satisfied, cf. [@Liu1987; @ChenLevermoreLiu1994; @SeiboldFlynnKasimovRosales2013]. Here, we are in the characteristic case. Moreover, shocks of the $2\times 2$ hyperbolic system are also shocks of the LWR model . Therefore, for the Cauchy problem, GARZ solutions converge to LWR solutions as $t\to\infty$, and/or as $\tau\to 0$.

![Riemann problem at right domain boundary. Consider a constant initial state $(\rho_\text{R},Q_\text{eq}(\rho_\text{R}))$ in the domain, and prescribed boundary data $(\rho_\text{R},\rho_\text{R}u_\text{R})$ with $u_\text{R} > U_\text{eq}(\rho_\text{R})$. The LWR model preserves the constant state (the boundary data is projected onto the equilibrium curve vertically along the dashed line). In contrast, the GARZ model generates a new state $(\rho_\text{M},Q_\text{eq}(\rho_\text{M}))$ at the boundary (projected onto the equilibrium curve along a ray through the origin), from which a shock moves into the domain, thus changing the initial state to the new boundary state.[]{data-label="fig:Riemann_problem_bc"}](fig_riemann_problem_bc){width=".75\textwidth"}

In contrast, for initial-boundary-value problems on a bounded domain $x\in [x_\text{L},x_\text{R}]$, this last property is in general not true. To highlight this fact we consider a simplified setting depicted in Fig. \[fig:Riemann\_problem\_bc\]. Let constant boundary data $\rho(x_\text{L})$, $u(x_\text{L})$, $\rho(x_\text{R})$, $u(x_\text{R})$ be given. With this data, we can solve the LWR model and the inhomogeneous GARZ model . Then, in general solutions to the latter problem do not converge 