>0$.

Moreover, we assume that the equilibrium speed $V:{\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$ is a non-increasing function ($V'(\rho )\le 0$) such that $V(\rho )=0$ for $\rho \ge \rho _{max}>0$, for some density value $\rho _{max}\in [R(w_{min}),R(w_{max})]$ being the maximal density at equilibrium (in particular, we have $\rho _{max}={\bar{R}}$ if $R(w_{min})=R(w_{max})$). Finally, the parameter $\tau >0$ is the relaxation time toward the equilibrium velocity.

Therefore, system (2.1) ensures the conservation of the number of vehicles through the first equation, and, under suitable hypotheses, the relaxation of the average speed v toward the equilibrium speed law $V=V(\rho )$ encoded in the second equation.

Notice that, setting ${{\mathcal {V}}}(\rho ,w)=w-p(\rho )$ for some “pressure” function p, system (2.1) corresponds to the Aw-Rascle-Zhang (ARZ) model [2, 41]. We also remark that, taking $w={\bar{w}}$ constant, we recover the classical Lighthill–Whitham–Richards (LWR) model [31, 35].

Under the above hypotheses, system (2.1) is strictly hyperbolic for $\rho >0$, with eigenvalues

$$\begin{aligned} \lambda _1 (\rho ,w) = {{\mathcal {V}}}(\rho ,w)+\rho \, \partial _\rho {{\mathcal {V}}}(\rho ,w), \qquad \lambda _2 (\rho ,w) = {{\mathcal {V}}}(\rho ,w), \end{aligned}$$

(2.4)

and corresponding eigenvectors

$$\begin{aligned} r_1(\rho ,w) =\begin{pmatrix} -1 \\ 0 \end{pmatrix}, \qquad r_2(\rho ,w) = \begin{pmatrix} \partial _w {{\mathcal {V}}}(\rho ,w) \\ -\partial _\rho {{\mathcal {V}}}(\rho ,w) \end{pmatrix}, \end{aligned}$$

(2.5)

with the first characteristic field being genuinely nonlinear and the second linearly degenerate. The associated Riemann invariants [14, Chapter 7.3] are

$$\begin{aligned} z_1 (\rho ,w) = {{\mathcal {V}}}(\rho ,w), \qquad z_2 (\rho ,w) = w. \end{aligned}$$

Since $\partial _\rho {{\mathcal {V}}}(\rho ,w)<0$ and ${{\mathcal {V}}}(0,w)=w$, the range of $v={{\mathcal {V}}}(\rho ,w)$ is given by $v \in [0,w]$ for any $w\in [w_{min},w_{max}]$. Therefore, the inverse function $\rho = {\mathcal {R}}(v,w)$ is uniquely defined in the invariant domain

$$\begin{aligned} {\mathcal {W}} : = \left\{ W = (v,w)\in {\mathbb {R}}^2 :0 \le v \le w, w \in [w_{min},w_{max}]\right\} , \end{aligned}$$

(2.6)

which corresponds to (2.2).

We recall that weak solutions of (2.1) with initial datum $U(0,x)=U_0(x)=(\rho _0(x),w_0(x))\in \mathbf {L^\infty }({\mathbb {R}},E)$ are defined as

Definition 1

Let $u_0=(\rho _0,\rho _0 w_0)^T\in \mathbf {L^1}({\mathbb {R}};{\mathbb {R}}_+^2)$ and $T>0$ be given. Then, $u = (\rho ,\rho w)^T\in \mathbf {C^{0}} \left( [0,T], \mathbf {L^1}({\mathbb {R}};{\mathbb {R}}_+^2)\right)$ is a weak solution to the Cauchy problem for (2.1) if, for all $\varphi \in \mathbf {C_c^{1}}\left( ]-\infty ,T[ \,\times {\mathbb {R}};{\mathbb {R}}\right)$, it holds

$$\begin{aligned} \int \limits _{\mathbb {R}}u_0(x) \varphi (0, x) \,\textrm{d}x + \int \limits _{0}^{T} \int \limits _{\mathbb {R}}\left[ u\partial _t \varphi + F(u) \partial _x\varphi \right] (t,x) \,\textrm{d}x \textrm{d}t + \int \limits _{0}^{T} \int \limits _{\mathbb {R}}G(u) \varphi (t,x) \,\textrm{d}x \textrm{d}t=0, \end{aligned}$$

(2.7)

where

$$\begin{aligned} F(u)= \begin{bmatrix} \rho v\\ \rho w v \end{bmatrix}, \qquad G(u)= \begin{bmatrix} 0\\ \rho \dfrac{V(\rho )-v}{\tau } \end{bmatrix}. \end{aligned}$$

For smooth solutions, system (2.1) is equivalent to:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho + \partial _x (\rho v) =0, \\ \partial _t v + \lambda _1(\rho ,w)\partial _x v =\dfrac{V(\rho )-v}{\tau }, \end{array}\right. } \qquad x\in {\mathbb {R}},~t>0, \end{aligned}$$

(2.8)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho + \partial _x (\rho v) =0, \\ \partial _t w + \lambda _2(\rho ,w)\partial _x w =\dfrac{V(\rho )-v}{\tau }, \end{array}\right. } \qquad x\in {\mathbb {R}},~t>0. \end{aligned}$$

(2.9)

We also recall that the sub-characteristic condition [8, 9] writes

$$\begin{aligned} \partial _\rho {{\mathcal {V}}}(\rho ,w) \le V'(\rho ) \le 0 \qquad \hbox {for }w \hbox {s.t.} {{\mathcal {V}}}(\rho ,w)=V(\rho ), \end{aligned}$$

(2.10)

see, e.g., Fig. 1.

If (2.10) is satisfied, it is known [23] that (entropy weak) solutions to (2.1) converge to the (entropy weak) solutions of

$$\begin{aligned} \partial _t \rho + \partial _x (\rho V(\rho )) =0, \qquad x\in {\mathbb {R}},~t>0, \end{aligned}$$

(2.11)

for $\tau \searrow 0$ or $t\rightarrow \infty$ (using the change of variables ${{\tilde{t}}}= t/\tau$ and ${{\tilde{x}}} = x/\tau$).

Fig. 1

Speed-density (left) and flow-density (right) fundamental diagrams for the GSOM model (2.1) corresponding to the choices (satisfying the sub-characteristic condition (2.10)): Top: ${{\mathcal {V}}}(\rho ,w)=w-\alpha \rho$, with $w\in [V_{\min },V_{\max }]$ and $V(\rho )= \beta (R-\rho )$ with $\alpha \ge \beta >0$; Bottom: ${{\mathcal {V}}}(\rho ,w)=w(R-\alpha \rho )$, with $w\in [V_{\min },V_{\max }]$ and $V(\rho )= \beta (R-\rho )$ with $\alpha \ge \beta >0$

The equilibrium curve $V(\rho )=V({\mathcal {R}}(v,w))$ is implicitly defined by

$$\begin{aligned} V({\mathcal {R}}(v,\phi (v))) =v. \end{aligned}$$

Differentiating w.r.to v, we obtain

$$\begin{aligned} V'(\rho ) \left( \partial _v {\mathcal {R}} + \partial _w {\mathcal {R}}\, \phi '(v)\right) =1, \end{aligned}$$

which gives

$$\begin{aligned} \phi '(v) = \frac{1}{\partial _w{\mathcal {R}}} \left( \frac{1}{V'(\rho )} - \partial _v {\mathcal {R}}\right) = \frac{1}{\partial _w {{\mathcal {V}}}} \left( 1 - \frac{\partial _\rho {{\mathcal {V}}}}{V'(\rho )}\right) . \end{aligned}$$

Since $\partial _w {{\mathcal {V}}}>0$, we get $\phi '(v)\le 0$ if and only if $\partial _\rho {{\mathcal {V}}}\le V'(\rho )$, which is exactly the sub-characteristic condition (2.10).

In this paper, we are interested in investigating the behavior of (2.1) when (2.10) is violated. Therefore, we assume that there exists a point $v_{cr}$, resp. $w_{cr}=\phi (v_{cr})$, $\rho _{cr}={\mathcal {R}}(v_{cr},w_{cr})$, such that $\phi '(v_{cr}) =0$, i.e.,

$$\begin{aligned} \partial _\rho {{\mathcal {V}}}(\rho _{cr},w_{cr}) =\partial _\rho {{\mathcal {V}}}(\rho _{cr},\phi (v_{cr})) = V'(\rho _{cr}). \end{aligned}$$

(2.12)

We will distinguish the two cases:

$$\begin{aligned} \phi '(v)< 0 \quad \hbox {if}~v<v_{cr}, \qquad \phi '(v)> 0 \quad \hbox {if}~v>v_{cr}, \end{aligned}$$

(2.13a)

and

$$\begin{aligned} \phi '(v)> 0 \quad \hbox {if}~v<v_{cr}, \qquad \phi '(v) < 0 \quad \hbox {if}~v>v_{cr}. \end{aligned}$$

(2.13b)

Moreover, we observe that, for $v>v_{cr}$ for case (2.13a), resp. $v<v_{cr}$ for case (2.13b), $\phi '(v)< 1$ if $V'({\mathcal {R}}(v,\phi (v))) > \partial _\rho {{\mathcal {V}}}({\mathcal {R}}(v,\phi (v)),\phi (v)) + V'({\mathcal {R}}(v,\phi (v))) \partial _w {{\mathcal {V}}}({\mathcal {R}}(v,\phi (v)),\phi (v))$.

