This paper deals with a macroscopic traffic flow model accounting for the boundedness of traffic acceleration.
The authors propose to couple the LWR model with a finite number of ordinary differential equations, each accounting for the trajectory of a leading vehicle, initially located at a downward jump in density, and accelerating at a constant rate. Hence these vehicles act as moving constraints, enforcing a zero flux along their trajectories, until they catch the downstream traffic.
Since situations with unbounded acceleration can only appear at downward jumps of density, they propose to couple the LWR model with a moving bottleneck consisting in a single vehicle accelerating at constant rate \(A > 0\) and originating at the jump location. They introduce \(I\) moving bottlenecks, where \(I \in \mathbb{N}\) is the finite number of downward jumps in the initial density \[\begin{array}{lll} &\partial_t\rho +\partial_x f(\rho)=0,&t>0,\,x\in\mathbb{R}\\ &f (\rho (t, y_i(t)))-\rho (t, y_i(t))\dot y_i(t) \leq 0,&t>0,\, i\in \{1,...,I\},\\ &\dot y_i(t) = \omega_i (t,y_i(t)),&t>0,\, i\in \{1,...,I\},\\ &\rho(0,x)=\rho_0(x),&x\in\mathbb{R},\\ &y_i(0)=y_i^0,&i\in \{1,...,I\}, \end{array}\] where \(\rho_0\) is a piecewise constant function, and for every \(y_i^0\) the initial datum has a downward jump (i.e., \(\rho_0(y_i^0-) > \rho_0(y_i^0+)),\) \[ \omega_i(t, y_i(t)) := \min \{ At + v_i^0, v(\rho (t, y_i(t)+)\},\qquad v_i^0 = v(\rho_0(y_i^0 -)), \] that is the initial speed of the moving bottleneck starting at position \(y_i^0\) and \(y_i (t)\) is the trajectory of the \(i-\)th moving bottleneck. They propose a wave-front tracking algorithm to construct approximate solutions and use this algorithm to prove the existence of entropy weak solutions to the associated Cauchy problem and provide some numerical simulations illustrating the solution behaviour.