In case (2.13a), the equilibrium curve intersects the vacuum line $v=w$ at ${\hat{v}} = {\hat{w}}$ implicitly defined by ${{\mathcal {V}}}(0,{\hat{w}})=V(0)$, see Fig. 2, left. We also define $\check{\rho }, {\check{v}}$ such that

$$\begin{aligned} {\check{v}}={{\mathcal {V}}}(\check{\rho },\hat{w})=V(\check{\rho }). \end{aligned}$$

(2.14)

In both cases (2.13), the equilibrium curve intersects the congestion line $v=0$ at ${\check{w}}$ implicitly defined by ${{\mathcal {V}}}({\mathcal {R}}({\check{w}}),{\check{w}})=0=V(\rho _{max})$, see Fig. 2. In case (2.13b), we define $\check{\rho }, {\check{v}}>0$ such that

$$\begin{aligned} {\check{v}}={{\mathcal {V}}}(\check{\rho },{\check{w}})=V(\check{\rho }), \end{aligned}$$

(2.15)

see Fig. 2, right.

Fig. 2

Examples of non-monotone equilibrium curves $w=\phi (v)$ and minimal invariant domains ${\textbf{E}}$ (continuous red line) and general invariant domains (dashed red line): left, convex case (2.13a); right, concave case (2.13b)

3 The Riemann problem

In this section, we analyze the properties of the solutions to Riemann problems, treating separately the transport and relaxation components of system (2.1), which will be used to construct the wave-front tracking approximate solutions in Sect. 4. Moreover, we aim at characterizing the invariant subsets of ${\mathcal {W}}$ given in (2.6). We refer to Sect. 8.1 for an illustration of solutions’ global behavior in unstable regimes.

Let us consider (2.1) with initial data

$$\begin{aligned} (\rho , w )(0,x) = {\left\{ \begin{array}{ll} U_L = (\rho _L, w_L) &{}\hbox {if } ~x<0, \\ U_R = (\rho _R, w_R) &{}\hbox {if } ~x>0, \end{array}\right. } \end{aligned}$$

(3.1)

and set $v_L = {{\mathcal {V}}}(\rho _L,w_L)$, $v_R = {{\mathcal {V}}}(\rho _R,w_R)$.

3.1 The homogeneous Riemann problem

We recall here the construction of weak (entropy) solutions of problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho + \partial _x (\rho v) =0, \\ \partial _t (\rho w) + \partial _x (\rho w v) =0, \end{array}\right. } \qquad x\in {\mathbb {R}},~t>0, \end{aligned}$$

(3.2)

with initial conditions. In general, solutions to (3.2),  (3.1), consist of two waves connected by an intermediate state. More precisely, the left state $U_L$ is connected to $U_M = (\rho _M, w_M)$ by a first family wave (rarefaction or shock), i.e., $z_2(\rho _L,w_L) = w_L=w_M=z_2(\rho _M, w_M)$, and $U_M$ is connected to $U_R$ by a contact discontinuity with $z_1(\rho _M,w_M)= v_M = {{\mathcal {V}}}(\rho _M, w_M) = v_R = z_1(\rho _R,w_R)$. Thus, the intermediate state $U_M$ is identified by the system of equations

$$\begin{aligned} {\left\{ \begin{array}{ll} w_M = w_L ,\\ v_M = v_R, \\ \rho _M = {\mathcal {R}}(v_R, w_L). \end{array}\right. } \end{aligned}$$

If $w_L \le v_R$, we set $\rho _M = 0$, meaning that $U_M$ corresponds to the vacuum state.

The propagation speed $\sigma$ of a shock wave between two states $U_-$ and $U_+$ is given by the Rankine–Hugoniot condition

$$\begin{aligned} \sigma (U_-, U_+) = \frac{\rho _+ v_+ - \rho _- v_-}{\rho _+ - \rho _-}. \end{aligned}$$

(3.3)

In this work, we will rely on the following definition of solutions of (3.2), (3.1), see also [15]. Since, due to the loss of strict hyperbolicity, the presence of vacuum states prevents uniqueness even if an entropy condition is enforced, we refer to [1] for an alternative construction and to [24, Remark 3] for a discussion about different entropy weak solutions. An example of two different solutions involving a vacuum right-hand state is depicted in Fig. 3.

Fig. 3

Two entropy admissible solutions to the Riemann problem (3.2)–(3.1) in two different coordinate systems: $W=(v,w)$ and $Y=(\rho ,\rho v)$. The solution considered in this work is the dashed blue, the one in [1] corresponds to the dotted red

Definition 2

For any $U_L, U_R \in \Omega$, the Riemann solver for (3.2), (3.1),

$$\begin{aligned} \mathcal{R}\mathcal{S}: \Omega \times \Omega \rightarrow \mathbf {C^0}\left( ]0,+\infty [\, ;{\textbf{L}}_{loc}^1({\mathbb {R}}; \Omega )\right) , \qquad (U_L,U_R) \mapsto \mathcal{R}\mathcal{S}(U_L,U_R) \end{aligned}$$

is defined as follows:

• 1. If $(v_R,w_R) \in {\mathcal {W}}$ and $v_L \ge v_R$, then

  $$\begin{aligned} \mathcal{R}\mathcal{S}(U_L, U_R)(t,x) = {\left\{ \begin{array}{ll} U_L &{}\hbox {if } ~x< \sigma (U_L, U_M)t, \\ U_M &{}\hbox {if } ~ \sigma (U_L, U_M)t< x< v_R t, \\ U_R &{}\hbox {if } ~ x > v_R t, \end{array}\right. } \end{aligned}$$

  with $\sigma$ defined in (3.3).

• 2. If $(v_L,w_L) \in {\mathcal {W}}{\setminus }\{v=w\}$, and $v_L \le v_R < w_L$, then

  $$\begin{aligned} \mathcal{R}\mathcal{S}(U_L, U_R)(t,x) = {\left\{ \begin{array}{ll} U_L &{}\hbox {if } ~ x< \lambda _1(\rho _L,w_L) t, \\ \hat{U} &{}\hbox {if } ~ \lambda _1(\rho _L,w_L) t< x< \lambda _1(\rho _M,w_M) t,\\ U_M &{}\hbox {if } ~ \lambda _1(\rho _M,w_M) t< x < v_R t ,\\ U_R &{}\hbox {if } ~ x > v_R t, \end{array}\right. } \end{aligned}$$

  with $\hat{U} = (\rho , w_L)$ solving $\lambda _1(\rho , w_L) = \dfrac{x}{t}$.

• 3. If $(v_L,w_L) \in {\mathcal {W}}\setminus \{v=w\}$ and $w_L \le v_R$, then

  $$\begin{aligned} \mathcal{R}\mathcal{S}(U_L, U_R)(t,x) = {\left\{ \begin{array}{ll} U_L &{}\hbox {if } ~ x< \lambda _1(\rho _L,w_L) t, \\ \hat{U} &{}\hbox {if } ~ \lambda _1(\rho _L,w_L) t< x< \lambda _1(0,w_L) t,\\ U_M &{}\hbox {if } ~ \lambda _1(0,w_L) t< x < v_R t ,\\ U_R &{}\hbox {if } ~ x > v_R t, \end{array}\right. } \end{aligned}$$

  with $\hat{U} = (\rho , w_L)$ solving $\lambda _1(\rho , w_L) = \dfrac{x}{t}$ and $U_M = (0, w_L)$.

• 4. If $v_L=w_L$ and $v_R=w_R$, then

  $$\begin{aligned} \mathcal{R}\mathcal{S}(U_L, U_R)(t,x) = {\left\{ \begin{array}{ll} U_L &{}\hbox {if } ~x< 0, \\ U_R &{}\hbox {if } ~x > 0. \end{array}\right. } \end{aligned}$$

  (3.4)

3.2 The relaxation Riemann problem

We now turn to the solution of the relaxation step, consisting of the system of ODEs

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho =0, \\ \partial _t v =\dfrac{V(\rho )-v}{\tau } = \partial _t w, \end{array}\right. } \qquad x\in {\mathbb {R}},~t>0, \end{aligned}$$

(3.5)

with Riemann-like initial datum (3.1), whose solutions are given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \rho (t,x)=\rho (0,x), \\ v(t,x) = V(\rho (0,x)) +\left( v(0,x) - V(\rho (0,x)) \right) e^{-t/\tau }, \end{array}\right. } \qquad x\in {\mathbb {R}},~t>0. \end{aligned}$$

(3.6)

We remark that, in the (v, w) coordinates, the curves $\rho ={\mathcal {R}}(v,w)=const$ are parallel to the first bisector [34, Section 5]. Indeed, we get:

$$\begin{aligned} {\mathcal {R}}(v,w)=const&\qquad \Longrightarrow \qquad \partial _v {\mathcal {R}}+\partial _w{\mathcal {R}}=0 \\ {{\mathcal {V}}}({\mathcal {R}}(v,w),w)=v&\qquad \Longrightarrow \qquad (\partial _\rho {{\mathcal {V}}}) (\partial _v {\mathcal {R}}) =1 \hbox { and } (\partial _\rho {{\mathcal {V}}}) (\partial _w {\mathcal {R}}) + \partial _w {{\mathcal {V}}}=0 \end{aligned}$$

Replacing $\partial _v{\mathcal {R}}=1/\partial _\rho {{\mathcal {V}}}$ and $\partial _w {\mathcal {R}}=-\partial _w {{\mathcal {V}}}/\partial _\rho {{\mathcal {V}}}$ in the first relation, we get $\partial _w {{\mathcal {V}}}=1$ on ${\mathcal {R}}(v,w)$ level curves. In particular, for $t\rightarrow +\infty$ (equivalently $\tau \searrow 0$), solutions to (3.5), (3.1), given by (3.6) tend to $\bar{\rho }_{L,R}= {\mathcal {R}}({\bar{v}}_{L,R},{\bar{w}}_{L,R})$, ${\bar{v}}_{L,R}=V(\rho _{L,R})$ with

$$\begin{aligned} {\bar{w}}_{L,R}&={\bar{v}}_{L,R}+w_{L,R}-v_{L,R}, \end{aligned}$$

no matter the sub-characteristic condition, see Fig. 4.

Invariant domains for this step must therefore include segments parallel to the line $v=w$ joining any point of the domain with the equilibrium curve.

Fig. 4

Illustration of the limit behavior of the solutions to the Riemann problem (3.5), (3.1), in two different coordinate systems: $W=(v,w)$ and $Y=(\rho ,\rho v)$

From the above analysis, we easily see that the invariant domains for the relaxed Riemann problem (2.1), (3.1) must have edges parallel to the Riemann invariant level curves and contain properly a portion of the equilibrium curve, see Fig. 2 and definitions (4.1), (4.2).

4 Wave-front tracking approximations

The construction of wave-front tracking (WFT) approximate solutions is based on a two-step process, that successively solves the homogeneous system (3.2) for a piece-wise constant initial datum, and then integrates the source term contained in the ODE (3.5), see [23] and references therein.

To define the invariant domains, we need to distinguish the two cases illustrated in Fig. 2, namely the case in which the equilibrium curve $w= \phi (v)$ is a convex function and the case in which it is concave.

Under the hypothesis (2.13a), invariant domains take the form

$$\begin{aligned} {\textbf{E}}={\textbf{E}}[0, \bar{\rho } ] =\left\{ U= (\rho ,w): \rho \in [0, \bar{\rho }], {{\mathcal {V}}}(\rho ,w) \in [\bar{v}, v_M], w \in [w_m, \bar{w}] \right\} , \end{aligned}$$

(4.1)

with $\bar{\rho }\ge \check{\rho }$, $v_M\ge {\hat{v}}$, $w_m\le w_{cr}$ and ${\bar{v}} ={{\mathcal {V}}}(\bar{\rho },{\bar{w}})=V(\bar{\rho })$, where $\check{\rho }, \hat{v}$ and $w_{cr}$ are chosen as in (2.12), (2.14). In particular, we observe that, in this case, all invariant domains must include vacuum states, see Fig. 2, left.

Under the hypothesis (2.13b), invariant domains take the form

$$\begin{aligned} {\textbf{E}}={\textbf{E}}[\bar{\rho }, R({\hat{w}}) ] =\left\{ U= (\rho ,w): \rho \in [\bar{\rho }, R(w_M)], {{\mathcal {V}}}(\rho ,w) \in [0,\bar{v}], w \in [\bar{w},w_M] \right\} , \end{aligned}$$

(4.2)

with $\bar{\rho }\le \check{\rho }$, $w_M\ge w_{cr}$ and ${\bar{v}} ={{\mathcal {V}}}(\bar{\rho },{\bar{w}})=V(\bar{\rho })$, where $\check{\rho }$ and $w_{cr}$ are chosen as in (2.12), (2.15). In particular, all invariant domains must include congestion states, see Fig. 2, right.

Let us consider initial data $U_0=(\rho _0, w_0):{\mathbb {R}}\rightarrow {\textbf{E}}$ such that $\textrm{TV}(w_0)+ \textrm{TV}({{\mathcal {V}}}(\rho _0,w_0)) < +\infty$. For any $T > 0$, we consider a sequence of time steps $\Delta t^\nu >0$, $\nu \in N$ such that $\Delta t^\nu \rightarrow 0$ and we partition the interval [0, T[ in intervals of the form $[n \Delta t^\nu , (n+1) \Delta t^\nu [$, $n \in {{\mathbb {N}}}$. We denote with $U^{\nu }(t,x)= (\rho ^\nu , w^\nu )(t,x)$, $t \in [0,T]$, $x \in {\mathbb {R}}$, the sequence of WFT approximate solutions of (2.1) constructed as detailed below:

• 1. Define a sequence of piece-wise constant functions $U_0^\nu =(\rho _0^\nu , w_0^\nu ) \in {\textbf{E}}$ satisfying

  $$\begin{aligned}&\textrm{TV}({{\mathcal {V}}}(\rho _0^\nu , w_0^\nu )) \le \textrm{TV}({{\mathcal {V}}}(\rho _0, w_0))\,,&\ {\left\| {{\mathcal {V}}}(\rho _0^\nu , w_0^\nu )- {{\mathcal {V}}}(\rho _0, w_0)\right\| }_{\mathbf {L^\infty }} \le \dfrac{1}{\nu }\,,&\ {\left\| \rho _0^\nu - \rho _0 \right\| }_{\mathbf {L^1}} \le \dfrac{1}{\nu }\,,\\&\textrm{TV}(w_0^\nu ) \le \textrm{TV}(w_0) \,,&\ {\left\| w_0^\nu - w_0 \right\| }_{\mathbf {L^\infty }} \le \dfrac{1}{\nu }\,,&\ {\left\| w_0^\nu - w_0 \right\| }_{\mathbf {L^1}} \le \dfrac{1}{\nu }\,, \end{aligned}$$

  and, for each $\nu \in {{\mathbb {N}}}$, the piece-wise constant function $U_0^\nu$ has a finite number of discontinuities.

• 2. Solve the homogeneous system (3.2) corresponding to the Riemann problems arising at discontinuities for $t \in [0,\Delta t^\nu [$ using the WFT method and name $U^\nu (t,.)$, $t \in [0,\Delta t^\nu [$, the corresponding piece-wise constant function [3].

• 3. At $t= \Delta t^\nu$, we define

  $$\begin{aligned}&\rho ^\nu ( \Delta t^\nu , \cdot )=\rho ^\nu ( \Delta t^\nu -, \cdot )\,,\\&w^\nu ( \Delta t^\nu , \cdot )=w^\nu ( \Delta t^\nu -, \cdot )+ \dfrac{\Delta t^\nu }{\tau } \left[ V(\rho ^\nu (\Delta t^\nu , \cdot ))- {{\mathcal {V}}}( U^\nu ( \Delta t^\nu -, \cdot ))\right] . \end{aligned}$$

  Note that $\rho$ is conserved during this second step, while w (and $v={{\mathcal {V}}}(\rho ,w)$) is updated according to (3.5). Observe that we also have

  $$\begin{aligned} {{\mathcal {V}}}( U^\nu ( \Delta t^\nu , \cdot ))={{\mathcal {V}}}( U^\nu ( \Delta t^\nu -, \cdot ))+ \dfrac{\Delta t^\nu }{\tau } \left[ V(\rho ^\nu (\Delta t^\nu , \cdot ))- {{\mathcal {V}}}( U^\nu ( \Delta t^\nu -, \cdot ))\right] , \end{aligned}$$

  (4.3)

  see (3.5).

• 4. Treat $U^\nu (\Delta t^\nu ,.)$ as a new piece-wise constant initial condition and repeat the previous steps 2–3 to define the solution $U^\nu (t,.)$ for each $t \in [0, T]$, for any $T > 0$ fixed.

4.1 $\mathbf {L^\infty }$ estimates

Proposition 1

For $\Delta t^\nu \le \tau /\max \left\{ 1, {\left\| \partial _w {{\mathcal {V}}}\right\| }_\infty \right\}$, the set ${\textbf{E}}$ is an invariant domain for the proposed WFT scheme.

Proof

Let us assume $U_0(x)\in {\textbf{E}}$ for all $x\in {\mathbb {R}}$. System (3.2) being of Temple class, Step 2 clearly preserves the inequalities on the Riemann invariants $w, {{\mathcal {V}}}(\rho ,w)$.

Concerning the relaxation Step 3., if $v_- \ge V(\rho )$ we have that

$$\begin{aligned} V (\rho )= \left( 1- \dfrac{\Delta t}{\tau } \right) V (\rho )+ \dfrac{\Delta t}{\tau } V (\rho ) \le \left( 1- \dfrac{\Delta t}{\tau } \right) v_-+ \dfrac{\Delta t}{\tau } V (\rho )= v_+ \le v_-, \end{aligned}$$

where we used (4.3) and the hypothesis $\Delta t \le \tau$. Hence, $v_+\in [{\bar{v}},v_M]$ (resp. $v_+\in [0,{\bar{v}}]$).

On the other hand

$$\begin{aligned} w_+=w_-+ \dfrac{\Delta t}{\tau }\left( V (\rho )- v_-\right) \le w_- \end{aligned}$$

and, developing

$$\begin{aligned} {{\mathcal {V}}}(\rho ,w_-) = {{\mathcal {V}}}(\rho ,\phi (V(\rho ))) + \partial _w {{\mathcal {V}}}(\rho , {{\tilde{w}}}) \left( w_- - \phi (V(\rho ))\right) = V(\rho )+ \partial _w {{\mathcal {V}}}(\rho , {{\tilde{w}}}) \left( w_- - \phi (V(\rho ))\right) \end{aligned}$$

for some ${{\tilde{w}}}$,

$$\begin{aligned} w_+&=w_- -\dfrac{\Delta t}{\tau } {{\mathcal {V}}}(\rho ,w_-) + \dfrac{\Delta t}{\tau }V (\rho ) \\&= w_- \left( 1 - \dfrac{\Delta t}{\tau }\partial _w {{\mathcal {V}}}(\rho , {{\tilde{w}}})\right) + \dfrac{\Delta t}{\tau } \partial _w {{\mathcal {V}}}(\rho , {{\tilde{w}}})\, \phi (V(\rho )) \ge \phi (V(\rho )), \end{aligned}$$

where we used $\Delta t\le \tau / {\left\| \partial _w {{\mathcal {V}}}\right\| }_\infty$. Therefore, $w_+ \in [w_m,{\bar{w}}]$ (resp. $w_+\in [{\bar{w}},w_M]$).

The case $v_- < V(\rho )$ is treated analogously. $\square$

4.2 $\textrm{BV}$ estimates

Proposition 2

Let us assume there exists $c>0$ such that $\partial _\rho {{\mathcal {V}}}(\rho ,w) \le -c$ for all $(\rho ,w)\in {\textbf{E}}$. For any $t>0$, the total variation of the Riemann invariants

$$\begin{aligned} \textrm{TV}(W(U^\nu (t,\cdot ))):= \textrm{TV}(w^\nu (t,\cdot )) + \textrm{TV}({{\mathcal {V}}}(U^\nu (t,\cdot ))) \end{aligned}$$

of the WFT approximate solution satisfies the uniform bound

$$\begin{aligned} \textrm{TV}(W(U^\nu (t,\cdot ))) \le \textrm{TV}(W(U_0)) \, e^{K t / \tau }, \end{aligned}$$

(4.4)

where $K:= \dfrac{2}{c} {{\left\| V'\right\| }_\infty } \max \left\{ 1, {\left\| \partial _w {{\mathcal {V}}}\right\| }_\infty \right\}$. In particular, when $\tau \searrow 0$, the estimate (4.4) blows up.

Proof

Notice that, when we solve the homogeneous system (3.2), the total variation of both Riemann invariants is non-increasing in time since we are dealing with a Temple-class system. We thus focus on the evolution of the total variation at step 3, corresponding to (3.5).

We recall that, at $t_n=n\Delta t^\nu$ and dropping the index $\nu$ for simplicity,

$$\begin{aligned} \rho (t_n +, x)&= \rho (t_n -,x),\\ v(t_n+,x)&= v(t_n-,x) + \dfrac{\Delta t}{\tau }(V(\rho )-v)(t_n-,x), \\ w(t_n+,x)&= w(t_n-,x) + \dfrac{\Delta t}{\tau }(V(\rho )-v)(t_n-,x). \end{aligned}$$

Therefore, at each jump in the approximate solution, we have

$$\begin{aligned} | v_r^+-v_l^+|&=\left| v_r^--v_l^- + \dfrac{\Delta t}{\tau } \left( V(\rho _r)- V(\rho _l)\right) - \dfrac{\Delta t}{\tau } (v_r^-- v_l^-) \right| \\&= \left| \left( 1- \dfrac{\Delta t}{\tau } \right) (v_r^--v_l^-) + \dfrac{\Delta t}{\tau } \left( V(\rho _r)- V(\rho _l)\right) \right| \\&\le \left( 1- \dfrac{\Delta t}{\tau } \right) \left| v_r^--v_l^-\right| + \dfrac{\Delta t}{\tau } {\left\| V'\right\| }_\infty \left| \rho _r- \rho _l \right| . \end{aligned}$$

From the relation $\rho ={\mathcal {R}}(v,w)$, we get

$$\begin{aligned} \left| \rho _r- \rho _l \right|&\le {\left\| \partial _v {\mathcal {R}}\right\| }_{\infty } {\left| v_r^- - v_l^-\right| } + {\left\| \partial _w {\mathcal {R}}\right\| }_{\infty } {\left| w_r^- - w_l^-\right| }\\&\le \frac{1}{c} {\left| v_r^- - v_l^-\right| } + \frac{{\left\| \partial _w {{\mathcal {V}}}\right\| }_\infty }{c} {\left| w_r^- - w_l^-\right| } \end{aligned}$$

by the relations $\partial _v {\mathcal {R}} = 1/\partial _\rho {{\mathcal {V}}}$ and $\partial _w {\mathcal {R}} = - \partial _w {{\mathcal {V}}}/\partial _\rho {{\mathcal {V}}}$. Hence

$$\begin{aligned} | v_r^+-v_l^+| \le \left( 1 + \dfrac{\Delta t}{\tau }\left( \frac{{\left\| V'\right\| }_\infty }{c} -1 \right) \right) \left| v_r^--v_l^-\right| + \dfrac{\Delta t}{\tau } \frac{{\left\| V'\right\| }_\infty {\left\| \partial _w {{\mathcal {V}}}\right\| }_\infty }{c} {\left| w_r^- - w_l^-\right| }. \end{aligned}$$

(4.5)

Similar estimates lead to

$$\begin{aligned} | w_r^+-w_l^+| \le \dfrac{\Delta t}{\tau }\left( \frac{{\left\| V'\right\| }_\infty }{c} +1 \right) \left| v_r^--v_l^-\right| + \left( 1+\dfrac{\Delta t}{\tau } \frac{{\left\| V'\right\| }_\infty {\left\| \partial _w {{\mathcal {V}}}\right\| }_\infty }{c} \right) {\left| w_r^- - w_l^-\right| }. \end{aligned}$$

(4.6)

Summing (4.5) and (4.6), we obtain

$$\begin{aligned} \textrm{TV}(W(U(t_n+,\cdot ))) \le \left( 1 + K \dfrac{\Delta t}{\tau }\right) \textrm{TV}(W(U(t_n-,\cdot )) \end{aligned}$$

with

$$\begin{aligned} K:= \frac{2}{c} \max \left\{ {{\left\| V'\right\| }_\infty }, {{\left\| V'\right\| }_\infty {\left\| \partial _w {{\mathcal {V}}}\right\| }_\infty } \right\} , \end{aligned}$$

which gives (4.4). $\square$

4.3 $\mathbf {L^1}$ Lipschitz continuity in time

To get the $\textrm{BV}$ estimates in space and time, following [7], see also [23, Lemma 2.3], we need the next result.

Proposition 3

Under the same assumption as in Proposition 2, let us assume $\textrm{TV}(W(U_0)) = M <+\infty$. Then, there exist a constant $C_M>0$ independent of $\tau$ and a constant $L_\tau >0$, such that, $\forall$ $a<b$ and $\forall$ $0\le s<t$, the WFT approximate solutions satisfy

$$\begin{aligned}&\int \limits _{a}^{b}| \rho ^\nu (t,x)- \rho ^\nu (s,x)| \le C_M e^{K t / \tau } (t-s) , \end{aligned}$$

(4.7)

$$\begin{aligned}&\int \limits _{a}^{b}| w^\nu (t,x)- w^\nu (s,x)| \le \left( C_M e^{K t / \tau } + L_\tau \right) (t-s + \Delta t) . \end{aligned}$$

(4.8)

In particular, both estimates above blow up as $\tau \searrow 0$.

Proof

Let s and $t \in {\mathbb {R}}$ such that $0 \le s<t$. If there are no time steps between s and t, (4.7) and (4.8) are true for any $L_\tau \ge 0$, as a direct application of Temple-class system properties, see [23]. We suppose now that there are $N+1$ time steps between s and t:

$$\begin{aligned} s \le k \Delta t \le (k+1) \Delta t \le \dots \le (N+k) \Delta t \le t, \end{aligned}$$

so that $N \Delta t \le t-s$.

Let $a<b$ given and $x \in \, ]a,b[$. We can then estimate

$$\begin{aligned}&{\left| \rho ^\nu (t,x)- \rho ^\nu (s,x)\right| } \\ =&\ {\left| \rho ^\nu (t,x)- \rho ^\nu ((N+k)\Delta t,x)+\sum _{i=k}^{k+N-1} [ \rho ^\nu ((i+1) \Delta t,x)- \rho ^\nu (i \Delta t,x) ] +\rho ^\nu (k \Delta t,x)- \rho ^\nu (s,x)\right| }\\ \le&\ {\left| \rho ^\nu (t,x)- \rho ^\nu ((N+k)\Delta t,x)\right| } +\sum _{i=k}^{k+N-1} {\left| \rho ^\nu ((i+1) \Delta t,x)- \rho ^\nu (i \Delta t,x) \right| }+{\left| \rho ^\nu (k \Delta t,x)- \rho ^\nu (s,x)\right| } . \end{aligned}$$

Since $\rho ^\nu$ does not change through the splitting process, we can apply the previous property between two consecutive time steps to obtain

$$\begin{aligned}&\int \limits _{a}^{b} {\left| \rho ^\nu (t,x)- \rho ^\nu (s,x) \right| } \textrm{d}x \\ \le&\ C_M e^{K t / \tau } \left[ t- (N+k) \Delta t + \sum _{i=k}^{N+k-1}\left( (i+1) \Delta t- i \Delta t\right) + k \Delta t- s\right] \\ =&\ C_M e^{K t / \tau } \left[ t- (N+k) \Delta t + (N+k) \Delta t- k \Delta t+ k \Delta t- s\right] = C_M e^{K t / \tau } (t-s). \end{aligned}$$

Concerning $w^\nu$, which is modified at each splitting step, we have to consider an additional term:

$$\begin{aligned}&\sum _{i=k}^{N+k} \int \limits _{a}^{b} {\left| w^\nu (i \Delta t+,x)- w^\nu (i \Delta t-,x) \right| } \textrm{d}x= \dfrac{\Delta t}{\tau } \sum _{i=k}^{N+k}\int \limits _{a}^{b} {\left| (V(\rho ^\nu )-v^\nu )(i \Delta t-,x) \right| }\textrm{d}x\\ \le&\ \dfrac{\Delta t}{\tau }(N+1) (b-a) \sup _{U^\nu \in {\textbf{E}}} {\left| V(\rho ^\nu )-v^\nu \right| }\\ \le&\ L_\tau (t-s+ \Delta t), \end{aligned}$$

with $L_\tau = \dfrac{b-a}{\tau } \sup _{U^\nu \in {\textbf{E}}} {\left| V(\rho ^\nu )-v^\nu \right| }$. Summing to the other terms, we get (4.8). $\square$

5 Existence of weak solutions

The uniform bounds derived in Sect. 4 allow to apply Helly’s theorem to state the existence of a subsequence of WFT approximate solutions, still denoted by $\{ W^\nu \}_\nu$, converging in $\mathbf {L^{1}_{loc}}$ to a function W. It now remains to prove that $u=u(W)$ is a weak solution of (2.1).

Theorem 1

Let $U_0:{\mathbb {R}}\rightarrow {\textbf{E}}$ with $\textrm{TV}(W(U_0))<+\infty$ and let W be the limit function of the sequence $\{ W^\nu \}_\nu$ of WFT approximate solutions as $\nu \rightarrow \infty$. Then, $u=u(W)$ is a weak solution of (2.1) with initial data $u_0=u(W(U_0))$ in the sense of Definition 1.

Proof

Let $T > 0$ be a given finite time horizon and consider $\varphi \in \mathbf {C_c^{1}}\left( ]-\infty ,T[ \, \times {\mathbb {R}};{\mathbb {R}}\right)$. We define $N^\nu$ so that $T= N^\nu \Delta t^\nu + \beta ^\nu$, $\beta ^\nu \in [0, \Delta t^\nu [$. Following [13], we observe that

$$\begin{aligned}&\int \limits _{k \Delta t^\nu }^{(k+1) \Delta t^\nu } \int \limits _{\mathbb {R}}\left[ u(W^\nu ) \partial _t \varphi + F(u(W^\nu )) \partial _x\varphi \right] (t,x) \,\textrm{d}x \textrm{d}t \\ =&\ \int \limits _{\mathbb {R}}\varphi ((k+1) \Delta t^\nu , x) u(W^\nu ((k+1) \Delta t^\nu -, x)) \ \textrm{d}x - \int \limits _{\mathbb {R}}\varphi (k\Delta t^\nu , x) u(W^\nu (k \Delta t^\nu +, x)) \ \textrm{d}x, \end{aligned}$$

since $u^\nu :=u(W^\nu )$ are weak solutions of the homogeneous system (3.2) in each interval $]k\Delta t, (k+1)\Delta t[ \, \times {\mathbb {R}}$ by construction. Therefore, remembering that $\rho ^\nu (k\Delta t +) = \rho ^\nu (k\Delta t -)$, we get

$$\begin{aligned}&\int \limits _{0}^{T} \int \limits _{\mathbb {R}}\left[ u(W^\nu )\partial _t \varphi + F(u(W^\nu )) \partial _x\varphi \right] (t,x) \,\textrm{d}x \textrm{d}t \\ =&\ \left( \sum _{k=0}^{N^\nu -1} \int \limits _{k \Delta t^\nu }^{(k+1) \Delta t^\nu } + \int \limits _{N^\nu \Delta t^\nu }^{T}\right) \int \limits _{\mathbb {R}}\left[ u(W^\nu ) \partial _t \varphi + F(u(W^\nu )) \partial _x\varphi \right] (t,x) \,\textrm{d}x \textrm{d}t \\ =&\ \sum _{k=0}^{N^\nu -1} \int \limits _{\mathbb {R}}\varphi ((k+1) \Delta t^\nu , x) u(W^\nu ((k+1) \Delta t^\nu -, x)) \ \textrm{d}x \\&- \sum _{k=0}^{N^\nu } \int \limits _{\mathbb {R}}\varphi (k\Delta t^\nu , x) \left[ u(W^\nu (k \Delta t^\nu -, x)) + \Delta t G(u(W^\nu (k \Delta t^\nu -, x)))\right] \textrm{d}x \\ =&\ - \sum _{k=0}^{N^\nu } \Delta t \int \limits _{\mathbb {R}}\varphi (k\Delta t^\nu , x) G(u(W^\nu (k \Delta t^\nu -, x))) \ \textrm{d}x - \int \limits _{\mathbb {R}}\varphi (0, x) u(W_0^\nu (x)) \ \textrm{d}x \end{aligned}$$

Passing to the limit as $\nu \rightarrow +\infty$ in the above equality, by Lebesgue dominated convergence theorem we obtain

$$\begin{aligned}&\int \limits _{0}^{T} \int \limits _{\mathbb {R}}\left[ u(W)\partial _t \varphi + F(u(W)) \partial _x\varphi \right] (t,x) \,\textrm{d}x \textrm{d}t \\ =&\ - \int \limits _{0}^{T} \int \limits _{\mathbb {R}}\varphi (t, x) G(u(W(t, x))) \ \textrm{d}x - \int \limits _{\mathbb {R}}\varphi (0, x) u(W_0(x)) \ \textrm{d}x \, , \end{aligned}$$

which concludes the proof. $\square$

Remark 1

In the present setting, the violation of the sub-characteristic condition (2.10) contradicts the existence of an entropy, entropy-flux pairs for system (2.1), i.e., functions $\eta ,q:{\mathbb {R}}_+^2\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned}&\nabla \eta ^T(z) D F(z)= \nabla ^T q(z),\\&\nabla \eta ^T(z)G(z) \le 0, \end{aligned}$$

for $z \in {\mathbb {R}}^2_+$. We refer to [8, Theorem 2.1] for details.

6 Periodic traveling wave solutions

In this section, we aim at investigating the behavior of traveling waves for system (2.1) violating the stability condition (2.10). We refer to [18, 32, 37] for a similar analysis conducted on other models.

We look for traveling wave solutions of (2.8) of the form $(\rho (\xi ),v(\xi ))$, where $\xi =(x-\sigma t)/\tau$:

$$\begin{aligned}&-\sigma \rho ' + (\rho v)' =0, \end{aligned}$$

(6.1a)

$$\begin{aligned}&-\sigma v' + \lambda _1(\rho ,w) v' ={V(\rho )-v}. \end{aligned}$$

(6.1b)

Integrating (6.1a), we get

$$\begin{aligned} \rho v = \sigma \rho + m \qquad \Longrightarrow \qquad \rho = \dfrac{m}{v-\sigma } = \dfrac{m}{{{\mathcal {V}}}(\rho ,w)-\sigma } \qquad \Longleftrightarrow \qquad {{\mathcal {V}}}(\rho ,w) = \sigma + \dfrac{m}{\rho }. \end{aligned}$$

for some $m\in {\mathbb {R}}$. From (6.1b), we then get

$$\begin{aligned} v' = \dfrac{V(\rho )-v}{\lambda _1(\rho ,w) -\sigma } = \dfrac{(v-\sigma )(V-v)}{(v-\sigma )^2 + m \,\partial _\rho {{\mathcal {V}}}} \qquad \hbox {or}\qquad \rho ' = \frac{\rho }{\sigma -v}\, \dfrac{V(\rho )-v}{\lambda _1(\rho ,w) -\sigma }. \end{aligned}$$

(6.2)

Note that, if (2.10) holds, then the denominator $\lambda _1(\rho ,w) -\sigma \not =0$, otherwise there exists a sonic point $v^\sigma$ such that

$$\begin{aligned} v^\sigma - \sigma = \pm \sqrt{-m\,\partial _\rho {{\mathcal {V}}}} \qquad \Longrightarrow \qquad v^\sigma = \sigma \pm \sqrt{-m\,\partial _\rho {{\mathcal {V}}}}. \end{aligned}$$

Since $\partial _\rho {{\mathcal {V}}}\le 0$, we need to take $m>0$. To prevent the right-hand side of (6.2) from blowing up at the sonic point, we require $v^{\sigma }=V(m/(v^\sigma -\sigma ))$.

In a periodic setting, two states $(\rho _\pm ,v_\pm )$ will be connected alternatively by a jump discontinuity satisfying the Rankine–Hugoniot condition

$$\begin{aligned} \sigma = \dfrac{\rho _+ v_+ - \rho _- v_-}{\rho _+ - \rho _-}, \end{aligned}$$

(6.3)

and a solution of (6.2). We recall that entropy admissible jump discontinuities must satisfy $v_- \ge v_+$. In particular, if $v_- > v_+$ (shock) we must have $w_-=w_+$. Therefore, for the solution of (6.2) going from $v_+$ to $v_-$, we must have $v'\ge 0$. Besides, we have $m=\rho _\pm (v_\pm -\sigma )\ge 0$ since $v_\pm \ge \sigma$. This confirms the existence of a sonic point $v^\sigma$. (Note that in the case of a contact discontinuity, it holds $v_-=v_+=\sigma$ and therefore $v'=0$).

Let us analyze (6.2) better. We can rewrite the denominator as

$$\begin{aligned} (v-\sigma )^2-(\sqrt{-m\,\partial _\rho {{\mathcal {V}}}})^2 = \left( (v-\sigma )-\sqrt{-m\,\partial _\rho {{\mathcal {V}}}}\right) \left( (v-\sigma )+\sqrt{-m\,\partial _\rho {{\mathcal {V}}}}\right) . \end{aligned}$$

Since $v > \sigma$, the denominator has a unique zero of multiplicity one at

$$\begin{aligned} v^\sigma = \sigma + \sqrt{-m\,\partial _\rho {{\mathcal {V}}}}. \end{aligned}$$

Solution of (6.2) being increasing along $\xi$, we have

$$\begin{aligned} v_+&< \sigma + \sqrt{-m\,\partial _\rho {{\mathcal {V}}}} \qquad \Longrightarrow \qquad (v_+-s \sigma )^2-(\sqrt{-m\,\partial _\rho {{\mathcal {V}}}})^2 <0, \\ v_-&> \sigma + \sqrt{-m\,\partial _\rho {{\mathcal {V}}}} \qquad \Longrightarrow \qquad (v_--\sigma )^2-(\sqrt{-m\,\partial _\rho {{\mathcal {V}}}})^2 >0. \end{aligned}$$

Therefore, to have $v'\ge 0$ in (6.2), the numerator must satisfy

$$\begin{aligned} V(\rho ) \le v={{\mathcal {V}}}(\rho ,w_\pm )\quad \hbox {for } \rho ^\sigma \le \rho \le \rho _+,\\ V(\rho ) \ge v={{\mathcal {V}}}(\rho ,w_\pm )\quad \hbox {for } \rho _-\le \rho \le \rho ^\sigma , \end{aligned}$$

see Fig. 5. In particular, it holds $V(\rho ^\sigma )=v^\sigma =\sigma +\dfrac{m}{\rho ^\sigma }$.

Fig. 5

Speed-density (left) and flow-density (right) representations of the curves involved in the construction of the traveling wave profiles, where we set ${{\mathcal {V}}}(\rho ,w)=w-\rho$, $V(\rho )=V_{max}\left( 1 - \exp (C(1-R/\rho ))\right)$

To summarize, the recipe to construct a periodic wave oscillating between states $(\rho _\pm ,v_\pm )$ (refereed to as “jamiton” in [18, 37]) is the following:

• 1. For a prescribed downstream state $(\rho _+,v_+)$ such that $v_+={{\mathcal {V}}}(\rho _+,w_+) > V(\rho _+)$ compute $\rho ^\sigma$ implicitly defined by $V(\rho ^\sigma )={{\mathcal {V}}}(\rho ^\sigma ,w_+)=v^\sigma$. Then, from the identities

  $$\begin{aligned} {\left\{ \begin{array}{ll} v_+ = \sigma + \dfrac{m}{\rho _+}, \\ v^\sigma = \sigma + \dfrac{m}{\rho ^\sigma }, \end{array}\right. } \end{aligned}$$

  we recover

  $$\begin{aligned} {\left\{ \begin{array}{ll} m=\dfrac{\rho ^\sigma \rho _+}{\rho _+ - \rho ^\sigma } (v^\sigma -v_+), \\ \sigma = v_+ \dfrac{\rho _+}{\rho _+ - \rho ^\sigma } + v^\sigma \dfrac{\rho ^\sigma }{\rho ^\sigma -\rho _+}. \end{array}\right. } \end{aligned}$$

• 2. Determine the upstream state $(\rho _-,v_-)$, with $v_-={{\mathcal {V}}}(\rho _-,w_+)$ using the Rankine–Hugoniot condition (6.3).

• 3. Integrate (6.2) from $v(0)=v_+$ to $v(\pi )=v_-$. The density is then given by

  $$\begin{aligned} \rho (\xi ) = \dfrac{m}{v(\xi )-\sigma }. \end{aligned}$$

  The period of the traveling wave is then given by $\pi$.

• 4. The total number of vehicles involved is computed by

  $$\begin{aligned} N=\int \limits _{0}^{\pi } \rho (\xi ) \, \textrm{d}\xi . \end{aligned}$$

See Fig. 6 for a representation of the traveling density and speed profiles.

Fig. 6

Density and speed profiles of the jamiton solution of (6.1)

7 Chapman–Enskog expansion

Another classical way of investigating the stability of equilibria of relaxation systems is to perform a formal expansion with respect to the relaxation parameter, see, e.g., [5, 8, 11].

We consider the equations in $(\rho ,v)$ coordinates (2.8), which we recall here

$$\begin{aligned}&\partial _t \rho + \partial _x (\rho v) =0, \end{aligned}$$

(7.1a)

$$\begin{aligned}&\partial _t v + \lambda _1(\rho ,w)\partial _x v =\dfrac{V(\rho )-v}{\tau }. \end{aligned}$$

(7.1b)

Assuming smooth solutions, from (7.1b) we recover:

$$\begin{aligned} v&= V(\rho ) - \tau \left( \partial _t v + \lambda _1(\rho ,w)\partial _x v\right) = V(\rho ) - \tau \left( V'(\rho )\partial _t \rho + \lambda _1(\rho ,w)\partial _x v\right) . \end{aligned}$$

Substituting (7.1a), i.e., $\partial _t \rho = -\partial _x (\rho v) = -\partial _x (\rho V(\rho )) + {{\mathcal {O}}}(\tau )$ and $\partial _x v = \partial _x V(\rho ) +{{\mathcal {O}}}(\tau )$, we get

$$\begin{aligned} v= V(\rho ) - \tau \left( \lambda _1(\rho ,w)\partial _x V(\rho )-V'(\rho )\partial _x (\rho V(\rho )) \right) + {{\mathcal {O}}}(\tau ^2). \end{aligned}$$

Replacing this last expression in (7.1a) and truncating to the first order, we get

$$\begin{aligned} \partial _t \rho + \partial _x (\rho V(\rho )) = \tau \partial _x\left( \rho V'(\rho )\left( \lambda _1(\rho ,w)-(\rho V(\rho ))'\right) \partial _x\rho \right) . \end{aligned}$$

(7.2)

Since $V'(\rho )\le 0$, the diffusive equation (7.2) is stable if and only if $\lambda _1(\rho ,w)\le (\rho V(\rho ))'$, i.e., (2.10) holds.

Alternatively, the formal Chapman–Enskog expansion in $v=\sum _{k=0}^\infty \tau ^k v_k$ leads to:

$$\begin{aligned}&\partial _t \rho + \sum _{k=0}^\infty \tau ^k \partial _x (\rho v_k) =0, \\&\sum _{k=0}^\infty \tau ^k\partial _t v_k + \lambda _1(\rho ,w) \sum _{k=0}^\infty \tau ^k \partial _x v_k =\dfrac{V(\rho )}{\tau } - \sum _{k=0}^\infty \tau ^{k-1} v_k . \end{aligned}$$

The second identity yields

$$\begin{aligned}&v_0 = V(\rho ), \\&\partial _t v_0 + \lambda _1(\rho ,w)\partial _x v_0 = -v_{1}, \\&\partial _t v_k + \lambda _1(\rho ,w)\partial _x v_k = -v_{k+1}. \end{aligned}$$

Replacing in the first identity, we get

$$\begin{aligned} \partial _t\rho +\partial _x(\rho V(\rho ))&= \partial _t\rho +\partial _x(\rho v_0)\\&= - \sum _{k=1}^\infty \tau ^k\partial _x(\rho v_k) \\&= \sum _{k=0}^\infty \tau ^{k+1} \partial _x\left( \rho \left( \partial _t v_k + \lambda _1(\rho ,w)\partial _x v_k\right) \right) , \end{aligned}$$

which, to the first order $k=0$, is

$$\begin{aligned} \partial _t\rho +\partial _x(\rho V(\rho ))&= \tau \partial _x\left( \rho \left( \partial _t v_0 + \lambda _1(\rho ,w)\partial _x v_0\right) \right) \nonumber \\&= \tau \partial _x\left( \rho V'(\rho )\left( \partial _t \rho + \lambda _1(\rho ,w) \partial _x \rho \right) \right) \nonumber \\&= \tau \partial _x\left( \rho V'(\rho )\left( \lambda _1(\rho ,w) - (\rho V(\rho ))'\right) \partial _x \rho \right) \end{aligned}$$

(7.3)

as in (7.2).

Considering traveling wave solutions of (7.2) of the form $\rho (t,x)=\rho (\xi )$, where $\xi = (x-\sigma t)/\tau$, we get

$$\begin{aligned} - \sigma \rho ' + (\rho V(\rho ))' = \left( \rho V'(\rho )\left( \lambda _1(\rho ,w)-(\rho V(\rho ))'\right) \rho '\right) '. \end{aligned}$$

(7.4)

By integration, we obtain

$$\begin{aligned} \rho ' = \frac{\rho (V(\rho )-\sigma ) -m}{\rho V'(\rho )\left( \lambda _1(\rho ,w)-(\rho V(\rho ))'\right) }, \end{aligned}$$

(7.5)

which is coherent with (6.2) setting $m=\rho (v-\sigma )$ and assuming $\sigma \sim (\rho V(\rho ))'$, which gives $\rho V'(\rho )=\sigma - V(\rho ) \sim \sigma - v$.

8 Numerical simulations

To illustrate the behavior of solutions of model (2.1) when the sub-characteristic condition (2.10) is violated, we compute approximate solutions obtained via a finite volume scheme with time splitting developed merging the Godunov scheme for (3.2) in its supply–demand formulation [27, 28], as described in [17, 21], and an explicit one-step Euler scheme accounting for the relaxation component (3.5).

We consider a space step $\Delta x$ and a time step $\Delta t$ satisfying the CFL condition [12]

$$\begin{aligned} \Delta t < \min \left\{ \frac{\Delta x}{\max \{{\left\| \lambda _1\right\| }_\infty , {\left\| \lambda _2\right\| }_\infty \}}\, \frac{\tau }{\max \{1, {\left\| \partial _w {{\mathcal {V}}}\right\| }_\infty \}} \right\} , \end{aligned}$$

and we set $x_{j+1/2}=j\Delta x$, $j\in {\mathbb {Z}}$, to be the cells interfaces, and $t^n=n \Delta t$, $n\in {{\mathbb {N}}}$, the time mesh.

Denoting by $u=(\rho ,y)^T=(\rho ,\rho w)^T$ the vector of the conservative variables (where we set $y=\rho w$), we construct a finite volume approximate solution of (2.1) of the form ${u^{\Delta x}}=(\rho ^{\Delta x},y^{\Delta x})^T$ with $\rho ^{\Delta x}(t,x)=\rho ^n_j$ and $y^{\Delta x}(t,x)=y^n_j$ for $(t,x)\in C^n_{j}=[t^n, t^{n+1}[\,\times [x_{j-1/2}, x_{j+1/2}[$. To this end, we approximate the initial data with piece-wise constant functions

$$\begin{aligned} \rho ^0_j=\frac{1}{\Delta x} \int \limits _{x_{j-1/2}}^{x_{j+1/2}}\rho ^0 (x) \, \textrm{d}x, \quad y^0_j=\frac{1}{\Delta x} \int \limits _{x_{j-1/2}}^{x_{j+1/2}} y^0(x) \, \textrm{d}x, \qquad \forall {j\in {\mathbb {Z}}}, \end{aligned}$$

and we iterate in time applying the following two steps:

• 1. We approximate (3.2) by the conservation formulas

  $$\begin{aligned} \rho _j^{n+1/2}&= \rho _j^n - \dfrac{\Delta t}{\Delta x} \left( F_{j+1/2}^{\rho ,n} - F_{j-1/2}^{\rho ,n} \right) , \end{aligned}$$

  (8.1a)
  $$\begin{aligned} y_j^{n+1/2}&= y_j^n - \dfrac{\Delta t}{\Delta x} \left( F_{j+1/2}^{y,n} - F_{j-1/2}^{y,n} \right) , \end{aligned}$$

  (8.1b)

  where $F_{j+1/2}^{\rho ,n}$ and $F_{j+1/2}^{y,n}$ are the flows, respectively, of $\rho$ and y at $x=x_{j+1/2}$ in the time interval $[t^n, t^{n+1}[$. Since the variable w is advected with $\rho v$, we directly get

  $$\begin{aligned} F_{j+1/2}^{y,n} = w_j^n F_{j+1/2}^{\rho ,n} \end{aligned}$$

  (8.2)

  in (8.1b). To compute $F_{j+1/2}^{\rho ,n}$, we define the corresponding demand and supply functions as

  $$\begin{aligned} D_j=D(\rho _j,w_j)&= {\left\{ \begin{array}{ll} Q(\rho _j,w_j) &{} \hbox {if}~\rho _j\le \rho _{cr}(w_j), \\ Q_{\max } (w_j) &{} \hbox {if}~\rho _j>\rho _{cr}(w_j), \end{array}\right. } \\ S_{j+1}=S(\rho _{j+1},w_{j+1};w_j)&= {\left\{ \begin{array}{ll} Q_{\max } (w_j) &{} \hbox {if}~\rho _{j+1/2}\le \rho _{cr}(w_j), \\ Q(\rho _{j+1/2},w_j) &{} \hbox {if}~\rho _{j+1/2} > \rho _{cr}(w_j), \end{array}\right. } \end{aligned}$$

  where $\rho _{cr}(w)\in \, ]0,R(w)[$ is the point of maximum of $Q(\cdot ,w)$, $Q_{\max }(w)=Q(\rho _{cr}(w),w)$ is the corresponding maximum, and $\rho _{j+1/2}$ is the density of the intermediate state in the solution of the Riemann problem corresponding to $u_j$ and $u_{j+1}$:

  $$\begin{aligned}&\rho _{j+1/2} \quad \hbox {such that}\quad {{\mathcal {V}}}(\rho _{j+1/2},w_j) = {{\mathcal {V}}}(\rho _{j+1},w_{j+1})&\hbox {if}~ {{\mathcal {V}}}(\rho _{j+1},w_{j+1}) \le w_j; \\&0&\hbox {if}~ {{\mathcal {V}}}(\rho _{j+1},w_{j+1}) > w_j. \end{aligned}$$

  We can thus set

  $$\begin{aligned} F_{j+1/2}^{\rho ,n} = \min \left\{ D(\rho _j^n,w_j^n), S(\rho _{j+1}^n,w_{j+1}^n;w_j^n) \right\} \end{aligned}$$

  (8.3)

  in (8.1), (8.2).

• 2. We then solve (3.5) by updating the approximate solution as

  $$\begin{aligned} \rho _j^{n+1}&= \rho _j^{n+1/2} , \\ w_j^{n+1}&= w_j^{n+1/2} + \dfrac{\Delta t}{\tau } \left( V(\rho _j^{n+1/2})-{{\mathcal {V}}}(\rho _j^{n+1/2},w_j^{n+1/2}) \right) , \\ y_j^{n+1}&= \rho _j^{n+1} w_j^{n+1}\,. \end{aligned}$$

In the following tests, we consider (2.1) with

$$\begin{aligned} {{\mathcal {V}}}(\rho ,w)&= w\left( 1-\frac{\rho }{\rho _{max}}\right) ,\\ V(\rho )&= V_{\max } \left( 1- \exp \left( \alpha \left( 1-\frac{\rho _{max}}{\rho } \right) \right) \right) , \end{aligned}$$

with $V_{\max }=25\,m/s$ the maximal speed, $\rho _{max}=1\,veh/m$ the maximal density and $\alpha =0.5$. Note that, in this case, we have $R(w)=\rho _{max}$ for all $w>0$.

As reference point, we choose $\rho ^*=0.6\,veh/m$ and $w^*\simeq 17.7\,m/s$, so that ${{\mathcal {V}}}(\rho ^*,w^*)=V(\rho ^*)\simeq 7\,m/s$ violates (2.10), see Fig. 7.

In the following sections, while we drop them for convenience, the space units will always be given in meters and the time in seconds.

Fig. 7

Left: plots of $V(\rho )$ (continuous line) and ${{\mathcal {V}}}(\rho ,w^*)$ (dashed line). Right: corresponding fluxes $\rho V(\rho )$ (continuous line) and $Q(\rho ,w^*)=\rho {{\mathcal {V}}}(\rho ,w^*)$ (dashed line). The curves intersect at $\rho =\rho ^*=0.6\,veh/m$ and $\partial _\rho {{\mathcal {V}}}(\rho ^*,w^*) > V'(\rho ^*)$

8.1 The Riemann problem

As a first illustration of instability and lack of convergence to the scalar LWR equation (2.11) as $\tau \searrow 0$, we couple (2.1) with Riemann-type initial data (3.1). We take $w_L=w_R=w^*$ and consider the two cases:

• Case A: $\rho _L=0.7>0.5=\rho _R$, corresponding to a rarefaction wave for (2.11);

• Case B: $\rho _L=0.5<0.7=\rho _R$, corresponding to a shock wave for (2.11).

The corresponding density plots are shown in Fig. 8. We observe that, as $\tau$ decreases, the solution of the GSOM system with relaxation develops an increasing number of bounded oscillations, which confirms the non-convergence to the corresponding LWR model. In particular, the total variation of the density component, but also $\textrm{TV}(W(U))$, which is showed in Fig. 9, increases as $\tau$ decreases, in accordance with (4.4), while the wave amplitudes are not impacted. We notice that the total variation increase is much larger in case A, that is, in the case of a rarefaction wave.

Fig. 8

Density profiles at $t=10$ of the solution to model (2.1), (3.1), with $\tau =0.01$ and $\tau =0.1$, and the solution of the LWR model (2.11) for cases A (left: $\rho _L=0.7$, $\rho _R=0.5$) and B (right: $\rho _L=0.5$, $\rho _R=0.7$), see Sect. 8.1

Fig. 9

Time evolution of the total variation $\textrm{TV}(W(U (t,\cdot )))= \textrm{TV}(w(t,\cdot )) + \textrm{TV}({{\mathcal {V}}}(U (t,\cdot )))$, for $t\in [0,10]$, of the solution to problem (2.1), (3.1), with $\tau =0.01$ and $\tau =0.1$ for cases A (left: $\rho _L=0.7$, $\rho _R=0.5$) and B (right: $\rho _L=0.5$, $\rho _R=0.7$), see Sect. 8.1

8.2 Stop-and-go waves

We consider (2.1) with initial data $U_0(x)=\left( \rho ^*+\theta (x),w^*\right)$, where $\theta (x)$ is a small perturbation of the density given by

$$\begin{aligned} \theta (x) = {\left\{ \begin{array}{ll} 0.05\left( \cos (2(x-{\bar{x}})) - \cos (x-{\bar{x}})\right) , &{} \hbox {if}~x\in [{\bar{x}},{\bar{x}}+2\pi ],\\ 0, &{} \hbox {elsewhere}, \end{array}\right. } \end{aligned}$$

(8.4)

not affecting the total mass locally around the space interval $[{\bar{x}},{\bar{x}}+2\pi ]$.

The solution is displayed in Fig. 10. We notice that the small initial perturbation gives rise to large but bounded oscillations traveling with negative speed, thus reproducing the formation, persistence and spread of stop-and-go waves commonly observed in traffic dynamics [38], see also [25, Section 3.3] for a similar behavior induced by larger perturbations.

Fig. 10

Solution of (2.1) corresponding to constant initial conditions with a small perturbation (8.4) of the density component located at ${\bar{x}}=90$. Left: density, Lagrangian attribute and speed components of the initial data (dashdotted line) and the solution at time $t=10$ (continuous line). Right: density evolution (heat map) for $x\in [0,100]$ and $t\in [0,15]$

8.3 Ring road

In the last example, we aim at reproducing the formation of stop-and-go waves on a ring road, as described in [38]. To this end, we consider (2.1) on a segment of length $L=100$ with periodic boundary conditions

$$\begin{aligned} U(t,L) = U(t,0),\qquad t>0, \end{aligned}$$

and initial data $U_0(x)=\left( \rho ^*,w^*+\omega (x)\right)$, where $\omega (x)$ is a small localized perturbation in the drivers’ behavior showed in Fig. 11:

$$\begin{aligned} \omega (x) = {\left\{ \begin{array}{ll} 10 \exp \left( \frac{5}{{\left| x-{\bar{x}}\right| }^2-1} \right) , &{} \hbox {if}~{\left| x-{\bar{x}}\right| }<1,\\ 0, &{} \hbox {elsewhere}, \end{array}\right. } \end{aligned}$$

(8.5)

mimicking the presence of some slightly more aggressive driver at $x={\bar{x}}$. We remark that the perturbation has magnitude of approximately 0.06, which is approximately $0.34\%$ of the baseline value $w^*$. Figure 12 shows that even such a small perturbation is able to drive the system away from equilibrium, and we can observe the formation of a stable periodic pattern consisting of two waves traveling with the same constant speed.

We notice that in all the examples showed in this section the solution’s density never exceeds the maximal density $\rho _{max}=1$, as prescribed by Proposition 1.

Fig. 11

Zoom of the Lagrangian attribute component of the initial datum considered in Sect. 8.3 obtained by perturbing the constant state $w^*$ with the function (8.5) at ${\bar{x}}=50$

Fig. 12

Solution of (2.1) corresponding to constant initial conditions with a small perturbation (8.5) of the Lagrangian attribute located at ${\bar{x}}=50$ (not visible at the scale of the plots). Left: density, Lagrangian attribute and speed components of the initial data (dashdotted line) and the solution at time $t=100$ (continuous line). Right: density evolution (heat map) for $x\in [0,100]$ and $t\in [0,100]$

9 Conclusion

In this paper, we provided an analytical study of the GSOM system with relaxation in super-characteristic regimes. We showed, with the help of numerical simulations, that the solutions can reproduce the formation and persistence of large but bounded oscillations, which can be interpreted as the stop-and-go waves observed in traffic flows. A similar analysis can probably be extended to other second order traffic models, such as [6, 10].

Models displaying such unstable behavior can constitute a good basis for designing control strategies that allow to dampen traffic oscillations in a macroscopic framework, in the spirit of [19, 20, 39]. Preliminary results in this direction can be found in [22].